Difference between revisions of "PDE Geometric Analysis seminar"
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(→PDE GA Seminar Schedule Fall 2018-Spring 2019) |
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===[[Previous PDE/GA seminars]]=== | ===[[Previous PDE/GA seminars]]=== | ||
− | ===[[Spring | + | ===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]=== |
+ | |||
+ | == PDE GA Seminar Schedule Fall 2018-Spring 2019 == | ||
+ | |||
− | |||
{| cellpadding="8" | {| cellpadding="8" | ||
!style="width:20%" align="left" | date | !style="width:20%" align="left" | date | ||
Line 10: | Line 12: | ||
!align="left" | title | !align="left" | title | ||
!style="width:20%" align="left" | host(s) | !style="width:20%" align="left" | host(s) | ||
− | |- | + | |
− | | | + | |- |
− | + | |August 31 (FRIDAY), | |
− | | | + | | Julian Lopez-Gomez (Complutense University of Madrid) |
− | + | |[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]] | |
− | + | | Rabinowitz | |
− | + | ||
− | + | |- | |
− | |[[# | + | |September 10, |
− | | | + | | Hiroyoshi Mitake (University of Tokyo) |
− | |- | + | |[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]] |
− | |September | + | |
− | + | ||
− | | | + | |
− | |[[# | + | |
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
| Tran | | Tran | ||
− | |- | + | |- |
− | |September | + | |September 12 and September 14, |
− | + | | Gunther Uhlmann (UWash) | |
− | | | + | |[[#Gunther Uhlmann | TBA ]] |
− | |[[# | + | | Li |
+ | |- | ||
+ | |September 17, | ||
+ | | Changyou Wang (Purdue) | ||
+ | |[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]] | ||
| Tran | | Tran | ||
− | |- | + | |- |
− | | | + | |Sep 28, Colloquium |
− | + | | [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU) | |
− | | | + | |[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]] |
− | |[[# | + | | Thiffeault |
− | | | + | |- |
− | |- | + | |October 1, |
− | |October | + | | Matthew Schrecker (UW) |
− | | | + | |[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]] |
− | | | + | | Kim and Tran |
− | | | + | |- |
− | |- | + | |October 8, |
− | |October | + | | Anna Mazzucato (PSU) |
− | | | + | |[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]] |
− | |[[# | + | | Li and Kim |
+ | |- | ||
+ | |October 15, | ||
+ | | Lei Wu (Lehigh) | ||
+ | |[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]] | ||
| Kim | | Kim | ||
+ | |- | ||
+ | |October 22, | ||
+ | | Annalaura Stingo (UCD) | ||
+ | |[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]] | ||
+ | | Mihaela Ifrim | ||
+ | |- | ||
+ | |October 29, | ||
+ | | Yeon-Eung Kim (UW) | ||
+ | |[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]] | ||
+ | | Kim and Tran | ||
+ | |- | ||
+ | |November 5, | ||
+ | | Albert Ai (UC Berkeley) | ||
+ | |[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]] | ||
+ | | Mihaela Ifrim | ||
+ | |- | ||
+ | |Nov 7 (Wednesday), Colloquium | ||
+ | | [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT) | ||
+ | |[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]] | ||
+ | | Feldman | ||
+ | |- | ||
+ | |December 3, ''' Time: 3:00, Room: B223 Van Vleck ''' | ||
+ | | Trevor Leslie (UW) | ||
+ | |[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]] | ||
+ | | Kim and Tran | ||
+ | |- | ||
+ | |December 10, ''' Time: 2:25, Room: B223 Van Vleck ''' | ||
+ | |Serena Federico (MIT) | ||
+ | |[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]] | ||
+ | | Mihaela Ifrim | ||
|- | |- | ||
− | | | + | |December 10, Colloquium, '''Time: 4:00''' |
− | | | + | | [https://math.mit.edu/~maxe/ Max Engelstein] (MIT) |
− | |[[# | + | |[[# Max Engelstein| The role of Energy in Regularity ]] |
− | | | + | | Feldman |
|- | |- | ||
− | | | + | |January 28, |
− | | | + | | Ru-Yu Lai (Minnesota) |
− | |[[# | + | |[[# Ru-Yu Lai | TBA ]] |
− | | Kim | + | | Li and Kim |
− | |- | + | |- |
− | | | + | | February 4, |
− | | | + | | Seokbae Yun (SKKU, long term visitor of UW-Madison) |
− | |[[# | + | |[[# Seokbae Yun | TBA ]] |
− | | | + | | Kim |
− | |- | + | |- |
− | | | + | | February 18, |
− | | | + | | Daniel Tataru (Berkeley) |
− | |[[# | + | |[[# Daniel Tataru | TBA ]] |
− | | | + | | Ifrim |
− | |- | + | |- |
− | | | + | |Time: TBD in February, |
| Xiaoqin Guo (UW) | | Xiaoqin Guo (UW) | ||
− | |[[#Xiaoqin Guo | | + | |[[#Xiaoqin Guo | TBA ]] |
− | | Kim | + | | Kim and Tran |
+ | |- | ||
+ | |Time: TBD in February, | ||
+ | | Wenjia Jing (Tsinghua University) | ||
+ | |[[#Wenjia Jing | TBA ]] | ||
+ | | Tran | ||
|- | |- | ||
− | | | + | |March 4 |
− | | | + | | Vladimir Sverak (Minnesota) |
− | |[[# | + | |[[#Vladimir Sverak | TBA(Wasow lecture) ]] |
− | | | + | | Kim |
− | |- | + | |- |
− | | | + | |March 11 |
− | | | + | | Jonathan Luk (Stanford) |
− | |[[# | + | |[[#Jonathan Luk | TBA ]] |
+ | | Kim | ||
+ | |- | ||
+ | |March 18, | ||
+ | | Spring recess (Mar 16-24, 2019) | ||
+ | |[[# | ]] | ||
+ | | | ||
+ | |- | ||
+ | |April 1 | ||
+ | | Zaher Hani (Michigan) | ||
+ | |[[#Zaher Hani | TBA ]] | ||
+ | | Ifrim | ||
+ | |- | ||
+ | |April 15, | ||
+ | | Yao Yao (Gatech) | ||
+ | |[[#Yao Yao | TBA ]] | ||
| Tran | | Tran | ||
− | |- | + | |- |
− | | | + | |April 22, |
− | | | + | | Jessica Lin (McGill University) |
− | |[[# | + | |[[#Jessica Lin | TBA ]] |
− | | | + | | Tran |
− | |- | + | |- |
− | | | + | |April 29, |
− | | | + | | ( ) |
− | |[[# | + | |[[# | TBA ]] |
− | | | + | | |
|} | |} | ||
− | ==Abstracts== | + | == Abstracts == |
− | === | + | ===Julian Lopez-Gomez=== |
− | + | Title: The theorem of characterization of the Strong Maximum Principle | |
− | + | Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan. | |
− | === | + | ===Hiroyoshi Mitake=== |
+ | Title: On approximation of time-fractional fully nonlinear equations | ||
− | + | Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) | |
− | |||
− | |||
− | + | ===Changyou Wang=== | |
− | + | Title: Some recent results on mathematical analysis of Ericksen-Leslie System | |
+ | Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years. | ||
− | === | + | ===Matthew Schrecker=== |
− | + | ||
− | + | Title: Finite energy methods for the 1D isentropic Euler equations | |
− | + | ||
− | + | Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations. | |
− | + | ||
− | + | ===Anna Mazzucato=== | |
+ | Title: On the vanishing viscosity limit in incompressible flows | ||
− | + | Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains. | |
− | + | ||
− | + | ===Lei Wu=== | |
− | + | Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects | |
− | Title: | + | |
− | Abstract: I will | + | Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition. |
− | |||
− | + | ===Annalaura Stingo=== | |
− | + | Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D | |
− | + | Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » . | |
+ | Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities. | ||
− | + | ===Yeon-Eung Kim=== | |
− | + | Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties | |
− | + | A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties. | |
− | + | ===Albert Ai=== | |
− | + | Title: Low Regularity Solutions for Gravity Water Waves | |
− | + | Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result. | |
− | + | ===Trevor Leslie=== | |
− | + | Title: Flocking Models with Singular Interaction Kernels | |
− | + | Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model. | |
− | + | ===Serena Federico=== | |
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | Title: Sufficient conditions for local solvability of some degenerate partial differential operators | |
− | + | Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable. | |
− | + | ===Max Engelstein=== | |
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | Title: The role of Energy in Regularity | |
+ | Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy. | ||
− | + | However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field. | |
− | + | We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur. | |
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + |
Latest revision as of 09:05, 10 December 2018
The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.
Contents
Previous PDE/GA seminars
Tentative schedule for Fall 2019-Spring 2020
PDE GA Seminar Schedule Fall 2018-Spring 2019
date | speaker | title | host(s) |
---|---|---|---|
August 31 (FRIDAY), | Julian Lopez-Gomez (Complutense University of Madrid) | The theorem of characterization of the Strong Maximum Principle | Rabinowitz |
September 10, | Hiroyoshi Mitake (University of Tokyo) | On approximation of time-fractional fully nonlinear equations | Tran |
September 12 and September 14, | Gunther Uhlmann (UWash) | TBA | Li |
September 17, | Changyou Wang (Purdue) | Some recent results on mathematical analysis of Ericksen-Leslie System | Tran |
Sep 28, Colloquium | Gautam Iyer (CMU) | Stirring and Mixing | Thiffeault |
October 1, | Matthew Schrecker (UW) | Finite energy methods for the 1D isentropic Euler equations | Kim and Tran |
October 8, | Anna Mazzucato (PSU) | On the vanishing viscosity limit in incompressible flows | Li and Kim |
October 15, | Lei Wu (Lehigh) | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects | Kim |
October 22, | Annalaura Stingo (UCD) | Global existence of small solutions to a model wave-Klein-Gordon system in 2D | Mihaela Ifrim |
October 29, | Yeon-Eung Kim (UW) | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties | Kim and Tran |
November 5, | Albert Ai (UC Berkeley) | Low Regularity Solutions for Gravity Water Waves | Mihaela Ifrim |
Nov 7 (Wednesday), Colloquium | Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems | Feldman |
December 3, Time: 3:00, Room: B223 Van Vleck | Trevor Leslie (UW) | Flocking Models with Singular Interaction Kernels | Kim and Tran |
December 10, Time: 2:25, Room: B223 Van Vleck | Serena Federico (MIT) | Sufficient conditions for local solvability of some degenerate partial differential operators | Mihaela Ifrim |
December 10, Colloquium, Time: 4:00 | Max Engelstein (MIT) | The role of Energy in Regularity | Feldman |
January 28, | Ru-Yu Lai (Minnesota) | TBA | Li and Kim |
February 4, | Seokbae Yun (SKKU, long term visitor of UW-Madison) | TBA | Kim |
February 18, | Daniel Tataru (Berkeley) | TBA | Ifrim |
Time: TBD in February, | Xiaoqin Guo (UW) | TBA | Kim and Tran |
Time: TBD in February, | Wenjia Jing (Tsinghua University) | TBA | Tran |
March 4 | Vladimir Sverak (Minnesota) | TBA(Wasow lecture) | Kim |
March 11 | Jonathan Luk (Stanford) | TBA | Kim |
March 18, | Spring recess (Mar 16-24, 2019) | ||
April 1 | Zaher Hani (Michigan) | TBA | Ifrim |
April 15, | Yao Yao (Gatech) | TBA | Tran |
April 22, | Jessica Lin (McGill University) | TBA | Tran |
April 29, | ( ) | TBA |
Abstracts
Julian Lopez-Gomez
Title: The theorem of characterization of the Strong Maximum Principle
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.
Hiroyoshi Mitake
Title: On approximation of time-fractional fully nonlinear equations
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.)
Changyou Wang
Title: Some recent results on mathematical analysis of Ericksen-Leslie System
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.
Matthew Schrecker
Title: Finite energy methods for the 1D isentropic Euler equations
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.
Anna Mazzucato
Title: On the vanishing viscosity limit in incompressible flows
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.
Lei Wu
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.
Annalaura Stingo
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » . Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.
Yeon-Eung Kim
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.
Albert Ai
Title: Low Regularity Solutions for Gravity Water Waves
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.
Trevor Leslie
Title: Flocking Models with Singular Interaction Kernels
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.
Serena Federico
Title: Sufficient conditions for local solvability of some degenerate partial differential operators
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.
Max Engelstein
Title: The role of Energy in Regularity
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.