相关论文: Large time behavior of strong solutions for stocha…

We investigate the large time behavior of solutions to the two-dimensional viscous Burgers equation $u_t+uu_x+uu_y=\Delta u$, toward a non-self-similar rarefaction wave of inviscid Burgers equation with two initial constant states,…

We prove the time-asymptotic stability of composite waves consisting of the superposition of a viscous shock and a rarefaction for the one-dimensional compressible barotropic Navier-Stokes equations. Our result solves a long-standing…

In this paper, we study the asymptotic stability of rarefaction waves for the compressible isentropic Navier-Stokes equations with density-dependent viscosity. First, a weak solution around a rarefaction wave to the Cauchy problem is…

In this paper, we study the asymptotic stability of viscous shock waves for Burgers' equation with fast diffusion $u_t+f(u)_x=\mu (u^m)_{xx}$ on $\mathbb{R} \times (0, +\infty)$ when $0<m<1$. For the proposed constant states $u_->u_+=0$,…

We are concerned with the large-time behavior of the solution to one-dimensional (1D) cubic non-convex scalar viscous conservation laws. Due to the inflection point of the cubic non-convex flux, the solution to the corresponding inviscid…

Dynamics of viscous shocks is considered in the modular Burgers equation, where the time evolution becomes complicated due to singularities produced by the modular nonlinearity. We prove that the viscous shocks are asymptotically stable…

We study a generalized 1d periodic SPDE of Burgers type: $$ \partial_t u =- A^\theta u + \partial_x u^2 + A^{\theta/2} \xi $$ where $\theta > 1/2$, $-A$ is the 1d Laplacian, $\xi$ is a space-time white noise and the initial condition $u_0$…

Time-asymptotic stability of generic Riemann solution, consisting of a rarefaction wave, a contact discontinuity and a shock, for the one-dimensional Boltzmann equation, has been a long-standing open problem in kinetic theory. In this…

This paper concerns with the large-time behaviors of the viscous shock profile and rarefaction wave under initial perturbations which tend to space-periodic functions at infinities for the one-dimensional compressible Navier-Stokes-Poisson…

We are concerned with the large-time behavior of the radially symmetric solution for multidimensional Burgers equation on the exterior of a ball $\mathbb{B}_{r_0}(0)\subset \mathbb{R}^n$ for $n\geq 3$ and some positive constant $r_0>0$,…

We prove that the stochastic Burgers equation on $\mathbf{R}^{d}$, $d<4$, forced by gradient noise that is white in time and smooth in space, admits spacetime-stationary solutions. These solutions are thus the gradients of solutions to the…

This work is devoted to investigating stochastic turbulence for the fluid flow in one-dimensional viscous Burgers equation perturbed by L\'evy space-time white noise with the periodic boundary condition. We rigorously discuss the regularity…

The KdV-Burgers equation is a canonical model describing the interplay between nonlinearity, viscosity and dispersion, and it admits viscous-dispersive shocks as traveling wave solutions. In this paper, we establish an $L^2$-contraction…

We define a notion of a viscous shock solution of the stochastic Burgers equation that connects "top" and "bottom" spatially stationary solutions of the same equation. Such shocks generally travel in space, but we show that they admit…

In the paper, the large time behavior of solutions of the Cauchy problem for the one dimensional fractal Burgers equation $u_t+(-\partial^2_x)^{\alpha/2} u+uu_x=0$ with $\alpha\in (1,2)$ is studied. It is shown that if the nondecreasing…

We consider reaction-diffusion equations that are stochastically forced by a small multiplicative noise term. We show that spectrally stable traveling wave solutions to the deterministic system retain their orbital stability if the…

In this paper, a rarefaction wave under space-periodic perturbation for the 3 times 3 rate-type viscoelastic system is considered. It is shown that if the initial perturbation around the rarefaction wave is suitably small, then the solution…

In this paper, we show the existence and uniqueness of the stationary solution $u(t,\omega)$ and stationary point $Y(\omega)$ of the differentiable random dynamical system $U:R\times L^2[0,1]\times \Omega\to L^2[0,1]$ generated by the…

In this paper we study the following Burgers equation du/dt + d/dx (u^2/2) = epsilon d^2u/dx^2 + f(x,t) where f(x,t)=dF/dx(x,t) is a random forcing function, which is periodic in x and white noise in t. We prove the existence and uniqueness…

The large-time behavior of solutions to Burgers equation with small viscosity is described using invariant manifolds. In particular, a geometric explanation is provided for a phenomenon known as metastability, which in the present context…