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Series: PDE Seminar

We will introduce a recently found channel of energy inequality for outgoing waves, which has been useful for semi-linear wave equations at energy criticality. Then we will explain an application of this channel of energy argument to the energy critical wave maps into the sphere. The main issue is to eliminate the so-called "null concentration of energy". We will explain why this is an important issue in the wave maps. Combining the absence of null concentration with suitable coercive property of energy near traveling waves, we show a universality property for the blow up of wave maps with energy that are just above the co-rotational wave maps. Difficulties with extending to arbitrarily large wave maps will also be discussed. This is joint work with Duyckaerts, Kenig and Merle.

Series: PDE Seminar

For a map u from a Riemann surface M to a Riemannian manifold and a>1, the a-energy functional is defined as E_a(u)=\int_M |\nabla u|^{2a}dx. We call u_a a sequence of Sacks-Uhlenbeck maps if u_a is a critical point of E_a, and sup_{a>1} E_a(u_a)<\infty. In this talk, when the target manifold is a standard sphere S^K, we prove the energy identity for a sequence of Sacks-Uhlenbeck maps during blowing up.

Series: PDE Seminar

We discuss the derivation and analysis of a family of 4th order nonlinear PDEs that arise in the study of crystal evolution. This is joint work with Jon Weare, Jianfeng Lu, Dio Margetis, Jian-Guo Liu and Anya Katsevich.

Series: PDE Seminar

I will discuss applications of the theory of gradient ﬂows to the dynamics of evolution equations. First, I will review how to obtain convergence rates towards equilibrium in the strictly convex case. Second, I will introduce a technique developed in collaboration with Moon-Jin Kang that allows one to obtain convergence rates towards equilibrium in some situations where convexity is not available. Finally, I will describe how these techniques were useful in the study of the dynamics of homogeneous Vicsek model and the Kuramoto-Sakaguchi equation. The contributions on the Kuramoto-Sakaguchi equation are based on a joint work with Seung-Yeal Ha, Young-Heon Kim, and Jinyeong Park. The contributions to the Vicsek model are based on works in collaboration with Alessio Figalli and Moon-Jin Kang.

Series: PDE Seminar

We consider stationary monotone mean-field games (MFGs) and study the existence of weak solutions. We introduce a regularized problem that preserves the monotonicity and prove the existence of solutions to the regularized problem. Next, using Minty's method, we establish the existence of solutions for the original MFGs. Finally, we examine the properties of these weak solutions in several examples.

Series: PDE Seminar

Linear wave solutions to the Charney-Hasegawa-Mima equation with periodic boundary conditions have two physical interpretations: Rossby (atmospheric) waves, and drift (plasma) waves in a tokamak. These waves display resonance in triads. In the case of infinite Rossby deformation radius, the set of resonant triads may be described as the set of integer solutions to a particular homogeneous Diophantine equation, or as the set of rational points on a projective surface. We give a rationalparametrization of the smooth points on this surface, answering the question: What are all resonant triads, and how may they be enumerated quickly? We also give a fiberwise description, yielding an algorithmic procedure to answer the question: For fixed $r \in \Q$, what are all wavevectors $(x,y)$ that resonate with a wavevector $(a,b)$ with $a/b = r$?

Series: PDE Seminar

Abstract: Abstract: Let p(x,xi) be a complex-valued polynomial of degree 2 on R^{2n}, and let P be the corresponding non-self-adjoint Weyl quantization. We will discuss some results on the relationship between the classical Hamilton flow exp(H_p) and the L^2 operator theory for the Schrödinger evolution exp(-iP), under a positivity condition of Melin and Sjöstrand.

Series: PDE Seminar

We study the quasilinear wave equation $\Box_{g} u = 0$, where the metric $g$ depends on $u$ and equals the Schwarzschild metric when u is identically 0. Under a couple of assumptions on the metric $g$ near the trapped set and the light cone, we prove global existence of solutions. This is joint work with Hans Lindblad.

Series: PDE Seminar

This talk is about the Dirac equation. We consider an electron modeled by awave function and evolving in the Coulomb field generated by a nucleus. Ina very rough way, this should be an equation of the form$$i\partial_t u = -\Delta u + V( \cdot - q(t)) u$$where $u$ represents the electron while $q(t)$ is the position of thenucleus. When one considers relativitic corrections on the dynamics of anelectron, one should replace the Laplacian in the equation by the Diracoperator. Because of limiting processes in the chemistry model from whichthis is derived, there is also a cubic term in $u$ as a correction in theequation. What is more, the position of the nucleus is also influenced bythe dynamics of the electron. Therefore, this equation should be coupledwith an equation on $q$ depending on $u$.I will present this model and give the first properties of the equation.Then, I will explain why it is well-posed on $H^2$ with a time of existencedepending only on the $H^1$ norm of the initial datum for $u$ and on theinitial datum for $q$. The linear analysis, namely the properties of thepropagator of the equation $i\partial_t u = D u + V( \cdot - q(t))$ where$D$ is the Dirac operator is based on works by Kato, while the non linearanalysis is based on a work by Cancès and Lebris.It is possible to have more than one nucleus. I will explain why.(Joint work with F. Cacciafesta, D. Noja and E. Séré)

Series: PDE Seminar

A major open problem in periodic homogenization of Hamilton-Jacobi equations is to understand deep properties of the effective Hamiltonian. In this talk, I will present some related works in both convex and non-convex situations. If time permits, relevant problems from applications in turbulent combustion and traffic flow will also be discussed.