DCDS
Global weak solutions to the Camassa-Holm equation
Zhenhua Guo Mina Jiang Zhian Wang Gao-Feng Zheng
The existence of a global weak solution to the Cauchy problem for a one-dimensional Camassa-Holm equation is established. In this paper, we assume that the initial condition $u_0(x)$ has end states $u_{\pm}$, which has much weaker constraints than that $u_0(x) \in H^1(\mathbb R)$ discussed in [30]. By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution as a limit of viscous approximation under the assumption $u_- < u_+$.
keywords: vanishing viscosity method Young measure. Camassa-Holm equation global weak solution
DCDS-B
On free boundary problem for compressible navier-stokes equations with temperature-dependent heat conductivity
Zilai Li Zhenhua Guo

We obtain the existence of global strong solution to the free boundary problem in 1D compressible Navier-Stokes system for the viscous and heat conducting ideal polytropic gas flow, when the heat conductivity depends on temperature in power law of Chapman-Enskog and the viscosity coefficient be a positive constant.

keywords: Compressible Navier-Stokes equations temperature-dependent heat conductivity free boundary global strong solution
CPAA
Zero dissipation limit to rarefaction wave with vacuum for a one-dimensional compressible non-Newtonian fluid
Li Fang Zhenhua Guo

In this paper, we study the zero dissipation limit toward rarefaction waves for solutions to a one-dimensional compressible non-Newtonian fluid for general initial data, whose far fields are connected by a rarefaction wave to the corresponding Euler equations with one end state being vacuum. Given a rarefaction wave with one-side vacuum state to the compressible Euler equations, we construct a sequence of solutions to the one-dimensional compressible non-Newtonian fluid which converge to the above rarefaction wave with vacuum as the viscosity coefficient $\epsilon$ tends to zero. Moreover, the uniform convergence rate is obtained, based on one fact that the viscosity constant can control the degeneracies caused by the vacuum in rarefaction waves and another fact that the energy estimates are obtained under some a priori assumption.

keywords: Compressible non-Newtonian fluid zero dissipation limit rarefaction wave vacuum
CPAA
Global weak solution to 3D compressible flows with density-dependent viscosity and free boundary
Mei Wang Zilai Li Zhenhua Guo

In this paper, we obtain the global weak solution to the 3D spherically symmetric compressible isentropic Navier-Stokes equations with arbitrarily large, vacuum data and free boundary when the shear viscosity $\mu$ is a positive constant and the bulk viscosity $\lambda(\rho)=\rho^\beta$ with $\beta>0$. The analysis of the upper and lower bound of the density is based on some well-chosen functionals. In addition, the free boundary can be shown to expand outward at an algebraic rate in time.

keywords: Navier-Stokes equations spherically symmetric density-dependent viscosity global weak solution free boundary
KRM
Global existence of weak solution to the free boundary problem for compressible Navier-Stokes
Zhenhua Guo Zilai Li
In this paper, the compressible Navier-Stokes system (CNS) with constant viscosity coefficients is considered in three space dimensions. we prove the global existence of spherically symmetric weak solutions to the free boundary problem for the CNS with vacuum and free boundary separating fluids and vacuum. In addition, the free boundary is shown to expand outward at an algebraic rate in time.
keywords: global existence free boundary. Navier-Stokes equations

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