Difference between revisions of "PDE Geometric Analysis seminar"

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===[[Previous PDE/GA seminars]]===
 
===[[Previous PDE/GA seminars]]===
===[[Spring 2018 | Tentative schedule for Spring 2018]]===
+
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===
 +
 
 +
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==
 +
 
  
== PDE GA Seminar Schedule Fall 2017 ==
 
 
{| 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)
|-  
+
 
|September 11
+
|-
|Mihaela Ifrim (UW)
+
|August 31 (FRIDAY),
|[[#Mihaela Ifrim|  Well-posedness and dispersive decay of small data solutions for the Benjamin-Ono equation]]
+
| Julian Lopez-Gomez (Complutense University of Madrid)
| Kim & Tran
+
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]
|-
+
| Rabinowitz
|September 18
+
 
|Longjie Zhang (University of Tokyo)  
+
|-
|[[#Longjie Zhang | On curvature flow with driving force starting as singular initial curve in the plane]]
+
|September 10,
| Angenent
+
| Hiroyoshi Mitake (University of Tokyo)
|-  
+
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]
|September 22,
+
VV 9th floor hall, 4:00pm
+
|Jaeyoung Byeon (KAIST)  
+
|[[#Jaeyoung Byeon| Colloquium: Patterns formation for elliptic systems with large interaction forces]]
+
|  Rabinowitz
+
|-
+
|September 25
+
| Tuoc Phan (UTK)
+
|[[#Tuoc Phan |  Calderon-Zygmund regularity estimates for weak solutions of quasi-linear parabolic equations with an application]]
+
 
| Tran
 
| Tran
|-  
+
|-
|September 26,  
+
|September 12 and September 14,
VV B139 4:00pm
+
| Gunther Uhlmann (UWash)
| Hiroyoshi Mitake (Hiroshima University)
+
|[[#Gunther Uhlmann | TBA ]]
|[[#Hiroyoshi Mitake Joint Analysis/PDE seminar: Derivation of multi-layered interface system and its application]]
+
| Li
 +
|- 
 +
|September 17,
 +
| Changyou Wang (Purdue)
 +
|[[#Changyou Wang Some recent results on mathematical analysis of Ericksen-Leslie System ]]
 
| Tran
 
| Tran
|-  
+
|-
|September 29,
+
|Sep 28, Colloquium
VV901 2:25pm
+
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)
| Dongnam Ko (CMU & SNU)
+
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]
|[[#Dongnam Ko | a joint seminar with ACMS: On the emergence of local flocking phenomena in Cucker-Smale ensembles ]]
+
| Thiffeault
| Shi Jin & Kim
+
|- 
|-  
+
|October 1,
|October 2
+
| Matthew Schrecker (UW)
| No seminar due to a KI-Net conference
+
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]
|
+
| Kim and Tran
|
+
|-
|-  
+
|October 8,
|October 9
+
| Anna Mazzucato (PSU)
| Sameer Iyer (Brown University)
+
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]
|[[#Sameer Iyer | Global-in-x Steady Prandtl Expansion over a Moving Boundary ]]
+
| 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 | TBA ]]
 +
| Kim and Tran
 +
|- 
 +
|November 5,
 +
| Albert Ai (UC Berkeley)
 +
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]
 +
| Mihaela Ifrim
 +
|- 
 +
|December 3,
 +
| Trevor Leslie (UW)
 +
|[[#Trevor Leslie | TBA ]]
 +
| Kim and Tran
 
|-  
 
|-  
|October 16
+
|December 10,
| Jingrui Cheng (UW)
+
|   ( )
|[[#Jingrui Cheng | A 1-D semigeostrophic model with moist convection ]]
+
|[[#  | TBA ]]
| Kim & Tran
+
|
 
|-  
 
|-  
|October 23
+
|January 28,
| Donghyun Lee (UW)
+
|   ( )
|[[#Donghyun Lee | The Vlasov-Poisson-Boltzmann system in bounded domains ]]
+
|[[#  | TBA ]]
| Kim & Tran
+
|   
|-
+
|-
|October 30
+
|Time: TBD,
| Myoungjean Bae (POSTECH)
+
| Jessica Lin (McGill University)
|[[#Myoungjean Bae |  3-D axisymmetric subsonic flows with nonzero swirl for the compressible Euler-Poisson system ]]
+
|[[#Jessica Lin | TBA ]]
Feldman
+
|-  
+
|November 6
+
| Jingchen Hu (USTC and UW)
+
|[[#Jingchen Hu |  Shock Reflection and Diffraction Problem with Potential Flow Equation ]]
+
| Kim & Tran
+
|-
+
|November 27
+
| Ru-Yu Lai (Minnesota)
+
|[[#Ru-Yu Lai |  TBD ]]
+
| Li
+
|-
+
|December 4
+
| Norbert Pozar (Kanazawa University)
+
|[[#Norbert Pozar | TBD ]]
+
 
| Tran
 
| 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 29,
 +
|  ( )
 +
|[[#  | TBA ]]
 +
 
|}
 
|}
  
==Abstracts==
+
== Abstracts ==
  
===Mihaela Ifrim===
+
===Julian Lopez-Gomez===
  
Well-posedness and dispersive decay of small data solutions for the Benjamin-Ono equation
+
Title: The theorem of characterization of the Strong Maximum Principle
  
Our goal is to take a first step toward understanding the long time dynamics of solutions for the Benjamin-Ono equation. While this problem is known to be both completely integrable and globally well-posed in $L^2$, much less seems to be known concerning its long time dynamics. We present that for small localized data the solutions have (nearly) dispersive dynamics almost globally in time. An additional objective is to revisit the $L^2$ theory for the Benjamin-Ono equation and provide a simpler, self-contained approach. This is joined work with Daniel Tataru.
+
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.
  
===Longjie Zhang===
+
===Hiroyoshi Mitake===
 +
Title: On approximation of time-fractional fully nonlinear equations
  
On curvature flow with driving force starting as singular initial curve in the plane
+
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.)
  
We consider a family of axisymmetric curves evolving by its mean curvature with driving force in the plane. However, the initial curve is oriented singularly at origin. We investigate this problem by level set method and give some criteria to judge whether the interface evolution is fattening or not. In the end, we can classify the solutions into three categories and provide the asymptotic behavior in each category. Our main tools in this paper are level set method and intersection number principle.
 
  
===Jaeyoung Byeon===
 
  
Title: Patterns formation for elliptic systems with large interaction forces
+
===Changyou Wang===
  
Abstract: Nonlinear elliptic systems arising from nonlinear Schroedinger systems have simple looking reaction terms. The corresponding energy for the reaction terms can be expressed as quadratic forms in terms of density functions.  The i, j-th entry of the matrix for the quadratic form represents the interaction force between the components i and j of the system. If the sign of an entry is positive, the force between the two components is attractive; on the other hand, if it is negative, it is repulsive. When the interaction forces between different components are large, the network structure of attraction and repulsion between components might produce several interesting patterns for solutions. As a starting point to study the general pattern formation structure for systems with a large number of components, I will first discuss the simple case of 2-component systems, and then the much more complex case of 3-component systems.
+
Title: Some recent results on mathematical analysis of Ericksen-Leslie System
 
+
 
+
===Tuoc Phan===
+
Calderon-Zygmund regularity estimates for weak solutions of quasi-linear parabolic equations with an application.
+
 
+
Abstract: In this talk, we first introduce a problem on the existence of global time smooth solutions for a system of cross-diffusion equations. We then recall some classical results on regularity theories, and show that to solve our problem, new results on regularity theory estimates of Calderon-Zygmund type for gradients of solutions to a class of parabolic equations in Lebesgue spaces are required. We then discuss a result on Calderon-Zygmnud type estimate in the concrete setting to solve our
+
mentioned problem regarding the system of cross-diffusion equations. The remaining part of the talk will be focused on some new generalized results on regularity gradient estimates for some general class of quasi-linear parabolic equations. Regularity estimates for gradients of solutions in Lorentz spaces will be presented. Ideas of the proofs for the results are given.
+
 
+
===Hiroyoshi Mitake===
+
Derivation of multi-layered interface system and its application
+
  
Abstract: In this talk, I will propose a multi-layered interface system which can be formally derived by the singular limit of the weakly coupled system of the Allen-Cahn equation. By using the level set approach, this system can be written as a quasi-monotone degenerate parabolic system. We give results of the well-posedness of viscosity solutions, and study the singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.
+
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===
  
===Dongnam Ko===
+
Title: Finite energy methods for the 1D isentropic Euler equations
On the emergence of local flocking phenomena in Cucker-Smale ensembles
+
  
Emergence of flocking groups are often observed in many complex network systems. The Cucker-Smale model is one of the flocking model, which describes the dynamics of attracting particles. This talk concerns time-asymptotic behaviors of Cucker-Smale particle ensembles, especially for mono-cluster and bi-cluster flockings. The emergence of flocking phenomena is determined by sufficient initial conditions, coupling strength, and communication weight decay. Our asymptotic analysis uses the Lyapunov functional approach and a Lagrangian formulation of the coupled system. We derive a system of differential inequalities for the functionals that measure the local fluctuations and group separations along particle trajectories. The bootstrapping argument is the key idea to prove the gathering and separating behaviors of Cucker-Smale particles simultaneously.
+
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.
  
===Sameer Iyer===
+
===Anna Mazzucato===
Title: Global-in-x Steady Prandtl Expansion over a Moving Boundary.
+
  
Abstract: I will outline the proof that steady, incompressible Navier-Stokes flows posed over the moving boundary, y = 0, can be decomposed into Euler and Prandtl flows globally in the tangential variable, assuming a sufficiently small velocity mismatch. The main obstacles in the analysis center around obtaining sharp decay rates for the linearized profiles and the remainders. The remainders are controlled via a high-order energy method, supplemented with appropriate embedding theorems, which I will present.
+
Title: On the vanishing viscosity limit in incompressible flows
  
===Jingrui Cheng===
+
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.
  
A 1-D semigeostrophic model with moist convection.
+
===Lei Wu===
  
We consider a simplified 1-D model of semigeostrophic system with moisture, which describes moist convection in a single column in the atmosphere. In general, the solution is non-continuous and it is nontrivial part of the problem to find a suitable definition of weak solutions. We propose a plausible definition of such weak solutions which describes the evolution of the probability distribution of the physical quantities, so that the equations hold in the sense of almost everywhere. Such solutions are constructed from a discrete scheme which obeys the physical principles. This is joint work with Mike Cullen, together with Bin Cheng, John Norbury and Matthew Turner.
+
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects
  
===Donghyun Lee===
+
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.
  
We construct a unique global-in-time solution to the Vlasov-Poisson-Boltzmann system in convex domains with the diffuse boundary condition. Moreover we prove an exponential convergence of distribution function toward the global Maxwellian.
 
  
===Myoungjean Bae===
+
===Annalaura Stingo===
  
3-D axisymmetric subsonic flows with nonzero swirl for the compressible Euler-Poisson system.
+
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D
  
I will present a recent result on the structural stability of 3-D axisymmetric subsonic flows with nonzero swirl for the steady compressible Euler–Poisson system in a cylinder supplemented with non-small boundary data. A special Helmholtz decomposition of the velocity field is introduced for 3-D axisymmetric flow with a nonzero swirl (=angular momentum density) component. This talk is based on a joint work with S. Weng (Wuhan University, China).
+
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. »
  
===Jingchen Hu===
+
===Albert Ai===
  
Shock Reflection and Diffraction Problem with Potential Flow Equation
+
Title: Low Regularity Solutions for Gravity Water Waves
  
In this talk, we will present our work on nonsymmetric shock reflection and diffraction problem, the equation concerned is potential flow equation, which is a simplification of Euler System, mainly based on the assumption that flow has no vortex. We showed in both nonsymmetric reflection case and diffraction case, that physically admissible solution does not exist. This implies that the formation of vortex is essential to maintain the structural stability of shock reflection and diffraction.
+
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.

Latest revision as of 18:03, 12 October 2018

The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.

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) TBA Kim and Tran
November 5, Albert Ai (UC Berkeley) Low Regularity Solutions for Gravity Water Waves Mihaela Ifrim
December 3, Trevor Leslie (UW) TBA Kim and Tran
December 10, ( ) TBA
January 28, ( ) TBA
Time: TBD, Jessica Lin (McGill 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 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. »

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.