Department Seminars & Colloquia
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Day 3. Passage from Boltzmann to Navier-Stokes
Despite its conceptual and practical importance, the rigorous derivation of the steady incompressible Navier-Stokes-Fourier system from the Boltzmann theory has been an outstanding open problem for general domains in 3D. We settle this open question in the affirmative, in the presence of a small external field and a small boundary temperature variation for the diffuse boundary condition. Avoiding the boundary layer correction to cope with general geometry, we employ a recent quantitative $L^2-L^infty$ approach with crucial $L^3$ estimates for the hydrodynamic part of the solution. Such gain of integrability is established via an extension lemma for the non-grazing component of the solution, which is critical to close our program in 3D. This talk is based on the joint work [4] with Esposito, Guo, and Marra.
We will begin with explaining the Poincare-Dulac normal form idea to prove the local well-posedness of nonlinear dispersive equations. Later, we will discuss with a particular example, quadratic derivative NLS. We develop an infinite iteration scheme of normal form reductions for dNLS. By combining this normal form procedure with the (modified) Cole-Hopf transformation, we prove unconditional global well-posedness in L^2(T), and more generally in certain Fourier-Lebesgue spaces FL^{s,p}(T), under the mean-zero and smallness assumptions. With this example, we observe a relation between normal form approach and canonical nonlinear transform.
Day 2. $L^2-L^infty$ bootstrap for the specular boundary condition
Due to the quadratic nonlinearity of the Boltzmann equation it is desired to have a $L^infty$ bound. Without the boundary the standard way to achieve it is some high order energy estimate. However it is known that solutions are not expected to have such high regularity [2,5,6]. In order to overcome this difficulty Guo developed $L^2-L^infty$ bootstrap argument in [7] for several boundary conditions. However for the specular BC his theorem is far to be complete since the analyticity and convexity of a domain is necessary in a crucial way. In the joint work with Donghyun Lee [1] we develop a new $L^2-L^infty$ bootstrap machinery for the specular BC in a $C^3$ convex domain with a small $C^{2,alpha}$ external potential. This consists with geometric decomposition of the derivatives of the trajectory and a triple iterations of the Duhamel principle. Note that without boundaries the work [3] provided $L^2-L^infty$ bootstrap with $C^3$ external potential. Our new method works for $C^{2,alpha}$ potential which would be crucial for many applications such as Vlasov-Poisson-Boltzmann system.
In this talk, we study the kinetic Fokker-Planck equation in general multi-dimensional bounded domains with inflow boundary condition. There has not been many results on the regularity of solutions when the spatial domain has a boundary. We will discuss the global well-posedness, interior and boundary regularity for the Fokker-Planck case, and compare it with some other kinetic equations
Day 1. Introductory and the coercivity estimate
We start with an introduction for the linearized Boltzmann equation around a Maxwellian. The linear operator $L$ is degenerated coercive with the 5-dimensional null space. In the application it is important to prove that $L$ is actually coercive operator if $f$ is a solution of the Boltzmann equation. We discuss the recent proof of such coercivity for the Boltzmann equation with the specular BC with external potential. This is based on a joint work [1] with Donghyun Lee.
In this talk, we discuss the mean field quantum fluctuation dynamics for a system of infinitely many fermions with delta pair interactions in the vicinity of an equilibrium solution (the Fermi sea) at zero temperature in two and three dimensions. Our work extends some recent important results of M. Lewin and J. Sabin, who address the corresponding problem for more regular pair interactions. This is a joint work with Thomas Chen and Natasa Pavlovic at University of Texas at Austin.