The areas of my research are in the field of applied pde. I have been studying two different types of problems : boundary problems of the Boltzmann equation and the stability/instability of free boundary problem in fluid dynamics. Now I start to study large amplitude solutions of the Boltzmann equation with Clement Mouhot.
Find my preprints in ARXIV
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In the study of the heat transfer in the Boltzmann theory, the basic problem is to construct solutions to the steady problem for the Boltzmann equation in a general bounded domain with diffuse reflection boundary conditions corresponding to a non isothermal temperature of the wall. Denoted by \delta the size of the temperature oscillations on the boundary, we develop a theory to characterize such a solution mathematically. We construct a unique solution F_s to the Boltzmann equation, which is dynamically asymptotically stable with exponential decay rate. Moreover, if the domain is convex and the temperature of the wall is continuous we show that F_s is continuous away from the grazing set. If the domain is non-convex, discontinuities can form and then propagate along the forward characteristics. We show that they actually form for a suitable smooth temperature profile. We remark that this solution differs from a local equilibrium Maxwellian, hence it is a genuine non equilibrium stationary solution. Our analysis is based on recent studies of the boundary value problems for the Boltzmann equation but with new constructive coercivity estimates for both steady and dynamic cases. A natural question in this setup is to determine if the general Fourier law, stating that the heat conduction vector q is proportional to the temperature gradient, is valid. As an application of our result we establish an expansion in \delta for F_s whose first order term F_1 satisfies a linear, parameter free equation. Consequently, we discover that if the Fourier law were valid for F_s, then the temperature of F_1 must be linear in a slab. Such a necessary condition contradicts available numerical simulations, leading to the prediction of break-down of the Fourier law in the kinetic regime.
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In rotationally symmetric domains, the Boltzmann equation with specular reflection boundary condition has a special type of equilibrium states called the rotational local Maxwellian which, unlike the uniform Maxwellian, has an additional term related to the angular momentum of the gas. In this paper, we consider the initial boundary value problem of the Boltzmann equation near the rotational local Maxwellian. Based on the L2-L1 framework of [12], we establish the global well-posedness and the convergence toward such equilibrium states.
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One week workshop in “Kinetic Theory and Computation”
Brown University , ICERM
October 30 / 2011 ~ November 12 / 2011
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http://arxiv.org/abs/1109.1798
Yanjin Wang, Ian Tice, Chanwoo Kim
Abstract: We consider the free boundary problem for two layers of immiscible, viscous, incompressible fluid in a uniform gravitational field, lying above a general rigid bottom in a three-dimensional horizontally periodic setting. We establish the global well-posedness of the problem both with and without surface tension. We prove that without surface tension the solution decays to the equilibrium state at an almost exponential rate; with surface tension, we show that the solution decays at an exponential rate. Our results include the case in which a heavier fluid lies above a lighter one, provided that the surface tension at the free internal interface is above a critical value, which we identify. This means that sufficiently large surface tension stabilizes the Rayleigh-Taylor instability in the nonlinear setting. As a part of our analysis, we establish elliptic estimates for the two-phase stationary Stokes problem.
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Postech, Korea
June 22~24
http://math.postech.ac.kr/~hjhwang/kinetic/
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The 2011 Annual Kinetic FRG Meeting
May 23~ 27 , 2011
Wisconsin – Madison
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Brown, DAM
May 12 ~ 14, 2011
http://www.dam.brown.edu/HyperbolicConf/
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