Global smooth solutions to the nonisothermal compressible fluid models of Korteweg type with large initial data

Large-time behavior of smooth solutions to the isothermal compressible fluid models of Korteweg type with large initial data

Global classical solutions to the one-dimensional compressible fluid models of Korteweg type with large initial data

This paper is concerned with the large-time behavior of solutions to the Cauchy problem of the one-dimensional compressible fluid models of Korteweg type with density- and temperature-dependent viscosity, capillarity, and heat conductivity coefficients, which models the motions of compressible viscous fluids with internal capillarity. We show that the combination of a viscous contact wave with two rarefaction waves is asymptotically stable with a large initial perturbation if the strength of the composite wave and the capillarity coefficient satisfy some smallness conditions. The proof is based on some refined L2-energy estimates to control the possible growth of the solutions caused by the high nonlinearity of the system, the interactions of waves from different families and large data, and the key ingredient is to derive the uniform positive lower and upper bounds on the specific volume and the temperature.

Asymptotic stability of strong rarefaction waves for the compressible fluid models of Korteweg type

We are interested in studying the Cauchy problem for the viscous shallow-water system in dimension N≥2, we show the existence of global strong solutions with large initial data on the irrotational part of the velocity for the scaling of the equations. More precisely our smallness assumption on the initial data is supercritical for the scaling of the equations. It allows us to give a first kind of answer to the problem of the existence of global strong solution with large initial energy data in dimension N=2. To do this, we introduce the notion of quasi-solutions which consists in solving the pressureless viscous shallow water system. We can obtain such solutions at least for irrotational data which are subject to regularizing effects both on the velocity and on the density. This smoothing effect is purely nonlinear and is crucial in order to build solution of the viscous shallow water system as perturbations of the “quasi-solutions”. Indeed the pressure term can be considered as a remainder term which becomes small in high frequencies for the scaling of the equations. To finish we prove the existence of global strong solution with large initial data when N≥2 provided that the Mach number is sufficiently large.

Existence of solutions for compressible fluid models of Korteweg type

In this paper we consider an initial boundary value problem for planar magnetohydrodynamic compressible flows. By assuming that the adiabatic constant γ \gamma is sufficiently close to 1 1 , we prove the existence and uniqueness of global strong solutions with large initial data when all the viscosity, heat conductivity, and diffusivity coefficients are constant. Moreover, the asymptotic behavior of solutions is also investigated.

In this paper, we consider the three-dimensional isentropic compressible fluid models of Korteweg type, called the compressible Navier--Stokes--Korteweg system. We mainly present the vanishing capillarity limit of the smooth solution to the initial value problem. Precisely, we first establish the uniform estimates of the global smooth solution with respect to the capillary coefficient $\kappa$. Then by the Lions--Aubin lemma, we show that the unique smooth solution of the three-dimensional Navier--Stokes--Korteweg system converges globally in time to the smooth solution of the three-dimensional Navier--Stokes system as $\kappa$ tends to zero. Also, we give the convergence rate estimates for any given positive time.

We show the blow-up of smooth solutions to a non-isothermal model of capillary compressible fluids in arbitrary space dimensions with initial density of compact support. This is an extension of Xin’s result [Xin, Z.: Blow-up of smooth solutions to the compressible Navier-Stokes equations with compact density. Comm. Pure Appl. Math., 51, 229–240 (1998)] to the capillary case but we do not need the condition that the entropy is bounded below. Moreover, from the proof of Theorem 1.2, we also obtain the exact relationship between the size of support of the initial density and the life span of the solutions. We also present a sufficient condition on the blow-up of smooth solutions to the compressible fluid models of Korteweg type when the initial density is positive but has a decay at infinity.

This paper considers the initial boundary problem to the planar compressible magnetohydrodynamic equations with large initial data and vacuum. The global existence and uniqueness of large strong solutions are established when the heat conductivity coefficient $\kappa(\theta)$ satisfies \begin{equation*} C_{1}(1+\theta^q)\leq \kappa(\theta)\leq C_2(1+\theta^q) \end{equation*} for some constants $q>0$, and $C_1,C_2>0$.

We prove the solution of the full compressible fluid models of Korteweg type with centered rarefaction wave data of large strength exists globally in time. As the viscosity, heat-conductivity and capillary coefficients tend to zero, the global solution converges to the centered rarefaction wave solution of the corresponding Euler equations uniformly when the initial perturbation is small. Our analysis is based on the energy method.

Global existence and optimal decay rate of the compressible Navier–Stokes–Korteweg equations with external force

Global Solutions of Nonlinear Magnetohydrodynamics with Large Initial Data

Nonlinear stability of traveling wave solutions for the compressible fluid models of Korteweg type

Existence of global strong solution for Korteweg system with large infinite energy initial data