Isentropic Navier-Stokes Equations Research Articles

The Regularity and Uniqueness of a Global Solution to the Isentropic Navier-Stokes Equation with Rough Initial Data

The Regularity and Uniqueness of a Global Solution to the Isentropic Navier-Stokes Equation with Rough Initial Data

Dissipative solutions to the compressible isentropic Navier–Stokes equations

Dissipative solutions to the compressible isentropic Navier–Stokes equations

Asymptotic stability of planar rarefaction waves under periodic perturbations for 3-d Navier-Stokes equations

In this paper, we study a Cauchy problem for the 3-d compressible isentropic Navier-Stokes equations, in which the initial data is a 3-d periodic perturbation around a planar rarefaction wave. We prove that the solution of the Cauchy problem exists globally in time and tends to the background rarefaction wave in the L∞(R3) space as t→+∞. The result reveals that even though the initial perturbation has infinite oscillations at the far field and is not integrable along any direction of space, the planar rarefaction wave is nonlinearly stable for the 3-d N-S equations. The key point is to construct a suitable ansatz (ρ˜,u˜) to carry the same oscillations as those of the solution (ρ,u) at the far field in the normal direction of the rarefaction wave, so that the difference (ρ−ρ˜,u−u˜) belongs to some Sobolev space and the energy method is available.

High order all-speed semi-implicit weighted compact nonlinear scheme for the isentropic Navier–Stokes equations

This paper presents a high order all-speed semi-implicit weighted compact nonlinear scheme (WCNS) for the isentropic Navier–Stokes system. To avoid the severe CFL stability restriction, the pressure and viscous terms are treated implicitly in time, while the other terms are treated explicitly in time. The third-order IMEX Runge–Kutta methods and the fifth-order WCNS are used for time discretization and spatial discretization, respectively. The generated linear equations of velocity components are solved by the GMRES iterative algorithm. Numerical results in one, two and three dimensions in both compressible and incompressible regimes are presented to show the performance of the designed scheme.

Linear stability of viscous shock wave for 1-D compressible Navier-Stokes equations with Maxwell’s law

In this paper, we consider the linear stability of traveling wave solutions for one-dimensional compressible isentropic Navier-Stokes equations with Maxwell’s Law. The global stability of traveling wave solution is established with shock-profile initial data for the linearized system. Anti-derivative and some delicate energy methods are explored to get the desired results. Moreover, the relaxation limit of traveling wave solution is also obtained.

Pointwise estimates of the solution to one dimensional compressible Naiver-Stokes equations in half space

<p style='text-indent:20px;'>In this paper, we study the global existence and pointwise behavior of classical solution to one dimensional isentropic Navier-Stokes equations with mixed type boundary condition in half space. Based on classical energy method for half space problem, the global existence of classical solution is established firstly. Through analyzing the quantitative relationships of Green's function between Cauchy problem and initial boundary value problem, we observe that the leading part of Green's function for the initial boundary value problem is composed of three items: delta function, diffusive heat kernel, and reflected term from the boundary. Then applying Duhamel's principle yields the explicit expression of solution. With the help of accurate estimates for nonlinear wave coupling and the elliptic structure of velocity, the pointwise behavior of the solution is obtained under some appropriate assumptions on the initial data. Our results prove that the solution converges to the equilibrium state at the optimal decay rate <inline-formula><tex-math id="M1">\begin{document}$ (1+t)^{-\frac{1}{2}} $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M2">\begin{document}$ L^\infty $\end{document}</tex-math></inline-formula> norm.</p>

Existence and nonlinear stability of steady-states to outflow problem for the full two-phase flow

The outflow problem for the viscous full two-phase flow model in a half line is investigated in the present paper. The existence, uniqueness and nonlinear stability of the steady-state are shown respectively corresponding to the supersonic, sonic or subsonic state at far field. This is different from the outflow problem for the isentropic Navier-Stokes equations, where there is no steady-state for the subsonic state. Furthermore, we obtain either exponential time decay rates for the supersonic state or algebraic time decay rates for supersonic and sonic states in weighted Sobolev spaces.

Local weak solution of the isentropic compressible Navier–Stokes equations

Whether the three dimensional isentropic compressible Navier–Stokes equations admit weak solutions for arbitrary initial data with adiabatic exponent γ > 1 remains a challenging problem. The only available results under γ > 1 were achieved by either assuming the initial data with small energy due to Hoff [J. Differ. Equations 120(1), 215–254 (1995)] or under the spherically symmetric condition by Jiang and Zhang [Commun. Math. Phys. 215, 559–581 (2001)] and Huang [J. Differ. Equations 262, 1341–1358 (2017)]. In this paper, we establish the existence of weak solutions with higher regularity of the three-dimensional periodic compressible isentropic Navier–Stokes equations in small time for the adiabatic exponent γ > 1 in the presence of vacuum. It can be viewed as a local version of Hoff’s work and also extends the result of Desjardins [Commun. Partial Differ. Equations 22(5–6), 977–1008 (1997)] by removing the assumption of γ > 3.

Weak Solutions for Compressible Isentropic Navier–Stokes Equations in Dimensions Three

Weak Solutions for Compressible Isentropic Navier–Stokes Equations in Dimensions Three

Stability of Large-Amplitude Viscous Shock Under Periodic Perturbation for 1-d Isentropic Navier–Stokes Equations

The stability of solutions under periodic perturbations for both inviscid and viscous conservation laws is an interesting and important problem. In this paper, a large-amplitude viscous shock under space-periodic perturbation for the isentropic Navier–Stokes equations is considered. It is shown that if the initial perturbation around the shock is suitably small and satisfies a zero-mass type condition (2.17), then the solution of the N–S equations tends to the viscous shock with a shift, which is partially determined by the periodic oscillations. In other words, the viscous shock is nonlinearly stable even though the perturbation oscillates at the far fields. The key point is to construct a suitable ansatz $$ (\tilde{v},\tilde{u}) $$ , which carries the same oscillations of the solution (v, u) at the far fields, so that the difference $$ (v-\tilde{v},u-\tilde{u}) $$ belongs to the $$ H^2(\mathbb {R}) $$ space for all $$ t\ge 0. $$

Analytical Solution to 1D Compressible Navier-Stokes Equations

There exist complex behavior of the solution to the 1D compressible Navier-Stokes equations in half space. We find an interesting phenomenon on the solution to 1D compressible isentropic Navier-Stokes equations with constant viscosity coefficient on x , t ∈ 0 , + ∞ × R + , that is, the solutions to the initial boundary value problem to 1D compressible Navier-Stokes equations in half space can be transformed to the solution to the Riccati differential equation under some suitable conditions.

Large time behavior of solutions to a two phase fluid model in [formula omitted

We establish the global existence and large time behavior of a unique classical solution for a two phase fluid model consisting of the compressible isothermal Euler equations coupled with compressible isentropic Navier-Stokes equations via a drag force, provided the initial data is sufficiently small in H4(R3)∩L1(R3). The main tools employed in the analysis are the spectrum analysis and energy method. Our results show that the classical solution converges to a given equilibrium state at algebra decay rate.

Large-time behaviors of the solution to 3D compressible Navier-Stokes equations in half space with Navier boundary conditions

<p style="text-indent:20px;">We are concerned with the large-time asymptotic behaviors towards the planar rarefaction wave to the three-dimensional (3D) compressible and isentropic Navier-Stokes equations in half space with Navier boundary conditions. It is proved that the planar rarefaction wave is time-asymptotically stable for the 3D initial-boundary value problem of the compressible Navier-Stokes equations in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^+\times \mathbb{T}^2 $\end{document}</tex-math></inline-formula> with arbitrarily large wave strength. Compared with the previous work [<xref ref-type="bibr" rid="b17">17</xref>, <xref ref-type="bibr" rid="b16">16</xref>] for the whole space problem, Navier boundary conditions, which state that the impermeable wall condition holds for the normal velocity and the fluid tangential velocity is proportional to the tangential component of the viscous stress tensor on the boundary, are crucially used for the stability analysis of the 3D initial-boundary value problem.</p>

Local weak solution of the isentropic compressible Navier-Stokes equations in a half-space

In this paper, we establish the local existence of weak solutions with higher regularity of the three-dimensional half-space compressible isentropic Navier-Stokes equations with the adiabatic exponent γ > 1 in the presence of vacuum. Here we do not need any smallness of the initial data.

Lie group analysis, exact solutions and conservation laws to compressible isentropic Navier–Stokes equation

The current study dedicated to the compressible isentropic Navier–Stokes equations in one-dimensional with a general pressure law. The Lie Group method is employed to reduce the compressible Navier–Stokes equations to a system of highly nonlinear ordinary differential equations with suitable similarity transformations. Consequently, with the help of exact solutions of reduced ordinary differential equations, similarity variable and similarity solutions, exact solutions of the main equation are obtained. Finally, using conservation laws multiplier, we find the complete set of local conservation laws of compressible isentropic Navier–Stokes equations for the arbitrary constant coefficients.

Vanishing viscosity limit to the planar rarefaction wave for the two-dimensional full compressible Navier-Stokes equations

The vanishing viscosity limit for the multi-dimensional compressible Navier-Stokes equations to the corresponding Euler equations is a difficult and challenging problem in the mathematics. Recently, L.Li, D.Wang and Y.Wang [17] verified that the solutions for the 2D compressible isentropic Navier-Stokes equations converge to the planar rarefaction wave solution for the corresponding 2D Euler equations as viscosity vanishes with a convergence rate ϵ1/6|ln⁡ϵ|. In this paper, the vanishing viscosity limit of 2D non-isentropic compressible Navier-Stokes equations is studied. In contrast to the work [17], the convergence rate for the 2D full compressible Navier-Stokes equations is improved to ϵ2/7|ln⁡ϵ|2 by choosing a different scaling argument and performing more detailed energy estimates.

Nonlinear stability of large amplitude viscous shock wave for general viscous gas

In the present paper, it is shown that the large amplitude viscous shock wave is nonlinearly stable for isentropic Navier-Stokes equations, in which the pressure could be general and includes γ-law, and the viscosity coefficient is a smooth function of density. The strength of shock wave could be arbitrarily large. The proof is given by introducing a new variable, which can formulate the original system into a new one, and the elementary energy method introduced in [21].

A fractional step method for computational aeroacoustics using weak imposition of Dirichlet boundary conditions

In this work we consider the approximation of the isentropic Navier–Stokes equations. The model we present is capable of taking into account acoustic and flow scales at once. After space and time discretizations have been chosen, it is very convenient from the computational point of view to design fractional step schemes in time so as to permit a segregated calculation of the problem unknowns. While these segregation schemes are well established for incompressible flows, much less is known in the case of isentropic flows. We discuss this issue in this article and, furthermore, we study the way to weakly impose Dirichlet boundary conditions via Nitsche’s method. In order to avoid spurious reflections of the acoustic waves, Nitsche’s method is combined with a non-reflecting boundary condition. Employing a purely algebraic approach to discuss the problem, some of the boundary contributions are treated explicitly and we explain how these are included in the different steps of the final algorithm. Numerical evidence shows that this explicit treatment does not have a significant impact on the convergence rate of the resulting time integration scheme. The equations of the formulation are solved using a subgrid scale technique based on a term-by-term stabilization.

Vanishing Viscosity Limit to the Planar Rarefaction Wave for the Two-Dimensional Compressible Navier–Stokes Equations

The vanishing viscosity limit of the two-dimensional (2D) compressible isentropic Navier-Stokes equations is studied in the case that the corresponding 2D inviscid Euler equations admit a planar rarefaction wave solution. It is proved that there exists a family of smooth solutions for the 2D compressible Navier-Stokes equations converging to the planar rarefaction wave solution with arbitrary strength for the 2D Euler equations. A uniform convergence rate is obtained in terms of the viscosity coefficients away from the initial time. In the proof, the hyperbolic wave is crucially introduced to recover the physical viscosities of the inviscid rarefaction wave profile, in order to rigorously justify the vanishing viscosity limit.

Convergence rate of solutions toward stationary solutions to a two-phase model with magnetic field in a half line

In this paper, we study an asymptotic behavior of a solution to the outflow problem for a two-phase model with magnetic field. Our idea mainly comes from [1] and [2] which investigate the asymptotic stability and convergence rates of stationary solutions to the outflow problem for an isentropic Navier–Stokes equation. For the two-phase model with magnetic field, we also obtain the asymptotic stability and convergence rates of global solutions towards corresponding stationary solutions if the initial perturbation belongs to the weighted Sobolev space. The proof is based on the weighted energy method.