As a typical example of hyperbolic conservation laws, the system of Euler equations representing the conservation of mass, momentum and energy has three basic wave patterns. They are two nonlinear waves called shock and rarefaction wave, and one linearly degenerate wave called contact discontinuity. These basic wave patterns have their counterparts both in other physical models for gas motions in equilibrium involving viscosity and heat conductivity, and gas motion in non-equilibrium. The stability of these basic wave patterns in the system of Navier-Stokes equations and the Boltzmann equation has been an active research topic. Even though the stability of the two nonlinear wave patterns has been extensively studied, the stability of the linearly degenerate contact wave was not solved until recently. In this paper, we will briefly present our recent results in [32] on the stability of the contact wave patterns for both the Navier-Stokes equations and the Boltzmann equation.

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In this paper we are concerned with the existence of a weak solution to the initial boundary value problem for the equation ∂ t u=Δ(Δu) −3 . This problem arises in the mathematical modeling of the evolution of a crystal surface. Existence of a weak solution u with Δu≥0 is obtained via a suitable substitution. Our investigations reveal the close connection between this problem and the equation ∂ t ρ+ρ 2 Δ 2 ρ 3 =0 , another crystal surface model first proposed by H. Al Hajj Shehadeh, R. V. Kohn and J. Weare in [1].

We investigate the nonlinear stability of the superposition of a viscous contact wave and two rarefaction waves for a one-dimensional (1D) bipolar Vlasov--Poisson--Boltzmann (VPB) system, which can be used to describe the transportation of charged particles under the additional electrostatic potential force. Based on a new micro-macro type of decomposition around the local Maxwellian related to the bipolar VPB system in the previous work [H.-L. Li, Y. Wang, T. Yang, and M.-Y. Zhong, <i>Arch. Ration. Mech. Anal.</i>, 228 (2018), pp. 39--127], we prove that the superposition of a viscous contact wave and two rarefaction waves is time-asymptotically stable to the 1D bipolar VPB system under some smallness conditions on the initial perturbations and wave strength, which implies that this typical superposition wave pattern is nonlinearly stable under the combined effects of the binary collisions, the electrostatic potential

This paper investigates the existence of a uniform in time $L^{\infty}$ bounded weak entropy solution for the quasilinear parabolic-parabolic Keller-Segel model with the supercritical diffusion exponent $0 < m < 2-\frac{2}{d}$ in the multi-dimensional space ${\mathbb{R}}^d$ under the condition that the $L^{\frac{d(2-m)}{2}}$ norm of initial data is smaller than a universal constant. Moreover, the weak entropy solution $u(x,t)$ satisfies mass conservation when $m > 1-\frac{2}{d}$. We also prove the local existence of weak entropy solutions and a blow-up criterion for general $L^1\cap L^{\infty}$ initial data.