# Difference between revisions of "PDE Geometric Analysis seminar"

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|November 9 | |November 9 | ||

− | | Javier Gomez | + | | Javier Gomez Serrano (Princeton) |

− | ||[[# Javier Gomez-Serrano | | + | ||[[# Javier Gomez-Serrano | Existence and regularity of rotating global solutions for active scalars ]] |

|Zlatos | |Zlatos | ||

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Abstract: W^{2,1} estimates for the Monge-Ampere equation \det D^2u = f in R^n were first obtained by De Philippis and Figalli in the case that f is bounded between positive constants. Motivated by applications to the semigeostrophic equation, we consider the case that f is bounded but allowed to be zero on some set. In this case there are simple counterexamples to W^{2,1} regularity in dimension n \geq 3 that have a Lipschitz singularity. In contrast, if n = 2 a classical theorem of Alexandrov on the propagation of Lipschitz singularities shows that solutions are C^1. We will discuss a counterexample to W^{2,1} regularity in two dimensions whose second derivatives have nontrivial Cantor part, and also a related result on the propagation of Lipschitz / log(Lipschitz) singularities that is optimal by example. | Abstract: W^{2,1} estimates for the Monge-Ampere equation \det D^2u = f in R^n were first obtained by De Philippis and Figalli in the case that f is bounded between positive constants. Motivated by applications to the semigeostrophic equation, we consider the case that f is bounded but allowed to be zero on some set. In this case there are simple counterexamples to W^{2,1} regularity in dimension n \geq 3 that have a Lipschitz singularity. In contrast, if n = 2 a classical theorem of Alexandrov on the propagation of Lipschitz singularities shows that solutions are C^1. We will discuss a counterexample to W^{2,1} regularity in two dimensions whose second derivatives have nontrivial Cantor part, and also a related result on the propagation of Lipschitz / log(Lipschitz) singularities that is optimal by example. | ||

+ | |||

+ | === Javier Gomez Serrano === | ||

+ | |||

+ | Existence and regularity of rotating global solutions for active scalars | ||

+ | |||

+ | A particular kind of weak solutions for a 2D active scalar equation are the so called patches, i.e., solutions for which the scalar is a step function taking one value inside a moving region and another in the complement. The evolution of such distribution is completely determined by the evolution of the boundary, allowing the problem to be treated as a non-local one dimensional equation for the contour. In this talk we will discuss the existence and regularity of uniformly rotating solutions for the vortex patch and generalized surface quasi-geostrophic (gSQG) patch equation. We will also outline the proof for the smooth (non patch) SQG case. Joint work with Angel Castro and Diego Cordoba. |

## Revision as of 10:02, 5 November 2015

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 Spring 2016

# Seminar Schedule Fall 2015

date | speaker | title | host(s) |
---|---|---|---|

September 7 (Labor Day) | |||

September 14 (special room: B115) | Hung Tran (Madison) | Some inverse problems in periodic homogenization of Hamilton--Jacobi equations | |

September 21 (special room: B115) | Eric Baer (Madison) | Optimal function spaces for continuity of the Hessian determinant as a distribution | |

September 28 | Donghyun Lee (Madison) | FLUIDS WITH FREE-SURFACE AND VANISHING VISCOSITY LIMIT | |

October 5 | Hyung-Ju Hwang (Postech & Brown Univ) | The Fokker-Planck equation in bounded domains | Kim |

October 12 | Minh-Binh Tran (Madison) | Nonlinear approximation theory for kinetic equations | |

October 19 | Bob Jensen (Loyola University Chicago) | Crandall-Lions Viscosity Solutions of Uniformly Elliptic PDEs | Tran |

October 26 | Luis Silvestre (Chicago) | A priori estimates for integral equations and the Boltzmann equation | Kim |

November 2 | Connor Mooney (UT Austin) | Counterexamples to Sobolev regularity for degenerate Monge-Ampere equations | Lin |

November 9 | Javier Gomez Serrano (Princeton) | Existence and regularity of rotating global solutions for active scalars | Zlatos |

November 16 | Yifeng Yu (UC Irvine) | TBA | Tran |

November 23 | Nam Le (Indiana) | TBA | Tran |

November 30 | Qin Li (Madison) | TBA | |

December 7 | Lu Wang (Madison) | TBA | |

December 14 | Christophe Lacave (Paris 7) | TBA | Zlatos |

# Abstract

### Hung Tran

Some inverse problems in periodic homogenization of Hamilton--Jacobi equations.

Abstract: We look at the effective Hamiltonian $\overline{H}$ associated with the Hamiltonian $H(p,x)=H(p)+V(x)$ in the periodic homogenization theory. Our central goal is to understand the relation between $V$ and $\overline{H}$. We formulate some inverse problems concerning this relation. Such type of inverse problems are in general very challenging. I will discuss some interesting cases in both convex and nonconvex settings. Joint work with Songting Luo and Yifeng Yu.

### Eric Baer

Optimal function spaces for continuity of the Hessian determinant as a distribution.

Abstract: In this talk we describe a new class of optimal continuity results for the action of the Hessian determinant on spaces of Besov type into the space of distributions on $\mathbb{R}^N$, obtained in collaboration with D. Jerison. Inspired by recent work of Brezis and Nguyen on the distributional Jacobian determinant, we show that the action is continuous on the Besov space $B(2-2/N,N)$ of fractional order, and that all continuity results in this scale of Besov spaces are consequences of this result. A key ingredient in the argument is the characterization of $B(2-2/N,N)$ as the space of traces of functions in the Sobolev space $W^{2,N}(\mathbb{R}^{N+2})$ on the subspace $\mathbb{R}^N$ (of codimension 2). The most elaborate part of the analysis is the construction of a counterexample to continuity in $B(2-2/N,p)$ with $p>N$. Tools involved in this step include the choice of suitable ``atoms" having a tensor product structure and Hessian determinant of uniform sign, formation of lacunary series of rescaled atoms, and delicate estimates of terms in the resulting multilinear expressions.

### Donghyun Lee

FLUIDS WITH FREE-SURFACE AND VANISHING VISCOSITY LIMIT.

Abstract : Free-boundary problems of incompressible fluids have been studied for several decades. In the viscous case, it is basically solved by Stokes regularity. However, the inviscid case problem is generally much harder, because the problem is purely hyperbolic. In this talk, we approach the problem via vanishing viscosity limit, which is a central problem of fluid mechanics. To correct boundary layer behavior, conormal Sobolev space will be introduced. In the spirit of the recent work by N.Masmoudi and F.Rousset (2012, non-surface tension), we will see how to get local regularity of incompressible free-boundary Euler, taking surface tension into account. This is joint work with Tarek Elgindi. If possible, we also talk about applying the similar technique to the free-boundary MHD(Magnetohydrodynamics). Especially, we will see that strong zero initial boundary condition is still valid for this coupled PDE. For the general boundary condition (for perfect conductor), however, the problem is still open.

### Hyung-Ju Hwang

The Fokker-Planck equation in bounded domains

abstract: In this talk, we consider the initial-boundary value problem for the Fokker-Planck equation in an interval or in a bounded domain with absorbing boundary conditions. We discuss a theory of well-posedness of classical solutions for the problem as well as the exponential decay in time, hypoellipticity away from the singular set, and the Holder continuity of the solutions up to the singular set. This is a joint work with J. Jang, J. Jung, and J. Velazquez.

### Minh-Binh Tran

Nonlinear approximation theory for kinetic equations

Abstract: Numerical resolution methods for the Boltzmann equation plays a very important role in the practical a theoretical study of the theory of rarefied gas. The main difficulty in the approximation of the Boltzmann equation is due to the multidimensional structure of the Boltzmann collision operator. The major problem with deterministic numerical methods using to solve Boltzmann equation is that we have to truncate the domain or to impose nonphysical conditions to keep the supports of the solutions in the velocity space uniformly compact. I n this talk, we will introduce our new way to make the connection between nonlinear approximation theory and kinetic theory. Our nonlinear wavelet approximation is nontruncated and based on an adaptive spectral method associated with a new wavelet filtering technique. The approximation is proved to converge and preserve many properties of the homogeneous Boltzmann equation. The nonlinear approximation solves the equation without having to impose non-physics conditions on the equation.

### Bob Jensen

Crandall-Lions Viscosity Solutions of Uniformly Elliptic PDEs

Abstract: I will discuss C-L viscosity solutions of uniformly elliptic partial differential equations for operators with only measurable spatial regularity. E.g., $L[u] = \sum a_{i\,j}(x)\,D_{i\,j}u(x)$ where $a_{i\,j}(x)$ is bounded, uniformly elliptic, and measurable in $x$. In general there isn't a meaningful extension of the C-L viscosity solution definition to operators with measurable spatial dependence. But under uniform ellipticity there is a natural extension. Though there isn't a general comparison principle in this context, we will see that the extended definition is robust and uniquely characterizes the ``right" solutions for such problems.

### Luis Silvestre

A priori estimates for integral equations and the Boltzmann equation.

Abstract: We will review some results on the regularity of general parabolic integro-differential equations. We will see how these results can be applied in order to obtain a priori estimates for the Boltzmann equation (without cutoff) modelling the evolution of particle density in a dilute gas. We derive a bound in L^infinity for the full Boltzmann equation, and Holder continuity estimates in the space homogeneous case.

### Connor Mooney

Counterexamples to Sobolev regularity for degenerate Monge-Ampere equations

Abstract: W^{2,1} estimates for the Monge-Ampere equation \det D^2u = f in R^n were first obtained by De Philippis and Figalli in the case that f is bounded between positive constants. Motivated by applications to the semigeostrophic equation, we consider the case that f is bounded but allowed to be zero on some set. In this case there are simple counterexamples to W^{2,1} regularity in dimension n \geq 3 that have a Lipschitz singularity. In contrast, if n = 2 a classical theorem of Alexandrov on the propagation of Lipschitz singularities shows that solutions are C^1. We will discuss a counterexample to W^{2,1} regularity in two dimensions whose second derivatives have nontrivial Cantor part, and also a related result on the propagation of Lipschitz / log(Lipschitz) singularities that is optimal by example.

### Javier Gomez Serrano

Existence and regularity of rotating global solutions for active scalars

A particular kind of weak solutions for a 2D active scalar equation are the so called patches, i.e., solutions for which the scalar is a step function taking one value inside a moving region and another in the complement. The evolution of such distribution is completely determined by the evolution of the boundary, allowing the problem to be treated as a non-local one dimensional equation for the contour. In this talk we will discuss the existence and regularity of uniformly rotating solutions for the vortex patch and generalized surface quasi-geostrophic (gSQG) patch equation. We will also outline the proof for the smooth (non patch) SQG case. Joint work with Angel Castro and Diego Cordoba.