New Progress Made in the Field of Partial Differential Equations

Editor's Note

Professor Liu Shuangqian from the School of Mathematics and Statistics of CCNU, Professor Yang Tong from the City University of Hong Kong, and Professor Duan Renjun from the Chinese University of Hong Kong recently had published a paper entitled "the Boltzmann equation for plane Couette flow".  Professor Liu Shuangqian served as the paper's corresponding author, and the esteemed Journal of the European Mathematical Society accepted it for publication. This is the first time our school's work has been published in this journal.

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A key issue in the study of the kinetic equation boundary problem is the steady state of a rarefied gas between two plates. Based on their earlier work on the stability of the Boltzmann equation for uniform shear flow, Professor Liu and his colleagues study the plane Couette flow of a rarefied gas between two parallel infinite plates in the paper. Assuming that the stationary state takes the specific form of F(y,vx−αy,vy,vz) with the x-component of the molecular velocity sheared linearly along the y-direction, such steady flow is governed by a boundary value problem on a steady nonlinear Boltzmann equation driven by an external shear force under the homogeneous non-moving diffuse reflection boundary condition. In case of the Maxwell molecule collisions, they establish the existence of spatially inhomogeneous non-equilibrium stationary solutions to the steady problem for any small enough shear rate α>0 via an elaborate perturbation approach based on Caflisch's decomposition together with L^2-L^\infty theory. The result indicates the polynomial tail at large velocities for the stationary distribution. Moreover, the large time asymptotic stability of the stationary solution with an exponential convergence is also obtained and as a consequence, the nonnegativity of the steady profile is justified.

This work provides a truly rigorous mathematical proof for the stability of the Couette flow of the Boltzmann equation on a finite pipe with a  physical boundary. The results of this work are an important step forward in demonstrating the connection between the shear flow of the Navier-Stokes system of equations and the Boltzmann equation's shear flow, which will positively affect the study of the stability of the shear flow of the Boltzmann equation.

The fine perturbation methods developed in this work, based on Caflisch’s decomposition and L^2-L^\infty theory, are expected to be applied to prove some effects like Landau damping and enhanced dissipation of the Boltzmann equation and provide new ideas to further prove Hilbert's sixth problem from the perspective of shear flow.

This research is supported by the National Natural Science Foundation of China and the Talent Program of Central China Normal University, etc.

DOI:10.48550/arXiv.2107.02458

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Editor:Cici(trainee)

Reviser:Steve(trainee)

Designer:Ashes(trainee)