Global existence of strong solutions of Navier–Stokes equations with non-Newtonian potential for one-dimensional isentropic compressible fluids

We consider strong solutions to the initial boundary value problems for the isentropic compressible Navier–Stokes equations in one dimension: $$\rho\left\{\begin{array}{lll} t+(\rho u)_x=0\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\, {\rm in}\,(0,T)\times(0,1)\\ (\rho u )_t+(\rho u^2)_x+\rho \Phi_x-(\mu( \rho )u_x)_x+P_x=0\quad\quad {\rm in}\,(0,T)\times(0,1) \\\left(\left(\frac{\delta(\Phi_x)^2\,+\,1}{(\Phi_x)^2\,+\,\delta}\right)^{\frac{2-p}{2}}\Phi_x\right)_x=4\pi g(\rho-\frac{1}{|\Omega|}\int\nolimits_\Omega \rho dx\,\,\,\, )\quad\,\, {\rm in}\,(0,T)\times(0,1)\end{array}\right.$$ Here, the Φ is a non-Newtonian potential and strong solutions of the problem and obtains the uniqueness under the compatibility condition.     

On the uniqueness of weak solutions for the 3D Navier–Stokes equations

In this paper, we improve some known uniqueness results of weak solutions for the 3D Navier–Stokes equations. The proof uses the Fourier localization technique and the losing derivative estimates.     

A criterion for the existence of strong solutions for the 3D Navier–Stokes equations

In this note we provide a criterion for the existence of globally defined solutions for any regular initial data for the 3D Navier–Stokes system in Serrin’s classes.     

Global solutions of the Navier–Stokes equations for isentropic flow with large external potential force

We prove the global-in-time existence of weak solutions to the Navier–Stokes equations of compressible isentropic flow in three space dimensions with adiabatic exponent γ ≥ 1. Initial data and solutions are small in L ² around a non-constant steady state with densities being positive and essentially bounded. No smallness assumption is imposed on the external forces when γ = 1. A great deal of information about partial regularity and large-time behavior is obtained.     

Asymptotic stability of contact waves for the compressible Navier–Stokes equations

In this article, we study the large-time asymptotic behaviour of contact wave for the Cauchy problem of one-dimensional compressible Navier–Stokes equations with zero viscosity. When the Riemann problem for the Euler system admits a contact discontinuity solution, we can construct a contact wave, which approximates the contact discontinuity on any finite-time interval for small heat conduction and then runs away from it for large time, and prove that it is nonlinearly stable provided that the strength of contact discontinuity and the perturbation of the initial data are suitably small.     

Large time behavior of the isentropic compressible Navier–Stokes–Maxwell system

In this paper, we study the large time behavior of the isentropic compressible Navier–Stokes–Maxwell system introduced by Jiang and Li (Nonlinearity 25(6):1735–1752, 2012) in the whole space \({{\mathbb{R}}^3}\) when the initial data are a small perturbation of some given constant state. We obtain the desired result through taking the refined analysis on the time decay property and Green’s function of the linearized system. Moreover, we also obtain the optimal time rate of the solution.     

Asymptotic behavior of solutions for the 1-D isentropic Navier–Stokes–Korteweg equations with free boundary

In this paper, we consider the problem with a gas–gas free boundary for the one dimensional isentropic compressible Navier–Stokes–Korteweg system. For shock wave, asymptotic profile of the problem is shown to be a shifted viscous shock profile, which is suitably away from the boundary, and prove that if the initial data around the shifted viscous shock profile and its strength are sufficiently small, then the problem has a unique global strong solution, which tends to the shifted viscous shock profile as time goes to infinity. Also, we show the asymptotic stability toward rarefaction wave without the smallness on the strength if the initial data around the rarefaction wave are sufficiently small.     

Ergodicity for the 3D stochastic Navier–Stokes equations

We consider the Kolmogorov equation associated with the stochastic Navier–Stokes equations in 3D, we prove existence of a solution in the strict or mild sense. The method consists in finding several estimates for the solutions u_m of the Galerkin approximations of u and their derivatives. These estimates are obtained with the help of an auxiliary Kolmogorov equation with a very irregular negative potential. Although uniqueness is not proved, we are able to construct a transition semigroup for the 3D Navier–Stokes equations. Furthermore, this transition semigroup has a unique invariant measure, which is ergodic and strongly mixing.     

A convergent FEM-DG method for the compressible Navier–Stokes equations

This paper presents a new numerical method for the compressible Navier–Stokes equations governing the flow of an ideal isentropic gas. To approximate the continuity equation, the method utilizes a discontinuous Galerkin discretization on piecewise constants and a basic upwind flux. For the momentum equation, the method is a new combined discontinuous Galerkin and finite element method approximating the velocity in the Crouzeix–Raviart finite element space. While the diffusion operator is discretized in a standard fashion, the convection and time-derivative are discretized using discontinuous Galerkin on the element average velocity and a Lax–Friedrich type flux. Our main result is convergence of the method to a global weak solution as discretization parameters go to zero. The convergence analysis constitutes a numerical version of the existence analysis of Lions and Feireisl.     

Regularity of weak solutions to 3D incompressible Navier–Stokes equations

In this paper, we establish some new local and global regularity properties for weak solutions of 3D non-stationary Navier–Stokes equations in the class of L ^r (0, T ; L ³(Ω)) with ${r \in [1, \infty)}In this paper, we establish some new local and global regularity properties for weak solutions of 3D non-stationary Navier–Stokes equations in the class of L ^r (0, T ; L ³(Ω)) with r ? [1, ￥){r \in [1, \infty)} , which are beyond Serrin’s condition.     

Existence of global weak solutions for a 3D Navier–Stokes–Poisson–Korteweg equations

The purpose of this work is to study the global-in-time existence of weak solutions of a viscous capillary model of plasma expressed as a so-called Navier–Stokes–Poisson–Korteweg model for large data in three-dimensional space. Using the compactness argument, we prove the existence of global weak solutions in the classical sense to such system with a cold pressure.     

Self-similar solutions of stationary Navier–Stokes equations

In this paper, we mainly study the existence of self-similar solutions of stationary Navier–Stokes equations for dimension $n=3,4$. For $n=3$, if the external force is axisymmetric, scaling invariant, $C^{1,\alpha }$ continuous away from the origin and small enough on the sphere $S^2$, we shall prove that there exists a family of axisymmetric self-similar solutions which can be arbitrarily large in the class $C_{loc}^{3,\alpha }({\mathbb{R}}^3\backslash 0)$. Moreover, for axisymmetric external forces without swirl, corresponding to this family, the momentum flux of the flow along the symmetry axis can take any real number. However, there are no regular ($U\in C_{loc}^{3,\alpha }({\mathbb{R}}^3\backslash 0)$) axisymmetric self-similar solutions provided that the external force is a large multiple of some scaling invariant axisymmetric F which cannot be driven by a potential. In the case of dimension 4, there always exists at least one self-similar solution to the stationary Navier–Stokes equations with any scaling invariant external force in $L^{4/3,\infty }({\mathbb{R}}^4)$.     

Existence of weak solutions for the generalized Navier–Stokes equations with damping

In this work we consider the generalized Navier–Stokes equations with the presence of a damping term in the momentum equation. The problem studied here derives from the set of equations which govern isothermal flows of incompressible and homogeneous non-Newtonian fluids. For the generalized Navier–Stokes problem with damping, we prove the existence of weak solutions by using regularization techniques, the theory of monotone operators and compactness arguments together with the local decomposition of the pressure and the Lipschitz-truncation method. The existence result proved here holds for any ${q > \frac{2N}{N+2}}$ and any σ > 1, where q is the exponent of the diffusion term and σ is the exponent which characterizes the damping term.     

A remark on weak solutions to the barotropic compressible quantum Navier–Stokes equations

In [A. Jüngel, Global weak solutions to compressible Navier–Stokes equations for quantum fluids, SIAM J. Math. Anal. 42 (2010) 1025–1045], Jüngel proved the global existence of the barotropic compressible quantum Navier–Stokes equations for when the viscosity constant is bigger than the scaled Planck constant. Recently, Dong [J. Dong, A note on barotropic compressible quantum Navier–Stokes equations, Nonlinear Anal. TMA 73 (2010) 854–856] extended Jüngel’s result to the case where the viscosity constant is equal to the scaled Planck constant by using a new estimate of the square root of the solutions. In this paper we show that Jüngel’s existence result still holds when the viscosity constant is bigger than the scaled Planck constant. Consequently, with our result, the existence for all physically interesting cases of the scaled Planck and viscosity constants is obtained.