https://www.math.wisc.edu/wiki/api.php?action=feedcontributions&user=Donghyun&feedformat=atomUW-Math Wiki - User contributions [en]2020-09-20T06:12:10ZUser contributionsMediaWiki 1.30.1https://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=15245PDE Geometric Analysis seminar2018-03-13T18:55:44Z<p>Donghyun: /* Abstracts */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2018 | Tentative schedule for Fall 2018]]===<br />
<br />
<br />
<br />
== PDE GA Seminar Schedule Spring 2018 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|January 29, '''3-3:50PM, B341 VV.'''<br />
| Dan Knopf (UT Austin)<br />
|[[#Dan Knopf | Non-K&auml;hler Ricci flow singularities that converge to K&auml;hler-Ricci solitons]]<br />
| Angenent<br />
|- <br />
|February 5, '''3-3:50PM, B341 VV.'''<br />
| Andreas Seeger (UW)<br />
|[[#Andreas Seeger | Singular integrals and a problem on mixing flows ]]<br />
| Kim & Tran<br />
|- <br />
|February 12<br />
| Sam Krupa (UT-Austin)<br />
|[[#Sam Krupa | Proving Uniqueness of Solutions for Burgers Equation Entropic for a Single Entropy, with Eye Towards Systems Case ]]<br />
| Lee <br />
|- <br />
|February 19<br />
| Maja Taskovic (UPenn)<br />
|[[#Maja Taskovic | Exponential tails for the non-cutoff Boltzmann equation ]]<br />
| Kim<br />
|- <br />
|February 26<br />
| Ashish Kumar Pandey (UIUC)<br />
|[[# | Instabilities in shallow water wave models ]]<br />
| Kim & Lee<br />
|- <br />
|March 5<br />
| Khai Nguyen (NCSU)<br />
|[[#Khai Nguyen | Burgers Equation with Some Nonlocal Sources ]]<br />
| Tran<br />
|- <br />
|March 12<br />
| Hongwei Gao (UCLA)<br />
|[[#Hongwei Gao | Stochastic homogenization of certain nonconvex Hamilton-Jacobi equations ]]<br />
| Tran<br />
|- <br />
|March 19<br />
| Huy Nguyen (Princeton)<br />
|[[#Huy Nguyen | Compressible fluids and active potentials ]]<br />
| Lee<br />
|-<br />
|March 26<br />
| <br />
|[[# | Spring recess (Mar 24-Apr 1, 2018) ]]<br />
| <br />
|-<br />
|April 2<br />
| In-Jee Jeong (Princeton)<br />
|[[#In-Jee Jeong | Singularity formation for the 3D axisymmetric Euler equations ]]<br />
| Kim<br />
|- <br />
|April 9<br />
| Jeff Calder (Minnesota)<br />
|[[#Jeff Calder | TBD ]]<br />
| Tran<br />
|- <br />
|April 21-22 (Saturday-Sunday)<br />
| Midwest PDE seminar<br />
|[[#Midwest PDE seminar | ]]<br />
| Angenent, Feldman, Kim, Tran.<br />
|- <br />
|April 25 (Wednesday)<br />
| Hitoshi Ishii (Wasow lecture)<br />
|[[#Hitoshi Ishii | TBD]]<br />
| Tran.<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Dan Knopf===<br />
<br />
Title: Non-K&auml;hler Ricci flow singularities that converge to K&auml;hler-Ricci solitons<br />
<br />
Abstract: We describe Riemannian (non-K&auml;hler) Ricci flow solutions that develop finite-time Type-I singularities whose parabolic dilations converge to a shrinking K&auml;hler–Ricci soliton singularity model. More specifically, the singularity model for these solutions is the “blowdown soliton” discovered by Feldman, Ilmanen, and Knopf in 2003. Our results support the conjecture that the blowdown soliton is stable under Ricci flow. This work also provides the first set of rigorous examples of non-K&auml;hler solutions of Ricci flow that become asymptotically K&auml;hler, in suitable space-time neighborhoods of developing singularities, at rates that break scaling invariance. These results support the conjectured stability of the subspace of K&auml;hler metrics under Ricci flow.<br />
<br />
===Andreas Seeger===<br />
<br />
Title: Singular integrals and a problem on mixing flows<br />
<br />
Abstract: The talk will be about results related to Bressan's mixing problem. We present an inequality for the change of a Bianchini semi-norm of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which one proves bounds on Hardy spaces. This is joint work with Mahir Hadžić, Charles Smart and Brian Street.<br />
<br />
===Sam Krupa===<br />
<br />
Title: Proving Uniqueness of Solutions for Burgers Equation Entropic for a Single Entropy, with Eye Towards Systems Case<br />
<br />
Abstract: For hyperbolic systems of conservation laws, uniqueness of solutions is still largely open. We aim to expand the theory of uniqueness for systems of conservation laws. One difficulty is that many systems have only one entropy. This contrasts with scalar conservation laws, where many entropies exist. It took until 1994 to show that one entropy is enough to ensure uniqueness of solutions for the scalar conservation laws (Panov). This single entropy result was proven again by De Lellis, Otto and Westdickenberg in 2004. These two proofs both rely on the special connection between Hamilton--Jacobi equations and scalar conservation laws in one space dimension. However, this special connection does not extend to systems. In our new work, we prove the single entropy result for scalar conservation laws without using Hamilton--Jacobi. Our proof lays out new techniques that are promising for showing uniqueness of solutions in the systems case. This is joint work with A. Vasseur.<br />
<br />
<br />
===Maja Taskovic===<br />
<br />
Title: Exponential tails for the non-cutoff Boltzmann equation<br />
<br />
Abstract: The Boltzmann equation models the motion of a rarefied gas, in which particles interact through binary collisions, by describing the evolution of the particle density function. The effect of collisions on the density function is modeled by a bilinear integral operator (collision operator) which in many cases has a non-integrable angular kernel. For a long time the equation was simplified by assuming that this kernel is integrable (the so called Grad's cutoff) with a belief that such an assumption does not affect the equation significantly. However, in the last 20 years it has been observed that a non-integrable singularity carries regularizing properties which motivates further analysis of the equation in this setting.<br />
<br />
We study behavior in time of tails of solutions to the Boltzmann equation in the non-cutoff regime by examining the generation and propagation of $L^1$ and $L^\infty$ exponentially weighted estimates and the relation between them. For this purpose we introduce Mittag-Leffler moments which can be understood as a generalization of exponential moments. An interesting aspect of this result is that the singularity rate of the angular kernel affects the order of tails that can be shown to propagate in time. This is based on joint works with Alonso, Gamba, Pavlovic and Gamba, Pavlovic.<br />
<br />
<br />
===Ashish Kumar Pandey===<br />
<br />
Title: Instabilities in shallow water wave models<br />
<br />
Abstract: Slow modulations in wave characteristics of a nonlinear, periodic traveling wave in a dispersive medium may develop non-trivial structures which evolve as it propagates. This phenomenon is called modulational instability. In the context of water waves, this phenomenon was observed by Benjamin and Feir and, independently, by Whitham in Stokes' waves. I will discuss a general mechanism to study modulational instability of periodic traveling waves which can be applied to several classes of nonlinear dispersive equations including KdV, BBM, and regularized Boussinesq type equations.<br />
<br />
<br />
===Khai Nguyen===<br />
<br />
Title: Burgers Equation with Some Nonlocal Sources<br />
<br />
Abstract: Consider the Burgers equation with some nonlocal sources, which were derived from models of nonlinear wave with constant frequency. This talk will present some recent results on the global existence of entropy weak solutions, priori estimates, and a uniqueness result for both Burgers-Poisson and Burgers-Hilbert equations. Some open questions will be discussed.<br />
<br />
===Hongwei Gao=== <br />
<br />
Title: Stochastic homogenization of certain nonconvex Hamilton-Jacobi equations<br />
<br />
Abstract: In this talk, we discuss the stochastic homogenization of certain nonconvex Hamilton-Jacobi equations. The nonconvex Hamiltonians, which are generally uneven and inseparable, are generated by a sequence of (level-set) convex Hamiltonians and a sequence of (level-set) concave Hamiltonians through the min-max formula. We provide a monotonicity assumption on the contact values between those stably paired Hamiltonians so as to guarantee the stochastic homogenization. If time permits, we will talk about some homogenization results when the monotonicity assumption breaks down.<br />
<br />
===Huy Nguyen===<br />
<br />
Title : Compressible fluids and active potentials<br />
<br />
Abstract: We consider a class of one dimensional compressible systems with degenerate diffusion coefficients. We establish the fact that the solutions remain smooth as long as the diffusion coefficients do not vanish, and give local and global existence results. The models include the barotropic compressible Navier-Stokes equations, shallow water systems and the lubrication approximation of slender jets. In all these models the momentum equation is forced by the gradient of a solution-dependent potential: the active potential. The method of proof uses the Bresch-Desjardins entropy and the analysis of the evolution of the active potential.<br />
<br />
===In-Jee Jeong===<br />
<br />
Title: Singularity formation for the 3D axisymmetric Euler equations<br />
<br />
Abstract: We consider the 3D axisymmetric Euler equations on exterior domains $\{ (x,y,z) : (1 + \epsilon|z|)^2 \le x^2 + y^2 \} $ for any $\epsilon > 0$ so that we can get arbitrarily close to the exterior of a cylinder. We construct a strong local well-posedness class, and show that within this class there exist compactly supported initial data which blows up in finite time. The local well-posedness class consists of velocities which are uniformly Lipschitz in space and have finite energy. Our results were inspired by recent works of Hou-Luo, Kiselev-Sverak, and many others, and the proof builds up on our previous works on 2D Euler and Boussinesq systems. This is joint work with Tarek Elgindi.</div>Donghyunhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=15244PDE Geometric Analysis seminar2018-03-13T18:54:27Z<p>Donghyun: </p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2018 | Tentative schedule for Fall 2018]]===<br />
<br />
<br />
<br />
== PDE GA Seminar Schedule Spring 2018 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|January 29, '''3-3:50PM, B341 VV.'''<br />
| Dan Knopf (UT Austin)<br />
|[[#Dan Knopf | Non-K&auml;hler Ricci flow singularities that converge to K&auml;hler-Ricci solitons]]<br />
| Angenent<br />
|- <br />
|February 5, '''3-3:50PM, B341 VV.'''<br />
| Andreas Seeger (UW)<br />
|[[#Andreas Seeger | Singular integrals and a problem on mixing flows ]]<br />
| Kim & Tran<br />
|- <br />
|February 12<br />
| Sam Krupa (UT-Austin)<br />
|[[#Sam Krupa | Proving Uniqueness of Solutions for Burgers Equation Entropic for a Single Entropy, with Eye Towards Systems Case ]]<br />
| Lee <br />
|- <br />
|February 19<br />
| Maja Taskovic (UPenn)<br />
|[[#Maja Taskovic | Exponential tails for the non-cutoff Boltzmann equation ]]<br />
| Kim<br />
|- <br />
|February 26<br />
| Ashish Kumar Pandey (UIUC)<br />
|[[# | Instabilities in shallow water wave models ]]<br />
| Kim & Lee<br />
|- <br />
|March 5<br />
| Khai Nguyen (NCSU)<br />
|[[#Khai Nguyen | Burgers Equation with Some Nonlocal Sources ]]<br />
| Tran<br />
|- <br />
|March 12<br />
| Hongwei Gao (UCLA)<br />
|[[#Hongwei Gao | Stochastic homogenization of certain nonconvex Hamilton-Jacobi equations ]]<br />
| Tran<br />
|- <br />
|March 19<br />
| Huy Nguyen (Princeton)<br />
|[[#Huy Nguyen | Compressible fluids and active potentials ]]<br />
| Lee<br />
|-<br />
|March 26<br />
| <br />
|[[# | Spring recess (Mar 24-Apr 1, 2018) ]]<br />
| <br />
|-<br />
|April 2<br />
| In-Jee Jeong (Princeton)<br />
|[[#In-Jee Jeong | Singularity formation for the 3D axisymmetric Euler equations ]]<br />
| Kim<br />
|- <br />
|April 9<br />
| Jeff Calder (Minnesota)<br />
|[[#Jeff Calder | TBD ]]<br />
| Tran<br />
|- <br />
|April 21-22 (Saturday-Sunday)<br />
| Midwest PDE seminar<br />
|[[#Midwest PDE seminar | ]]<br />
| Angenent, Feldman, Kim, Tran.<br />
|- <br />
|April 25 (Wednesday)<br />
| Hitoshi Ishii (Wasow lecture)<br />
|[[#Hitoshi Ishii | TBD]]<br />
| Tran.<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Dan Knopf===<br />
<br />
Title: Non-K&auml;hler Ricci flow singularities that converge to K&auml;hler-Ricci solitons<br />
<br />
Abstract: We describe Riemannian (non-K&auml;hler) Ricci flow solutions that develop finite-time Type-I singularities whose parabolic dilations converge to a shrinking K&auml;hler–Ricci soliton singularity model. More specifically, the singularity model for these solutions is the “blowdown soliton” discovered by Feldman, Ilmanen, and Knopf in 2003. Our results support the conjecture that the blowdown soliton is stable under Ricci flow. This work also provides the first set of rigorous examples of non-K&auml;hler solutions of Ricci flow that become asymptotically K&auml;hler, in suitable space-time neighborhoods of developing singularities, at rates that break scaling invariance. These results support the conjectured stability of the subspace of K&auml;hler metrics under Ricci flow.<br />
<br />
===Andreas Seeger===<br />
<br />
Title: Singular integrals and a problem on mixing flows<br />
<br />
Abstract: The talk will be about results related to Bressan's mixing problem. We present an inequality for the change of a Bianchini semi-norm of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which one proves bounds on Hardy spaces. This is joint work with Mahir Hadžić, Charles Smart and Brian Street.<br />
<br />
===Sam Krupa===<br />
<br />
Title: Proving Uniqueness of Solutions for Burgers Equation Entropic for a Single Entropy, with Eye Towards Systems Case<br />
<br />
Abstract: For hyperbolic systems of conservation laws, uniqueness of solutions is still largely open. We aim to expand the theory of uniqueness for systems of conservation laws. One difficulty is that many systems have only one entropy. This contrasts with scalar conservation laws, where many entropies exist. It took until 1994 to show that one entropy is enough to ensure uniqueness of solutions for the scalar conservation laws (Panov). This single entropy result was proven again by De Lellis, Otto and Westdickenberg in 2004. These two proofs both rely on the special connection between Hamilton--Jacobi equations and scalar conservation laws in one space dimension. However, this special connection does not extend to systems. In our new work, we prove the single entropy result for scalar conservation laws without using Hamilton--Jacobi. Our proof lays out new techniques that are promising for showing uniqueness of solutions in the systems case. This is joint work with A. Vasseur.<br />
<br />
<br />
===Maja Taskovic===<br />
<br />
Title: Exponential tails for the non-cutoff Boltzmann equation<br />
<br />
Abstract: The Boltzmann equation models the motion of a rarefied gas, in which particles interact through binary collisions, by describing the evolution of the particle density function. The effect of collisions on the density function is modeled by a bilinear integral operator (collision operator) which in many cases has a non-integrable angular kernel. For a long time the equation was simplified by assuming that this kernel is integrable (the so called Grad's cutoff) with a belief that such an assumption does not affect the equation significantly. However, in the last 20 years it has been observed that a non-integrable singularity carries regularizing properties which motivates further analysis of the equation in this setting.<br />
<br />
We study behavior in time of tails of solutions to the Boltzmann equation in the non-cutoff regime by examining the generation and propagation of $L^1$ and $L^\infty$ exponentially weighted estimates and the relation between them. For this purpose we introduce Mittag-Leffler moments which can be understood as a generalization of exponential moments. An interesting aspect of this result is that the singularity rate of the angular kernel affects the order of tails that can be shown to propagate in time. This is based on joint works with Alonso, Gamba, Pavlovic and Gamba, Pavlovic.<br />
<br />
<br />
===Ashish Kumar Pandey===<br />
<br />
Title: Instabilities in shallow water wave models<br />
<br />
Abstract: Slow modulations in wave characteristics of a nonlinear, periodic traveling wave in a dispersive medium may develop non-trivial structures which evolve as it propagates. This phenomenon is called modulational instability. In the context of water waves, this phenomenon was observed by Benjamin and Feir and, independently, by Whitham in Stokes' waves. I will discuss a general mechanism to study modulational instability of periodic traveling waves which can be applied to several classes of nonlinear dispersive equations including KdV, BBM, and regularized Boussinesq type equations.<br />
<br />
<br />
===Khai Nguyen===<br />
<br />
Title: Burgers Equation with Some Nonlocal Sources<br />
<br />
Abstract: Consider the Burgers equation with some nonlocal sources, which were derived from models of nonlinear wave with constant frequency. This talk will present some recent results on the global existence of entropy weak solutions, priori estimates, and a uniqueness result for both Burgers-Poisson and Burgers-Hilbert equations. Some open questions will be discussed.<br />
<br />
===Hongwei Gao=== <br />
<br />
Title: Stochastic homogenization of certain nonconvex Hamilton-Jacobi equations<br />
<br />
Abstract: In this talk, we discuss the stochastic homogenization of certain nonconvex Hamilton-Jacobi equations. The nonconvex Hamiltonians, which are generally uneven and inseparable, are generated by a sequence of (level-set) convex Hamiltonians and a sequence of (level-set) concave Hamiltonians through the min-max formula. We provide a monotonicity assumption on the contact values between those stably paired Hamiltonians so as to guarantee the stochastic homogenization. If time permits, we will talk about some homogenization results when the monotonicity assumption breaks down.<br />
<br />
===In-Jee Jeong===<br />
<br />
Title: Singularity formation for the 3D axisymmetric Euler equations<br />
<br />
Abstract: We consider the 3D axisymmetric Euler equations on exterior domains $\{ (x,y,z) : (1 + \epsilon|z|)^2 \le x^2 + y^2 \} $ for any $\epsilon > 0$ so that we can get arbitrarily close to the exterior of a cylinder. We construct a strong local well-posedness class, and show that within this class there exist compactly supported initial data which blows up in finite time. The local well-posedness class consists of velocities which are uniformly Lipschitz in space and have finite energy. Our results were inspired by recent works of Hou-Luo, Kiselev-Sverak, and many others, and the proof builds up on our previous works on 2D Euler and Boussinesq systems. This is joint work with Tarek Elgindi.</div>Donghyunhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=15039PDE Geometric Analysis seminar2018-02-06T21:38:47Z<p>Donghyun: </p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2018 | Tentative schedule for Fall 2018]]===<br />
<br />
<br />
<br />
== PDE GA Seminar Schedule Spring 2018 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|January 29, '''3-3:50PM, B341 VV.'''<br />
| Dan Knopf (UT Austin)<br />
|[[#Dan Knopf | Non-K&auml;hler Ricci flow singularities that converge to K&auml;hler-Ricci solitons]]<br />
| Angenent<br />
|- <br />
|February 5, '''3-3:50PM, B341 VV.'''<br />
| Andreas Seeger (UW)<br />
|[[#Andreas Seeger | Singular integrals and a problem on mixing flows ]]<br />
| Kim & Tran<br />
|- <br />
|February 12<br />
| Sam Krupa (UT-Austin)<br />
|[[#Sam Krupa | Proving Uniqueness of Solutions for Burgers Equation Entropic for a Single Entropy, with Eye Towards Systems Case ]]<br />
| Lee <br />
|- <br />
|February 19<br />
| Maja Taskovic (UPenn)<br />
|[[#Maja Taskovic | TBD ]]<br />
| Kim<br />
|- <br />
|February 26<br />
| Ashish Kumar Pandey (UIUC)<br />
|[[# | Instabilities in shallow water wave models ]]<br />
| Kim & Lee<br />
|- <br />
|March 5<br />
| Khai Nguyen (NCSU)<br />
|[[#Khai Nguyen | TBD ]]<br />
| Tran<br />
|- <br />
|March 12<br />
| Hongwei Gao (UCLA)<br />
|[[#Hongwei Gao | TBD ]]<br />
| Tran<br />
|- <br />
|March 19<br />
| Huy Nguyen (Princeton)<br />
|[[#Huy Nguyen | TBD ]]<br />
| Lee<br />
|-<br />
|March 26<br />
| <br />
|[[# | Spring recess (Mar 24-Apr 1, 2018) ]]<br />
| <br />
|-<br />
|April 2<br />
| <br />
|[[# | TBD ]]<br />
| <br />
|- <br />
|April 9<br />
| reserved<br />
|[[# | TBD ]]<br />
| Tran<br />
|- <br />
|April 21-22 (Saturday-Sunday)<br />
| Midwest PDE seminar<br />
|[[#Midwest PDE seminar | ]]<br />
| Angenent, Feldman, Kim, Tran.<br />
|- <br />
|April 25 (Wednesday)<br />
| Hitoshi Ishii (Wasow lecture)<br />
|[[#Hitoshi Ishii | TBD]]<br />
| Tran.<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Dan Knopf===<br />
<br />
Title: Non-K&auml;hler Ricci flow singularities that converge to K&auml;hler-Ricci solitons<br />
<br />
Abstract: We describe Riemannian (non-K&auml;hler) Ricci flow solutions that develop finite-time Type-I singularities whose parabolic dilations converge to a shrinking K&auml;hler–Ricci soliton singularity model. More specifically, the singularity model for these solutions is the “blowdown soliton” discovered by Feldman, Ilmanen, and Knopf in 2003. Our results support the conjecture that the blowdown soliton is stable under Ricci flow. This work also provides the first set of rigorous examples of non-K&auml;hler solutions of Ricci flow that become asymptotically K&auml;hler, in suitable space-time neighborhoods of developing singularities, at rates that break scaling invariance. These results support the conjectured stability of the subspace of K&auml;hler metrics under Ricci flow.<br />
<br />
===Andreas Seeger===<br />
<br />
Title: Singular integrals and a problem on mixing flows<br />
<br />
Abstract: The talk will be about results related to Bressan's mixing problem. We present an inequality for the change of a Bianchini semi-norm of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which one proves bounds on Hardy spaces. This is joint work with Mahir Hadžić, Charles Smart and Brian Street.<br />
<br />
===Sam Krupa===<br />
<br />
Title: Proving Uniqueness of Solutions for Burgers Equation Entropic for a Single Entropy, with Eye Towards Systems Case<br />
<br />
Abstract: For hyperbolic systems of conservation laws, uniqueness of solutions is still largely open. We aim to expand the theory of uniqueness for systems of conservation laws. One difficulty is that many systems have only one entropy. This contrasts with scalar conservation laws, where many entropies exist. It took until 1994 to show that one entropy is enough to ensure uniqueness of solutions for the scalar conservation laws (Panov). This single entropy result was proven again by De Lellis, Otto and Westdickenberg in 2004. These two proofs both rely on the special connection between Hamilton--Jacobi equations and scalar conservation laws in one space dimension. However, this special connection does not extend to systems. In our new work, we prove the single entropy result for scalar conservation laws without using Hamilton--Jacobi. Our proof lays out new techniques that are promising for showing uniqueness of solutions in the systems case. This is joint work with A. Vasseur.<br />
<br />
===Ashish Kumar Pandey===<br />
<br />
Title: Instabilities in shallow water wave models<br />
<br />
Abstract: Slow modulations in wave characteristics of a nonlinear, periodic traveling wave in a dispersive medium may develop non-trivial structures which evolve as it propagates. This phenomenon is called modulational instability. In the context of water waves, this phenomenon was observed by Benjamin and Feir and, independently, by Whitham in Stokes' waves. I will discuss a general mechanism to study modulational instability of periodic traveling waves which can be applied to several classes of nonlinear dispersive equations including KdV, BBM, and regularized Boussinesq type equations.</div>Donghyunhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=15015PDE Geometric Analysis seminar2018-02-05T12:44:21Z<p>Donghyun: /* PDE GA Seminar Schedule Spring 2018 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2018 | Tentative schedule for Fall 2018]]===<br />
<br />
<br />
<br />
== PDE GA Seminar Schedule Spring 2018 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|January 29, '''3-3:50PM, B341 VV.'''<br />
| Dan Knopf (UT Austin)<br />
|[[#Dan Knopf | Non-K&auml;hler Ricci flow singularities that converge to K&auml;hler-Ricci solitons]]<br />
| Angenent<br />
|- <br />
|February 5, '''3-3:50PM, B341 VV.'''<br />
| Andreas Seeger (UW)<br />
|[[#Andreas Seeger | Singular integrals and a problem on mixing flows ]]<br />
| Kim & Tran<br />
|- <br />
|February 12<br />
| Sam Krupa (UT-Austin)<br />
|[[#Sam Krupa | Proving Uniqueness of Solutions for Burgers Equation Entropic for a Single Entropy, with Eye Towards Systems Case ]]<br />
| Lee <br />
|- <br />
|February 19<br />
| Maja Taskovic (UPenn)<br />
|[[#Maja Taskovic | TBD ]]<br />
| Kim<br />
|- <br />
|February 26<br />
| Ashish Kumar Pandey (UIUC)<br />
|[[# | TBD ]]<br />
| Kim & Lee<br />
|- <br />
|March 5<br />
| Khai Nguyen (NCSU)<br />
|[[#Khai Nguyen | TBD ]]<br />
| Tran<br />
|- <br />
|March 12<br />
| Hongwei Gao (UCLA)<br />
|[[#Hongwei Gao | TBD ]]<br />
| Tran<br />
|- <br />
|March 19<br />
| Huy Nguyen (Princeton)<br />
|[[#Huy Nguyen | TBD ]]<br />
| Lee<br />
|-<br />
|March 26<br />
| <br />
|[[# | Spring recess (Mar 24-Apr 1, 2018) ]]<br />
| <br />
|-<br />
|April 2<br />
| <br />
|[[# | TBD ]]<br />
| <br />
|- <br />
|April 9<br />
| reserved<br />
|[[# | TBD ]]<br />
| Tran<br />
|- <br />
|April 21-22 (Saturday-Sunday)<br />
| Midwest PDE seminar<br />
|[[#Midwest PDE seminar | ]]<br />
| Angenent, Feldman, Kim, Tran.<br />
|- <br />
|April 25 (Wednesday)<br />
| Hitoshi Ishii (Wasow lecture)<br />
|[[#Hitoshi Ishii | TBD]]<br />
| Tran.<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Dan Knopf===<br />
<br />
Title: Non-K&auml;hler Ricci flow singularities that converge to K&auml;hler-Ricci solitons<br />
<br />
Abstract: We describe Riemannian (non-K&auml;hler) Ricci flow solutions that develop finite-time Type-I singularities whose parabolic dilations converge to a shrinking K&auml;hler–Ricci soliton singularity model. More specifically, the singularity model for these solutions is the “blowdown soliton” discovered by Feldman, Ilmanen, and Knopf in 2003. Our results support the conjecture that the blowdown soliton is stable under Ricci flow. This work also provides the first set of rigorous examples of non-K&auml;hler solutions of Ricci flow that become asymptotically K&auml;hler, in suitable space-time neighborhoods of developing singularities, at rates that break scaling invariance. These results support the conjectured stability of the subspace of K&auml;hler metrics under Ricci flow.<br />
<br />
===Andreas Seeger===<br />
<br />
Title: Singular integrals and a problem on mixing flows<br />
<br />
Abstract: The talk will be about results related to Bressan's mixing problem. We present an inequality for the change of a Bianchini semi-norm of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which one proves bounds on Hardy spaces. This is joint work with Mahir Hadžić, Charles Smart and Brian Street.<br />
<br />
===Sam Krupa===<br />
<br />
Title: Proving Uniqueness of Solutions for Burgers Equation Entropic for a Single Entropy, with Eye Towards Systems Case<br />
<br />
Abstract: For hyperbolic systems of conservation laws, uniqueness of solutions is still largely open. We aim to expand the theory of uniqueness for systems of conservation laws. One difficulty is that many systems have only one entropy. This contrasts with scalar conservation laws, where many entropies exist. It took until 1994 to show that one entropy is enough to ensure uniqueness of solutions for the scalar conservation laws (Panov). This single entropy result was proven again by De Lellis, Otto and Westdickenberg in 2004. These two proofs both rely on the special connection between Hamilton--Jacobi equations and scalar conservation laws in one space dimension. However, this special connection does not extend to systems. In our new work, we prove the single entropy result for scalar conservation laws without using Hamilton--Jacobi. Our proof lays out new techniques that are promising for showing uniqueness of solutions in the systems case. This is joint work with A. Vasseur.</div>Donghyunhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=14880PDE Geometric Analysis seminar2018-01-27T00:30:32Z<p>Donghyun: /* Abstracts */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2018 | Tentative schedule for Fall 2018]]===<br />
<br />
<br />
<br />
== PDE GA Seminar Schedule Spring 2018 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|January 29, '''3-3:50PM, B341 VV.'''<br />
| Dan Knopf (UT Austin)<br />
|[[#Dan Knopf | Non-K&auml;hler Ricci flow singularities that converge to K&auml;hler-Ricci solitons]]<br />
| Angenent<br />
|- <br />
|February 5<br />
| Andreas Seeger (UW)<br />
|[[#Andreas Seeger | TBD ]]<br />
| Kim & Tran<br />
|- <br />
|February 12<br />
| Sam Krupa (UT-Austin)<br />
|[[#Sam Krupa | Proving Uniqueness of Solutions for Burgers Equation Entropic for a Single Entropy, with Eye Towards Systems Case ]]<br />
| Lee <br />
|- <br />
|February 19<br />
| Maja Taskovic (UPenn)<br />
|[[#Maja Taskovic | TBD ]]<br />
| Kim<br />
|- <br />
|March 5<br />
| Khai Nguyen (NCSU)<br />
|[[#Khai Nguyen | TBD ]]<br />
| Tran<br />
|- <br />
|March 12<br />
| Hongwei Gao (UCLA)<br />
|[[#Hongwei Gao | TBD ]]<br />
| Tran<br />
|- <br />
|March 19<br />
| Huy Nguyen (Princeton)<br />
|[[#Huy Nguyen | TBD ]]<br />
| Lee<br />
|-<br />
|April 9<br />
| reserved<br />
|[[# | TBD ]]<br />
| Tran<br />
|- <br />
|April 21-22 (Saturday-Sunday)<br />
| Midwest PDE seminar<br />
|[[#Midwest PDE seminar | ]]<br />
| Angenent, Feldman, Kim, Tran.<br />
|- <br />
|April 25 (Wednesday)<br />
| Hitoshi Ishii (Wasow lecture)<br />
|[[#Hitoshi Ishii | TBD]]<br />
| Tran.<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Dan Knopf===<br />
<br />
Title: Non-K&auml;hler Ricci flow singularities that converge to K&auml;hler-Ricci solitons<br />
<br />
Abstract: We describe Riemannian (non-K&auml;hler) Ricci flow solutions that develop finite-time Type-I singularities whose parabolic dilations converge to a shrinking K&auml;hler–Ricci soliton singularity model. More specifically, the singularity model for these solutions is the “blowdown soliton” discovered by Feldman, Ilmanen, and Knopf in 2003. Our results support the conjecture that the blowdown soliton is stable under Ricci flow. This work also provides the first set of rigorous examples of non-K&auml;hler solutions of Ricci flow that become asymptotically K&auml;hler, in suitable space-time neighborhoods of developing singularities, at rates that break scaling invariance. These results support the conjectured stability of the subspace of K&auml;hler metrics under Ricci flow.<br />
<br />
<br />
<br />
===Sam Krupa===<br />
<br />
Title: Proving Uniqueness of Solutions for Burgers Equation Entropic for a Single Entropy, with Eye Towards Systems Case<br />
<br />
Abstract: For hyperbolic systems of conservation laws, uniqueness of solutions is still largely open. We aim to expand the theory of uniqueness for systems of conservation laws. One difficulty is that many systems have only one entropy. This contrasts with scalar conservation laws, where many entropies exist. It took until 1994 to show that one entropy is enough to ensure uniqueness of solutions for the scalar conservation laws (Panov). This single entropy result was proven again by De Lellis, Otto and Westdickenberg in 2004. These two proofs both rely on the special connection between Hamilton--Jacobi equations and scalar conservation laws in one space dimension. However, this special connection does not extend to systems. In our new work, we prove the single entropy result for scalar conservation laws without using Hamilton--Jacobi. Our proof lays out new techniques that are promising for showing uniqueness of solutions in the systems case. This is joint work with A. Vasseur.</div>Donghyunhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=14879PDE Geometric Analysis seminar2018-01-27T00:27:58Z<p>Donghyun: /* PDE GA Seminar Schedule Spring 2018 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2018 | Tentative schedule for Fall 2018]]===<br />
<br />
<br />
<br />
== PDE GA Seminar Schedule Spring 2018 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|January 29, '''3-3:50PM, B341 VV.'''<br />
| Dan Knopf (UT Austin)<br />
|[[#Dan Knopf | Non-K&auml;hler Ricci flow singularities that converge to K&auml;hler-Ricci solitons]]<br />
| Angenent<br />
|- <br />
|February 5<br />
| Andreas Seeger (UW)<br />
|[[#Andreas Seeger | TBD ]]<br />
| Kim & Tran<br />
|- <br />
|February 12<br />
| Sam Krupa (UT-Austin)<br />
|[[#Sam Krupa | Proving Uniqueness of Solutions for Burgers Equation Entropic for a Single Entropy, with Eye Towards Systems Case ]]<br />
| Lee <br />
|- <br />
|February 19<br />
| Maja Taskovic (UPenn)<br />
|[[#Maja Taskovic | TBD ]]<br />
| Kim<br />
|- <br />
|March 5<br />
| Khai Nguyen (NCSU)<br />
|[[#Khai Nguyen | TBD ]]<br />
| Tran<br />
|- <br />
|March 12<br />
| Hongwei Gao (UCLA)<br />
|[[#Hongwei Gao | TBD ]]<br />
| Tran<br />
|- <br />
|March 19<br />
| Huy Nguyen (Princeton)<br />
|[[#Huy Nguyen | TBD ]]<br />
| Lee<br />
|-<br />
|April 9<br />
| reserved<br />
|[[# | TBD ]]<br />
| Tran<br />
|- <br />
|April 21-22 (Saturday-Sunday)<br />
| Midwest PDE seminar<br />
|[[#Midwest PDE seminar | ]]<br />
| Angenent, Feldman, Kim, Tran.<br />
|- <br />
|April 25 (Wednesday)<br />
| Hitoshi Ishii (Wasow lecture)<br />
|[[#Hitoshi Ishii | TBD]]<br />
| Tran.<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Dan Knopf===<br />
<br />
Title: Non-K&auml;hler Ricci flow singularities that converge to K&auml;hler-Ricci solitons<br />
<br />
Abstract: We describe Riemannian (non-K&auml;hler) Ricci flow solutions that develop finite-time Type-I singularities whose parabolic dilations converge to a shrinking K&auml;hler–Ricci soliton singularity model. More specifically, the singularity model for these solutions is the “blowdown soliton” discovered by Feldman, Ilmanen, and Knopf in 2003. Our results support the conjecture that the blowdown soliton is stable under Ricci flow. This work also provides the first set of rigorous examples of non-K&auml;hler solutions of Ricci flow that become asymptotically K&auml;hler, in suitable space-time neighborhoods of developing singularities, at rates that break scaling invariance. These results support the conjectured stability of the subspace of K&auml;hler metrics under Ricci flow.</div>Donghyunhttps://www.math.wisc.edu/wiki/index.php?title=Spring_2018&diff=14732Spring 20182018-01-06T23:48:28Z<p>Donghyun: </p>
<hr />
<div>== PDE GA Seminar Schedule Spring 2018 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|January 29<br />
| Dan Knopf (UT Austin)<br />
|[[#Dan Knopf | Non-K&auml;hler Ricci flow singularities that converge to K&auml;hler-Ricci solitons]]<br />
| Angenent<br />
|- <br />
|February 5<br />
| Andreas Seeger (UW)<br />
|[[#Andreas Seeger | TBD ]]<br />
| Kim & Tran<br />
|- <br />
|February 12<br />
| Sam Krupa (UT-Austin)<br />
|[[#Sam Krupa | TBD ]]<br />
| Lee <br />
|- <br />
|February 19<br />
| Maja Taskovic (UPenn)<br />
|[[#Maja Taskovic | TBD ]]<br />
| Kim<br />
|- <br />
|March 5<br />
| Khai Nguyen (NCSU)<br />
|[[#Khai Nguyen | TBD ]]<br />
| Tran<br />
|- <br />
|March 12<br />
| Hongwei Gao (UCLA)<br />
|[[#Hongwei Gao | TBD ]]<br />
| Tran<br />
|- <br />
|March 19<br />
| Huy Nguyen (Princeton)<br />
|[[#Huy Nguyen | TBD ]]<br />
| Lee<br />
|-<br />
|April 21-22 (Saturday-Sunday)<br />
| Midwest PDE seminar<br />
|[[#Midwest PDE seminar | ]]<br />
| Angenent, Feldman, Kim, Tran.<br />
|- <br />
|April 25 (Wednesday)<br />
| Hitoshi Ishii (Wasow lecture)<br />
|[[#Hitoshi Ishii | TBD]]<br />
| Tran.<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Dan Knopf===<br />
<br />
Title: Non-K&auml;hler Ricci flow singularities that converge to K&auml;hler-Ricci solitons<br />
<br />
Abstract: We describe Riemannian (non-K&auml;hler) Ricci flow solutions that develop finite-time Type-I singularities whose parabolic dilations converge to a shrinking K&auml;hler–Ricci soliton singularity model. More specifically, the singularity model for these solutions is the “blowdown soliton” discovered by Feldman, Ilmanen, and Knopf in 2003. Our results support the conjecture that the blowdown soliton is stable under Ricci flow. This work also provides the first set of rigorous examples of non-K&auml;hler solutions of Ricci flow that become asymptotically K&auml;hler, in suitable space-time neighborhoods of developing singularities, at rates that break scaling invariance. These results support the conjectured stability of the subspace of K&auml;hler metrics under Ricci flow.</div>Donghyunhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=12626PDE Geometric Analysis seminar2016-10-28T17:22:30Z<p>Donghyun: /* Abstracts */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2016 | Tentative schedule for Spring 2017]]===<br />
<br />
= PDE GA Seminar Schedule Fall 2016 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 12<br />
| Daniel Spirn (U of Minnesota)<br />
|[[#Daniel Spirn | Dipole Trajectories in Bose-Einstein Condensates ]]<br />
| Kim<br />
|-<br />
|September 19<br />
| Donghyun Lee (UW-Madison)<br />
|[[#Donghyun Lee | The Boltzmann equation with specular boundary condition in convex domains ]]<br />
| Feldman<br />
|-<br />
|September 26<br />
| Kevin Zumbrun (Indiana)<br />
|[[# Kevin Zumbrun | A Stable Manifold Theorem for a class of degenerate evolution equations ]]<br />
| Kim <br />
|-<br />
|October 3<br />
| Will Feldman (UChicago )<br />
|[[#Will Feldman | Liquid Drops on a Rough Surface ]]<br />
| Lin & Tran<br />
|-<br />
|October 10<br />
| Ryan Hynd (UPenn)<br />
|[[#Ryan Hynd | Extremal functions for Morrey’s inequality in convex domains ]]<br />
| Feldman<br />
|-<br />
|October 17<br />
| Gung-Min Gie (Louisville)<br />
|[[# Gung-Min Gie | Boundary layer analysis of some incompressible flows ]]<br />
| Kim<br />
|-<br />
|October 24<br />
| Tau Shean Lim (UW Madison)<br />
|[[#Tau Shean Lim | Traveling Fronts of Reaction-Diffusion Equations with Ignition Media and Levy Operators ]]<br />
| Kim & Tran<br />
|-<br />
|October 31 ('''Special time and room''': B313VV, 3PM-4PM)<br />
| Tarek Elgindi ( Princeton)<br />
|[[# | Propagation of Singularities in Incompressible Fluids ]]<br />
| Lee & Kim<br />
|-<br />
|November 7<br />
| Adrian Tudorascu (West Virginia)<br />
|[[# Adrian Tudorascu | Hamilton-Jacobi equations in the Wasserstein space of probability measures ]]<br />
| Feldman<br />
|-<br />
|November 14<br />
| Alexis Vasseur ( UT-Austin)<br />
|[[# | ]]<br />
| Feldman<br />
|-<br />
|November 21<br />
| Minh-Binh Tran (UW Madison )<br />
|[[# | Quantum Kinetic Problems ]]<br />
| Hung Tran<br />
|-<br />
|November 28<br />
| David Kaspar (Brown)<br />
|[[# | ]]<br />
|Tran<br />
|-<br />
|December 5<br />
| Brian Weber (University of Pennsylvania)<br />
|[[# | TBA ]]<br />
|Bing Wang<br />
|-<br />
|December 12<br />
| <br />
|[[# | ]]<br />
| <br />
|}<br />
<br />
=Abstracts=<br />
<br />
===Daniel Spirn===<br />
<br />
Dipole Trajectories in Bose-Einstein Condensates<br />
<br />
Bose-Einstein condensates (BEC) are a state of matter in which supercooled atoms condense into the lowest possible quantum state. One interesting important feature of BECs are the presence of vortices that form when the condensate is stirred with lasers. I will discuss the behavior of these vortices, which interact with both the confinement potential and other vortices. I will also discuss a related inverse problem in which the features of the confinement can be extracted by the propagation of vortex dipoles.<br />
<br />
===Donghyun Lee===<br />
<br />
The Boltzmann equation with specular reflection boundary condition in convex domains<br />
<br />
I will present a recent work (https://arxiv.org/abs/1604.04342) with Chanwoo Kim on the global-wellposedness and stability of the Boltzmann equation in general smooth convex domains.<br />
<br />
===Kevin Zumbrun===<br />
<br />
TITLE: A Stable Manifold Theorem for a class of degenerate evolution equations <br />
<br />
ABSTRACT: We establish a Stable Manifold Theorem, with consequent exponential decay to equilibrium, for a class <br />
<br />
of degenerate evolution equations $Au'+u=D(u,u)$ with A bounded, self-adjoint, and one-to-one, but not invertible, and <br />
<br />
$D$ a bounded, symmetric bilinear map. This is related to a number of other scenarios investigated recently for which the <br />
<br />
associated linearized ODE $Au'+u=0$ is ill-posed with respect to the Cauchy problem. The particular case studied here <br />
<br />
pertains to the steady Boltzmann equation, yielding exponential decay of large-amplitude shock and boundary layers.<br />
<br />
<br />
<br />
===Will Feldman===<br />
<br />
Liquid Drops on a Rough Surface<br />
<br />
I will discuss the problem of determining the minimal energy shape of a liquid droplet resting on a rough solid surface. The shape of a liquid drop on a solid is strongly affected by the micro-structure of the surface on which it rests, where the surface inhomogeneity arises through varying chemical composition and surface roughness. I will explain a macroscopic regularity theory for the free boundary which allows to study homogenization, and more delicate properties like the size of the boundary layer induced by the surface roughness. <br />
<br />
The talk is based on joint work with Inwon Kim. A remark for those attending the weekend conference: this talk will attempt to have as little as possible overlap with I. Kim's conference talks. <br />
<br />
===Ryan Hynd===<br />
<br />
Extremal functions for Morrey’s inequality in convex domains<br />
<br />
A celebrated result in the theory of Sobolev spaces is Morrey's inequality, which establishes the continuous embedding of the continuous functions in certain Sobolev spaces. Interestingly enough the equality case of this inequality has not been thoroughly investigated (unless the underlying domain is R^n). We show that if the underlying domain is a bounded convex domain, then the extremal functions are determined up to a multiplicative factor. We will explain why the assertion is false if convexity is dropped and why convexity is not necessary for this result to hold. <br />
<br />
===Gung-Min Gie ===<br />
<br />
Boundary layer analysis of some incompressible flows<br />
<br />
The motions of viscous and inviscid fluids are modeled respectively by the Navier-Stokes and Euler equations. Considering the Navier-Stokes equations at vanishing viscosity as a singular perturbation of the Euler equations, one major problem, still essentially open, is to verify if the Navier-Stokes solutions converge as the viscosity tends to zero to the Euler solution in the presence of physical boundary. In this talk, we study the inviscid limit and boundary layers of some simplified Naiver-Stokes equations by either imposing a certain symmetry to the flow or linearizing the model around a stationary Euler flow. For the examples, we systematically use the method of correctors proposed earlier by J. L. Lions and construct an asymptotic expansion as the sum of the Navier-Stokes solution and the corrector. The corrector, which corrects the discrepancies between the boundary values of the viscous and inviscid solutions, is in fact an (approximating) solution of the corresponding Prandtl type equations. The validity of our asymptotic expansions is then confirmed globally in the whole domain by energy estimates on the difference of the viscous solution and the proposed expansion. This is a joint work with J. Kelliher, M. Lopes Filho, A. Mazzucato, and H. Nussenzveig Lopes.<br />
<br />
===Tau Shean Lim===<br />
<br />
Traveling Fronts of Reaction-Diffusion Equations with Ignition Media and Levy Operators<br />
<br />
We discuss traveling front solutions u(t,x) = U(x-ct) of reaction-diffusion equations u_t = Lu + f(u) with ignition media f and diffusion operators L generated by symmetric Levy processes X_t. Existence and uniqueness of fronts are well-known in the case of classical diffusion (i.e., Lu = Laplacian(u)) and non-local diffusion (Lu = J*u - u). Our work extends these results to general Levy operators. In particular, we show that a strong diffusivity in the underlying process (in the sense that the first moment of X_1 is infinite) prevents formation of fronts, while a weak diffusivity gives rise to a unique (up to translation) front U and speed c>0.<br />
<br />
===Tarek M. ELgindi===<br />
<br />
Propagation of Singularities in Incompressible Fluids<br />
<br />
We will discuss some recent results on the local and global stability of certain singular solutions to the incompressible 2d Euler equation. We will begin by giving a brief overview of the classical and modern results on the 2d Euler equation--particularly related to well-posedness theory in critical spaces. Then we will present a new well-posedness class which allows for merely Lipschitz continuous velocity fields and non-decaying vorticity. This will be based upon some interesting estimates for singular integrals on spaces with L^\infty scaling. After that we will introduce a class of scale invariant solutions to the 2d Euler equation and describe some of their remarkable properties including the existence of pendulum-like quasi periodic solutions and infinite-time cusp formation in vortex patches with corners. This is a joint work with I. Jeong. <br />
<br />
<br />
===Adrian Tudorascu===<br />
<br />
Hamilton-Jacobi equations in the Wasserstein space of probability measures <br />
<br />
In 2008 Gangbo, Nguyen and Tudorascu showed that certain variational solutions of the Euler-Poisson system in 1D can be regarded as optimal paths for the value-function giving the viscosity solution of some (infinite-dimensional) Hamilton-Jacobi equation whose phase-space is the Wasserstein space of Borel probability measures with finite second moment. At around the same time, Lasry, Lions, and others became interested in such Hamilton-Jacobi equations (HJE) in connection with their developing theory of Mean-Field games. A different approach (less intrinsic than ours) to the notion of viscosity solution was preferred, one that made an immediate connection between HJE in the Wasserstein space and HJE in Hilbert spaces (whose theory was well-studied and fairly well-understood). At the heart of the difference between these approaches lies the choice of the sub/supper-differential in the context of the Wasserstein space (i.e. the interpretation of ``cotangent space'' to this ``pseudo-Riemannian'' manifold) . In this talk I will start with a brief introduction to Mean-Field games and Optimal Transport, then I will discuss the challenges we encounter in the analysis of (our intrinsic) viscosity solutions of HJE in the Wasserstein space. Based on joint work with W. Gangbo.</div>Donghyunhttps://www.math.wisc.edu/wiki/index.php?title=Fall_2016&diff=12155Fall 20162016-08-31T21:07:15Z<p>Donghyun: </p>
<hr />
<div>= Seminar Schedule Fall 2016 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 12<br />
| Daniel Spirn (U of Minnesota)<br />
|[[# | ]]<br />
| Kim<br />
|-<br />
|September 19<br />
| Donghyun Lee (UW-Madison)<br />
|[[# | The Boltzmann equation with specular boundary condition in convex domains ]]<br />
| Feldman<br />
|-<br />
|September 26<br />
| Kevin Zumbrun (Indiana)<br />
|[[# | ]]<br />
| Kim <br />
|-<br />
|October 3<br />
| Will Feldman (UChicago )<br />
|[[# | ]]<br />
| Lin & Tran<br />
|-<br />
|October 10<br />
| Ryan Hynd (UPenn)<br />
|[[# | ]]<br />
| Feldman<br />
|-<br />
|October 17<br />
| Gung-Min Gie (Louisville)<br />
|[[# | ]]<br />
| Kim<br />
|-<br />
|October 24<br />
| ( )<br />
|[[# | ]]<br />
|<br />
|-<br />
|October 31<br />
| Tarek Elgindi ( Princeton)<br />
|[[# | Propagation of Singularities in Incompressible Fluids ]]<br />
| Lee & Kim<br />
|-<br />
|November 7<br />
| Adrian Tudorascu (West Virginia)<br />
|[[# | ]]<br />
| Feldman<br />
|-<br />
|November 14<br />
| Alexis Vasseur ( UT-Austin)<br />
|[[# | ]]<br />
| Feldman<br />
|-<br />
|November 21<br />
| Minh-Binh Tran (UW Madison )<br />
|[[# | Quantum Kinetic Problems ]]<br />
| Hung Tran<br />
|-<br />
|November 28<br />
| ( )<br />
|[[# | ]]<br />
|<br />
|-<br />
|December 5<br />
| Brian Weber(University of Pennsylvania)<br />
|[[# | TBA ]]<br />
|Bing Wang<br />
|-<br />
|December 12<br />
| David Kaspar (Brown)<br />
|[[# | ]]<br />
| Tran</div>Donghyunhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=11682PDE Geometric Analysis seminar2016-03-28T16:11:32Z<p>Donghyun: /* Abstracts */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2016 | Tentative schedule for Fall 2016]]===<br />
<br />
= Seminar Schedule Spring 2016 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 25<br />
||Tianling Jin (HKUST and Caltech)<br />
|[[#Tianling Jin | Holder gradient estimates for parabolic homogeneous p-Laplacian equations ]]<br />
| Zlatos<br />
|-<br />
|February 1<br />
|Russell Schwab (Michigan State University)<br />
|[[#Russell Schwab | Neumann homogenization via integro-differential methods ]]<br />
| Lin<br />
|-<br />
|February 8 <br />
|Jingrui Cheng (UW Madison)<br />
|[[#Jingrui Cheng | Semi-geostrophic system with variable Coriolis parameter ]]<br />
| Tran & Kim<br />
|-<br />
|February 15 <br />
|Paul Rabinowitz (UW Madison)<br />
|[[#Paul Rabinowitz | On A Double Well Potential System ]]<br />
| Tran & Kim<br />
|-<br />
|February 22 <br />
|Hong Zhang (Brown)<br />
|[[#Hong Zhang | On an elliptic equation arising from composite material ]]<br />
| Kim<br />
|-<br />
|February 29<br />
|Aaron Yip (Purdue university) <br />
|[[#Aaron Yip | Discrete and Continuous Motion by Mean Curvature in Inhomogeneous Media ]]<br />
| Tran<br />
|-<br />
|March 7<br />
|Hiroyoshi Mitake (Hiroshima university) <br />
||[[#Hiroyoshi Mitake | Selection problem for fully nonlinear equations]]<br />
| Tran<br />
|-<br />
|March 15<br />
|Nestor Guillen (UMass Amherst)<br />
|[[#Nestor Guillen | Min-max formulas for integro-differential equations and applications ]]<br />
| Lin<br />
|-<br />
|March 21 (Spring Break)<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|March 28<br />
|Ryan Denlinger (Courant Institute)<br />
|[[#Ryan Denlinger | The propagation of chaos for a rarefied gas of hard spheres in vacuum ]]<br />
| Lee<br />
|-<br />
|April 4<br />
| <br />
||[[# | ]]<br />
| <br />
|-<br />
|April 11<br />
|Misha Feldman (UW)<br />
|[[#Misha Feldman | ]]<br />
|<br />
|-<br />
|April 14: 2:25 PM in VV 901-Joint with Probability Seminar<br />
|Jessica Lin (UW-Madison)<br />
|[[#Jessica Lin | Optimal Quantitative Estimates in Stochastic Homogenization for Elliptic Equations in Nondivergence Form ]]<br />
|-<br />
|April 18<br />
|Sergey Bolotin (UW)<br />
|[[#Sergey Bolotin | Degenerate billiards in celestial mechanics]]<br />
|<br />
|-<br />
|April 21-24, KI-Net conference: Boundary Value Problems and Multiscale Coupling Methods for Kinetic Equations<br />
|Link: http://www.ki-net.umd.edu/content/conf?event_id=493<br />
|-<br />
|April 25<br />
| Moon-Jin Kang (UT-Austin)<br />
|[[# | ]]<br />
| Kim<br />
|-<br />
|May 2<br />
| <br />
|[[# | ]]<br />
|<br />
|}<br />
<br />
=Abstracts=<br />
<br />
===Tianling Jin===<br />
<br />
Holder gradient estimates for parabolic homogeneous p-Laplacian equations<br />
<br />
We prove interior Holder estimates for the spatial gradient of viscosity solutions to the parabolic homogeneous p-Laplacian equation <br />
u_t=|\nabla u|^{2-p} div(|\nabla u|^{p-2}\nabla u),<br />
where 1<p<\infty. This equation arises from tug-of-war like stochastic games with white noise. It can also be considered as the parabolic p-Laplacian equation in non divergence form. This is joint work with Luis Silvestre.<br />
<br />
===Russell Schwab===<br />
<br />
Neumann homogenization via integro-differential methods<br />
<br />
In this talk I will describe how one can use integro-differential methods to attack some Neumann homogenization problems-- that is, describing the effective behavior of solutions to equations with highly oscillatory Neumann data. I will focus on the case of linear periodic equations with a singular drift, which includes (with some regularity assumptions) divergence equations with non-co-normal oscillatory Neumann conditions. The analysis focuses on an induced integro-differential homogenization problem on the boundary of the domain. This is joint work with Nestor Guillen.<br />
<br />
===Jingrui Cheng===<br />
<br />
Semi-geostrophic system with variable Coriolis parameter.<br />
<br />
The semi-geostrophic system (abbreviated as SG) is a model of large-scale atmospheric/ocean flows. Previous works about the SG system have been restricted to the case of constant Coriolis force, where we write the equation in "dual coordinates" and solve. This method does not apply for variable Coriolis parameter case. We develop a time-stepping procedure to overcome this difficulty and prove local existence and uniqueness of smooth solutions to SG system. This is joint work with Michael Cullen and Mikhail Feldman.<br />
<br />
<br />
===Paul Rabinowitz===<br />
<br />
On A Double Well Potential System<br />
<br />
We will discuss an elliptic system of partial differential equations of the form<br />
\[<br />
-\Delta u + V_u(x,u) = 0,\;\;x \in \Omega = \R \times \mathcal{D}\subset \R^n, \;\;\mathcal{D} \; bounded \subset \R^{n-1}<br />
\]<br />
\[<br />
\frac{\partial u}{\partial \nu} = 0 \;\;on \;\;\partial \Omega,<br />
\]<br />
with $u \in \R^m$,\; $\Omega$ a cylindrical domain in $\R^n$, and $\nu$ the outward pointing normal to $\partial \Omega$. <br />
Here $V$ is a double well potential with $V(x, a^{\pm})=0$ and $V(x,u)>0$ otherwise. When $n=1, \Omega =\R^m$ and \eqref{*} is a Hamiltonian system of ordinary differential equations. <br />
When $m=1$, it is a single PDE that arises as an Allen-Cahn model for phase transitions. We will <br />
discuss the existence of solutions of \eqref{*} that are heteroclinic from $a^{-}$ to $a^{+}$ or homoclinic to $a^{-}$,<br />
i.e. solutions that are of phase transition type.<br />
<br />
This is joint work with Jaeyoung Byeon (KAIST) and Piero Montecchiari (Ancona).<br />
<br />
===Hong Zhang===<br />
<br />
On an elliptic equation arising from composite material<br />
<br />
I will present some recent results on second-order divergence type equations with piecewise constant coefficients. This problem arises in the study of composite materials with closely spaced interface boundaries, and the classical elliptic regularity theory are not applicable. In the 2D case, we show that any weak solution is piecewise smooth without the restriction of the underling domain where the equation is satisfied. This completely answers a question raised by Li and Vogelius (2000) in the 2D case. Joint work with Hongjie Dong.<br />
<br />
===Aaron Yip===<br />
<br />
Discrete and Continuous Motion by Mean Curvature in Inhomogeneous Media<br />
<br />
The talk will describe some results on the behavior of solutions of motion by mean curvature in inhomogeneous media. Emphasis will be put on the pinning and de-pinning transition, continuum limit of discrete spin systems and the motion of interface between patterns.<br />
<br />
<br />
===Hiroyoshi Mitake===<br />
<br />
Selection problem for fully nonlinear equations<br />
<br />
Recently, there was substantial progress on the selection problem on the ergodic problem for Hamilton-Jacobi equations, which was open during almost 30 years. In the talk, I will first show a result on the convex Hamilton-Jacobi equation, then tell important problems which still remain. Next, I will mainly focus on a recent joint work with H. Ishii (Waseda U.), and H. V. Tran (U. Wisconsin-Madison) which is about the selection problem for fully nonlinear, degenerate elliptic partial differential equations. I will present a new variational approach for this problem.<br />
<br />
===Nestor Guillen===<br />
<br />
Min-max formulas for integro-differential equations and applications<br />
<br />
We show under minimal assumptions that a nonlinear operator satisfying what is known as a "global comparison principle" can be represented by a min-max formula in terms of very special linear operators (Levy operators, which involve drift-diffusion and integro-differential terms). Such type of formulas have been very useful in the theory of second order equations -for instance, by allowing the representation of solutions as value functions for differential games. Applications include results on the structure of Dirichlet-to-Neumann mappings for fully nonlinear second order elliptic equations.<br />
<br />
===Ryan Denlinger===<br />
<br />
The propagation of chaos for a rarefied gas of hard spheres in vacuum<br />
<br />
We are interested in the rigorous mathematical justification of<br />
Boltzmann's equation starting from the deterministic evolution of<br />
many-particle systems. O. E. Lanford was able to derive Boltzmann's<br />
equation for hard spheres, in the Boltzmann-Grad scaling, on a short<br />
time interval. Improvements to the time in Lanford's theorem have so far<br />
either relied on a small data hypothesis, or have been restricted to<br />
linear regimes. We revisit the small data regime, i.e. a sufficiently<br />
dilute gas of hard spheres dispersing into vacuum; this is a regime<br />
where strong bounds are available globally in time. Subject to the<br />
existence of such bounds, we give a rigorous proof for the propagation<br />
of Boltzmann's ``one-sided'' molecular chaos.</div>Donghyunhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=10960PDE Geometric Analysis seminar2016-01-10T16:55:48Z<p>Donghyun: /* Seminar Schedule Spring 2016 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2016 | Tentative schedule for Fall 2016]]===<br />
<br />
= Seminar Schedule Spring 2016 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 25<br />
||Tianling Jin (HKUST and Caltech)<br />
|[[# Tianling Jin | Holder gradient estimates for parabolic homogeneous p-Laplacian equations ]]<br />
| Zlatos<br />
|-<br />
|February 1<br />
|Russell Schwab (Michigan State University)<br />
|[[# Russell Schwab | TBA ]]<br />
| Lin<br />
|-<br />
|February 8 <br />
|Jingrui Cheng (UW Madison)<br />
|[[# Jingrui Cheng | ]]<br />
|<br />
|-<br />
|February 15 <br />
| <br />
|[[# | ]]<br />
| <br />
|-<br />
|February 22 <br />
| Hong Zhang (Brown)<br />
|[[# Hong Zhang | ]]<br />
| Kim<br />
|-<br />
|February 29<br />
|Aaron Yip (Purdue university) <br />
|[[# Aaron Yip | TBD ]]<br />
| Tran<br />
|-<br />
|March 7<br />
|Hiroyoshi Mitake (Hiroshima university) <br />
||[[# Hiroyoshi Mitake | TBD ]]<br />
| Tran<br />
|-<br />
|March 15<br />
| Nestor Guillen (UMass Amherst)<br />
|[[#Nestor Guillen | TBA ]]<br />
| Lin<br />
|-<br />
|March 21 (Spring Break)<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|March 28<br />
| Ryan Denlinger (Courant Institute)<br />
|[[#Ryan Denlinger | The propagation of chaos for a rarefied gas of hard spheres in vacuum ]]<br />
| Lee<br />
|-<br />
|April 4<br />
| <br />
||[[# | ]]<br />
| <br />
|-<br />
|April 11<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|April 18<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|April 25<br />
| Moon-Jin Kang (UT-Austin)<br />
|[[# | ]]<br />
| Kim<br />
|-<br />
|May 2<br />
| <br />
|[[# | ]]<br />
|<br />
|}<br />
<br />
=Abstracts=<br />
<br />
===Tianling Jin===<br />
<br />
Holder gradient estimates for parabolic homogeneous p-Laplacian equations<br />
<br />
We prove interior Holder estimates for the spatial gradient of viscosity solutions to the parabolic homogeneous p-Laplacian equation <br />
u_t=|\nabla u|^{2-p} div(|\nabla u|^{p-2}\nabla u),<br />
where 1<p<\infty. This equation arises from tug-of-war like stochastic games with white noise. It can also be considered as the parabolic p-Laplacian equation in non divergence form. This is joint work with Luis Silvestre.</div>Donghyunhttps://www.math.wisc.edu/wiki/index.php?title=Spring_2016&diff=10823Spring 20162015-12-03T21:46:35Z<p>Donghyun: </p>
<hr />
<div>= Seminar Schedule Spring 2016 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 25<br />
||Tianling Jin (HKUST)<br />
|[[# Tianling Jin | TBA ]]<br />
| Zlatos<br />
|-<br />
|February 1<br />
|Russell Schwab (Michigan State University)<br />
|[[# Russell Schwab | TBA ]]<br />
| Lin<br />
|-<br />
|February 8 <br />
|Jingrui Cheng (UW Madison)<br />
|[[# Jingrui Cheng | ]]<br />
|<br />
|-<br />
|February 15<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|February 22 <br />
| Hong Zhang (Brown)<br />
|[[# Hong Zhang | ]]<br />
| Kim<br />
|-<br />
|February 29<br />
|Aaron Yip (Purdue university) <br />
|[[# Aaron Yip | TBD ]]<br />
| Tran<br />
|-<br />
|March 7<br />
| <br />
||[[# | ]]<br />
| <br />
|-<br />
|March 15<br />
| Nestor Guillen (UMass Amherst)<br />
|[[#Nestor Guillen | TBA ]]<br />
| Lin<br />
|-<br />
|March 21 (Spring Break)<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|March 28<br />
| Ryan Denlinger (Courant Institute)<br />
|[[#Ryan Denlinger | TBA ]]<br />
| Lee<br />
|-<br />
|April 4<br />
| <br />
||[[# | ]]<br />
| <br />
|-<br />
|April 11<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|April 18<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|April 25<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|May 2<br />
| <br />
|[[# | ]]<br />
|</div>Donghyun