On the inelastic Boltzmann equation for soft potentials with diffusion

Fei Meng; Fang Liu

doi:10.3934/cpaa.2020233

Article Contents

2020, Volume 19, Issue 11: 5197-5217. Doi: 10.3934/cpaa.2020233

This issue Previous Article A non-autonomous bifurcation problem for a non-local scalar one-dimensional parabolic equation Next Article On the Cauchy problem of 3D nonhomogeneous incompressible nematic liquid crystal flows with vacuum

On the inelastic Boltzmann equation for soft potentials with diffusion

Fei Meng^1, and
Fang Liu^2, ,

1.
School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, China
2.
Department of Mathematics, School of Science, Nanjing University of Science and Technology, Nanjing 210094, China

^*Corresponding author
^*Corresponding author

Received: December 31, 2019

Revised: April 30, 2020

Early access: September 2020

Published: November 2020

The work is supported by NSF grant of China (No.11501292) and NUPTSF (Grant No. NY218091)

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Abstract

We are concerned with the Cauchy problem of the inelastic Boltzmann equation for soft potentials, with a Laplace term representing the random background forcing. The inelastic interaction here is characterized by the non-constant restitution coefficient. We prove that under the assumption that the initial datum has bounded mass, energy and entropy, there exists a weak solution to this equation. The smoothing effect of weak solutions is also studied. In addition, it is shown the solution is unique and stable with respect to the initial datum provided that the initial datum belongs to $ L^{2}(R^{3}) $.

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Mathematics Subject Classification: Primary: 76P05, 76T25; Secondary: 74E20.

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