Compressible Navier Stokes Poisson System Research Articles

Mass concentration phenomenon in the 3D bipolar compressible Navier–Stokes–Poisson system

Mass concentration phenomenon in the 3D bipolar compressible Navier–Stokes–Poisson system

Stability of the planar rarefaction wave to three-dimensional full compressible Navier–Stokes–Poisson system

A full compressible Navier–Stokes–Poisson system models the motion of viscous ions under the effect of variable temperature and plays important roles in the study of self-gravitational viscous gaseous stars and in simulations of charged particles in semiconductor devices and plasmas physics. We establish the time-asymptotic nonlinear stability of a planar rarefaction wave to the initial value problem of a three-dimensional full compressible Navier–Stokes–Poisson equation when the initial data are a small perturbation of the planar rarefaction wave. The proof is given by a delicate energy method, which involves highly non-trivial a priori bounds due to the effect of the self-consistent electric field. This appears as the first result on the nonlinear stability of wave patterns to the full compressible Navier–Stokes–Poisson system in multi-dimensions.

On the global well-posedness for a multi-dimensional compressible Navier–Stokes–Poisson system

This article is dedicated to the study of the Cauchy problem for compressible Navier–Stokes–Poisson system in spatial dimensions two and higher. We prove the global well-posedness when the initial data are close to a stable equilibrium state in critical Lp framework.

Zero-electron-mass limit of the compressible Navier–Stokes–Poisson equations with well/ill-prepared initial data

In this study, we consider the viscous compressible Navier–Stokes–Poisson equations, which consist of the balance laws for electron density and moment, and a Poisson equation for the electrostatic potential. The limit of vanishing electron mass of this system with both well/ill-prepared initial data on the whole space is rigorously justified within the framework of local smooth solution. We first make use of the symmetric hyperbolic–parabolic structure of the compressible Navier–Stokes–Poisson equation to obtain uniform estimate in the short time, by which we show uniform existence of local classical solution to the compressible Navier–Stokes–Poisson equation in $\mathbb {R}^d(d\geq 1)$. Further, with uniform estimate of time derivatives, we show the zero-electron-mass limit of the solutions for the compressible Navier–Stokes–Poisson equation with well-prepared initial data in $\mathbb {R}^d(d\geq 1)$ by using Aubin's lemma. A detailed spectral analysis on the linearized system is done so that we are able to prove the zero-electron-mass limit of the solutions with ill-prepared initial data in $\mathbb {R}^d(d\geq 3)$, where the convergence occurs away from the time $t=0$. Finally, note that the dissipation mechanism for the linearized compressible Navier–Stokes–Poisson system is different from that of the compressible Euler equations in Grenier (Commun. Partial Diff. Eqns.21 (1996), 363–394); Grenier (Commun. Pure Appl. Math.50 (1997), 821–865); Ukai (J. Math. Kyoto Univ.26 (1986), 323–331), or that of the compressible Euler–Poisson equations in Ali and Chen (Nonlinearity24 (2011), 2745–2761), since its eigenvalues are somehow similar to that of heat equation, and the fundamental solution contains a part behaving like the heat kernel, thus a big difficulty is the singularity of the heat kernel at $t=0$.

Weak-strong uniqueness principle for compressible barotropic self-gravitating fluids

The aim of this work is to prove the weak–strong uniqueness principle for the compressible Navier–Stokes–Poisson system on an exterior domain, with an isentropic pressure of the type p(ϱ)=aϱγ and allowing the density to be close or equal to zero. In particular, the result will be first obtained for an adiabatic exponent γ∈[9/5,2] and afterwards, this range will be slightly enlarged via pressure estimates “up to the boundary”, deduced relaying on boundedness of a proper singular integral operator.

Stability of stationary solutions for the unipolar isentropic compressible Navier–Stokes–Poisson system

The stationary solutions to the outflow problem for unipolar isentropic Navier–Stokes–Poisson system in a half line (0, ∞) have recently been shown to be asymptotically stable in13 and26, provided that all the L2 norms of initial perturbations and their derivatives are small. The main purpose of this paper is to study the asymptotic stability of the stationary solutions under large initial perturbations. First, for the outflow problem, we show that the stationary solutions are asymptotically stable, provided that only the L2 norm of initial perturbation is small. Next, for the inflow problem, we show asymptotic stability under a large initial perturbation in the H1 norm. The main ingredient in the proofs is a careful analysis in order to control the growth induced by nonlinearities of the system in a priori estimates.

Stability of Non-constant Equilibrium Solutions for the Full Compressible Navier–Stokes–Maxwell System

In this article we consider a Cauchy problem for the full compressible Navier–Stokes–Maxwell system arising from viscosity plasmas. This system is quasilinear hyperbolic–parabolic. With the help of techniques of symmetrizers and the smallness of non-constant equilibrium solutions, we establish that global smooth solutions exist and converge to the equilibrium solution as the time approaches infinity. This result is obtained for initial data close to the steady-states. As a byproduct, we obtain the global stability of solutions near the equilibrium states for the full compressible Navier–Stokes–Poisson system in a three-dimensional torus.

Optimal time-decay rates of the compressible Navier–Stokes–Poisson system in <inline-formula><tex-math id="M1">$ \mathbb R^3 $</tex-math></inline-formula>

<p style='text-indent:20px;'>We are concerned with the Cauchy problem of the 3D compressible Navier–Stokes–Poisson system. Compared to the previous related works, the main purpose of this paper is two–fold: First, we prove the optimal decay rates of the higher spatial derivatives of the solution. Second, we investigate the influences of the electric field on the qualitative behaviors of solution. More precisely, we show that the density and high frequency part of the momentum of the compressible Navier–Stokes–Poisson system have the same <inline-formula><tex-math id="M2">$ L^2 $</tex-math></inline-formula> decay rates as the compressible Navier–Stokes equation and heat equation, but the <inline-formula><tex-math id="M3">$ L^2 $</tex-math></inline-formula> decay rate of the momentum is slower due to the effect of the electric field.</p>

Stationary solutions of outflow problem for full compressible Navier–Stokes–Poisson system: existence, stability and convergence rate

Stationary solutions of outflow problem for full compressible Navier–Stokes–Poisson system: existence, stability and convergence rate

Convergence of the two‐fluid compressible Navier–Stokes–Poisson system to the incompressible Euler equations

In this paper, we consider the combined quasi‐neutral and inviscid limits of the two‐fluid compressible Navier–Stokes–Poisson system in the unbounded domain with the ill‐prepared initial data. We prove that the weak solutions of the compressible Navier–Stokes–Poisson system converge to the strong solution of the incompressible Euler equation as long as the latter exists. Moreover, the convergence rates are also obtained.

Weak-strong uniqueness for the Navier–Stokes–Poisson equations

This paper is devoted to study the weak-strong uniqueness property of compressible Navier–Stokes–Poisson system. By means of relative entropy method, we prove the result that the weak solution coincides with the strong solution, emanating from the same initial data, as long as the latter exists.

The combined quasineutral and low Mach number limit of the Navier–Stokes–Poisson system

In this paper, the quasineutral limit of the compressible Navier–Stokes–Poisson system in the critical $$L^p$$ -type Besov space is considered. More precisely, we will show that the solution of compressible Navier–Stokes–Poisson equations will converge to that of incompressible Navier–Stokes equations in the $$L^p$$ framework when the Debye length is proportional to the Mach number and tends to zero. Moreover, the convergence rate will be obtained.

A sharp time-weighted inequality for the compressible Navier–Stokes–Poisson system in the critical Lp framework

The compressible Navier–Stokes–Poisson system takes the form of usual Navier–Stokes equations coupled with the self-consistent Poisson equation, which is used to simulate the transport of charged particles under the electrostatic potential force. In this paper, we focus on the large-time behavior of global strong solutions in the Lp critical Besov spaces. Inspired by the dissipative effect arising from Poisson potential, we formulate a new regularity assumption of low frequencies and then establish the sharp time-weighted inequality, which leads to the optimal time-decay estimates of strong solutions. Indeed, we see that the decay of density is faster at the half rate than that of velocity, which is a different ingredient in comparison with the situation of compressible Navier–Stokes equations. Our proof mainly depends on tricky and non classical Besov product estimates with respect to various Sobolev embeddings.

Stability of Non-constant Equilibrium Solutions for Bipolar Full Compressible Navier–Stokes–Maxwell Systems

We study the stability of smooth solutions near non-constant equilibrium states for a bipolar full compressible Navier–Stokes–Maxwell system in a three-dimensional torus $$\mathbb {T}= (\mathbb {R}/\mathbb {Z})^3$$ . This system is quasilinear hyperbolic-parabolic. In the first part, by using the maximum principle, we find a non-constant steady state solution with small amplitude for this system. In the second part, with the help of suitable choices of symmetrizers and classic energy estimates, we prove that global smooth solutions exist and converge to the non-constant steady states as the time goes to infinity. As a byproduct, we obtain the global stability for the bipolar full compressible Navier–Stokes–Poisson system.

Optimal decay rate for the compressible Navier–Stokes–Poisson system in the critical Lp framework

In this paper, we consider the large time behavior of global solutions to the initial value problem for the compressible Navier–Stokes–Poisson system in theLpcritical framework and in any dimensionN≥3. We obtain the time decay rates, not only for Lebesgue spaces, but also for a family of Besov norms with negative or nonnegative regularity exponents, which improves the decay results in high Sobolev regularity. The proof is mainly based on the Littlewood–Paley theory and refined time weighted inequalities in Fourier space.

Existence and stability of stationary solution to compressible Navier–Stokes–Poisson equations in half line

In this paper, we investigate the asymptotic stability of the stationary solution to the outflow problem for the compressible Navier–Stokes–Poisson system in a half line. We show the existence of the stationary solution with the aid of the stable manifold theory. The time asymptotic stability of the stationary solution is obtained by the elementary energy method. Furthermore, for the supersonic flow at spatial infinity, we also obtain an algebraic and an exponential decay rate, when the initial perturbation belongs to the corresponding weighted Sobolev space. The proof is based on a time and space weighted energy method.

On nonlinear asymptotic stability of the Lane–Emden solutions for the viscous gaseous star problem

This paper proves the nonlinear asymptotic stability of the Lane–Emden solutions for spherically symmetric motions of viscous gaseous stars if the adiabatic constant γ lies in the stability range (4/3,2). It is shown that for small perturbations of a Lane–Emden solution with same mass, there exists a unique global (in time) strong solution to the vacuum free boundary problem of the compressible Navier–Stokes–Poisson system with spherical symmetry for viscous stars, and the solution captures the precise physical behavior that the sound speed is C1/2-Hölder continuous across the vacuum boundary provided that γ lies in (4/3,2). The key is to establish the global-in-time regularity uniformly up to the vacuum boundary, which ensures the large time asymptotic uniform convergence of the evolving vacuum boundary, density and velocity to those of the Lane–Emden solution with detailed convergence rates, and detailed large time behaviors of solutions near the vacuum boundary. In particular, it is shown that every spherical surface moving with the fluid converges to the sphere enclosing the same mass inside the domain of the Lane–Emden solution with a uniform convergence rate and the large time asymptotic states for the vacuum free boundary problem (1.1.2) are determined by the initial mass distribution and the total mass. To overcome the difficulty caused by the degeneracy and singular behavior near the vacuum free boundary and coordinates singularity at the symmetry center, the main ingredients of the analysis consist of combinations of some new weighted nonlinear functionals (involving both lower-order and higher-order derivatives) and space–time weighted energy estimates. The constructions of these weighted nonlinear functionals and space–time weights depend crucially on the structures of the Lane–Emden solution, the balance of pressure and gravitation, and the dissipation. Finally, the uniform boundedness of the acceleration of the vacuum boundary is also proved.

Global existence of weak solution for the compressible Navier–Stokes–Poisson system for gaseous stars

We study the multi-dimensional compressible Navier–Stokes–Poisson system with γ-law pressure and density-dependent viscosity coefficients in the simulation of the motion of gaseous stars. The global existence of spherically symmetric weak solutions to the free boundary value problem for the Navier–Stokes–Poisson system for γ∈(65,43] is shown for arbitrarily large initial data with compact support.

Local strong solution of Navier–Stokes–Poisson equations with degenerated viscosity coefficient

In this paper, we investigate the vacuum free boundary problem of a compressible Navier–Stokes–Poisson system with density‐dependent viscosity. By introducing Eulerian and Lagrange energy, we obtain a local in time well‐posedness of the strong solution to the Navier–Stokes–Poisson system in a spherically symmetric case. Copyright © 2015 John Wiley & Sons, Ltd.

Decay of the non-isentropic Navier–Stokes–Poisson equations

We establish the time decay rates of the solution to the Cauchy problem for the non-isentropic compressible Navier–Stokes–Poisson system via a refined pure energy method. In particular, the optimal decay rates of the higher-order spatial derivatives of the solution are obtained. As a corollary, we also obtain the usual Lp−L2(1<p≤2) type of the optimal decay rates. The Ḣ−s(0≤s<3/2) negative Sobolev norms are shown to be preserved along time evolution and enhance the decay rates. We use a family of scaled energy estimates with minimum derivative counts and interpolations among them without linear decay analysis.