Nonlinear stability of shock profiles to Burgers' equation with critical fast diffusion and singularity
arxiv(2024)
摘要
In this paper we propose the first framework to study Burgers' equation
featuring critical fast diffusion in form of u_t+f(u)_x = (ln u)_xx. The
solution possesses a strong singularity when u=0 hence bringing technical
challenges. The main purpose of this paper is to investigate the asymptotic
stability of viscous shocks, particularly those with shock profiles vanishing
at the far field x=+∞. To overcome the singularity, we introduce some
weight functions and show the nonlinear stability of shock profiles through the
weighted energy method. Numerical simulations are also carried out in different
cases of fast diffusion with singularity, which illustrate and confirm our
theoretical results.
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