Isentropic Compressible Navier Research Articles

Wave propagation for the isentropic compressible Navier–Stokes/Allen–Cahn system

We study the Cauchy problem for the three‐dimensional isentropic compressible Navier–Stokes/Allen–Cahn system, which models the phase transitions in two‐component patterns interacting with a compressible fluid. Under the assumption that the initial perturbation is small and decays spatially, we establish the global existence and the pointwise behavior of strong solutions to this nonconserved system. To deal with the source terms involving the phase variable, we employ the Green's function and space‐time weighted estimates. The analysis shows that the phase variable mainly contains the diffusion wave with exponential decaying amplitude over time, and consequently the density and momentum of the compressible fluid adhere to a generalized Huygens principle.

Local Weak Solution of the Isentropic Compressible Navier–Stokes Equations with Variable Viscosity

Local Weak Solution of the Isentropic Compressible Navier–Stokes Equations with Variable Viscosity

Self-similar solutions and large time behavior to the 2D isentropic compressible Navier–Stokes equations with vacuum free boundary and rotation

This paper is concerned with a class of self-similar analytical solutions to the 2D isentropic compressible Navier–Stokes equations with vacuum free boundary for polytropic gases. It is shown that the free boundary will grow linearly in time (see (1.11)), moreover, both angular velocity uϕ and its derivatives, and the derivative of radial velocity ur will tend to zero as t→+∞, while radial velocity itself is bounded. In particular, it is interesting to see that when the adiabatic exponent γ>2 and for a suitably large time, the rotation plays a dominant role in accelerating the growth of the free boundary compared with the effect of pressure.

Global Solutions of 3D Isentropic Compressible Navier–Stokes Equations with Two Slow Variables

Global Solutions of 3D Isentropic Compressible Navier–Stokes Equations with Two Slow Variables

Boundary layers for a fluid–particle interaction system with density-dependent viscosity and cylindrical symmetry

In this paper, we study the initial boundary value problem for a cylindrical symmetry fluid–particle interaction system in three dimensions. The boundary layer phenomena is investigated when the shear viscosity μ=κρβ goes to zero. Furthermore, we establish the boundary layer thickness of the order O(κα) for more general initial data when 0<α<12 and give the optimal boundary-layer thickness for the system with more general initial data. As a byproduct, this work improves the corresponding results in Yao et al. (2011) for isentropic compressible Navier–Stokes equations where 0<α<14.

Global wellposedness and asymptotic behavior of axisymmetric strong solutions to the vacuum free boundary problem of isentropic compressible Navier–Stokes equations

Global wellposedness and asymptotic behavior of axisymmetric strong solutions to the vacuum free boundary problem of isentropic compressible Navier–Stokes equations

Global existence and large time behaviour for the pressureless Euler–Naver–Stokes system in ℝ3

We investigate the global Cauchy problem for a two–phase flow model consisting of the pressureless Euler equations coupled with the isentropic compressible Navier–Stokes equations through a drag forcing term. This model was first derived by Choi–Kwon [J. Differential Equations, 261(1) (2016), pp. 654–711] by taking the hydrodynamic limit of the Vlasov/compressible Navier–Stokes equations. Under the assumption that the initial perturbation is sufficiently small, Choi–Kwon [J. Differential Equations, 261(1) (2016), pp. 654–711] established the global well–posedness and large time behaviour for the three dimensional periodic domain $\mathbb {T}^3$. However, up to now, the global well–posedness and large time behaviour for the three dimensional Cauchy problem still remain unsolved. In this paper, we resolve this problem by proving the global existence and optimal decay rates of classic solutions for the three dimensional Cauchy problem when the initial data is near its equilibrium. One of key observations here is that to overcome the difficulties arising from the absence of pressure in the Euler equations, we make full use of the drag forcing term and the dissipative structure of the Navier–Stokes equations to closure the energy estimates of the variables for the pressureless Euler equations.

Convergence of the relaxed compressible Navier–Stokes equations to the incompressible Navier–Stokes equations

In this paper, we study the combined low Mach number and the relaxation limits of the isentropic compressible Navier–Stokes equations with revised Maxwell’s law satisfying Galilean invariance. It is shown that, for the ill-prepared initial data, the solutions of the relaxed compressible Navier–Stokes equations converge to that of the incompressible Navier–Stokes equations as the Mach number and relaxation parameters tend to zero.

Global existence of strong solutions to the 3D isentropic compressible Navier–Stokes equations with density‐dependent viscosities

We consider the Cauchy problem to the three‐dimensional isentropic compressible Navier–Stokes system with density‐dependent viscosities. Under the condition that the lower bound of initial density is sufficiently large, we prove that strong solutions exist globally in time. The initial data can be arbitrarily large.

Dissipative solutions to the compressible isentropic Navier–Stokes equations

Dissipative solutions to the compressible isentropic Navier–Stokes equations

On the regularity criteria for the three‐dimensional compressible Navier–Stokes system in Lorentz spaces

In this paper, we are concerned with the continuation criteria to the 3D isentropic compressible Navier–Stokes equations in Lorentz spaces. We prove that there exists a positive constant such that no blowup occurs at time in this system provided that the supernorm of the density is bounded and the space‐time Lorentz spaces norm with is small. As a direct application, we established some Serrin's blow‐up criteria in the Lorentz spaces.

On an inhomogeneous slip-inflow boundary value problem for a steady viscous compressible channel flow

We prove the existence and uniqueness of a strong solution to the steady isentropic compressible Navier–Stokes equations with inflow boundary condition for density and mixed boundary conditions for the velocity around a shear flow. In particular, the Dirichlet boundary conditions on the inflow and outflow part of the boundary and the full Navier boundary conditions on the wall for the velocity field are considered. For our result, there are no restrictions on the amplitude of friction coefficients α, and only the assumption that the viscosity coefficient μ is appropriately large is required. One of the substantial ingredients of our proof is an elegant transformation induced by the flow field. With the help of this transformation, we can overcome the difficulties caused by the hyperbolicity of the continuity equation, establish the a priori estimates for a linearized system and apply the fixed point argument.

Global-in-time dynamics of the two-phase fluid model in a bounded domain

In this work, we study the global existence of strong solutions and large-time behavior of a two-phase fluid model in a bounded domain. The model consists of the isothermal Euler equations and the isentropic compressible Navier–Stokes equations, coupled via the drag force. It was derived in Choi and Jung (2021) from a kinetic-fluid model describing the dynamics of particles subject to local alignment force and Brownian noises immersed in a compressible viscous fluid. For this system, we extend the local existence theory for strong solutions developed in Choi and Jung (2021) to obtain the global existence of strong solutions to the system. Moreover, we use the Lyapunov functional associated with the system to get large-time behavior estimates for global classical solutions.

Zero-viscosity-capillarity limit to the planar rarefaction wave for the 2D compressible Navier–Stokes–Korteweg equations

We shall consider the two-dimensional (2D) isentropic Navier–Stokes–Korteweg equations which are used to model compressible fluids with internal capillarity. Formally, the 2D isentropic Navier–Stokes–Korteweg equations converge, as the viscosity and the capillarity vanish, to the corresponding 2D inviscid Euler equations, and we do justify this for the case that the corresponding 2D inviscid Euler equations admit a planar rarefaction wave solution. More precisely, it is proved that there exists a family of smooth solutions for the 2D isentropic compressible Navier–Stokes–Korteweg 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 coefficient and the capillarity away from the initial time. The key ingredients of our proof are the re-scaling technique and energy estimate, in which we also introduce the hyperbolic wave to recover the physical viscosities and capillarity of the inviscid rarefaction wave profile.

Large-time behavior of the energy to the isentropic compressible Navier–Stokes equations with vacuum

In the present paper, we study the large-time behavior of the energy to the isentropic compressible Navier–Stokes equations in R3. More exactly, under the assumption that the initial data are of small energy but possibly large oscillations, and in addition that the initial density belongs to L1(R3), we derive the convergence rates of the kinetic energy and the potential energy to the Cauchy problem of the system with vacuum.

Stability properties of the steady state for the isentropic compressible Navier–Stokes equations with density dependent viscosity in bounded intervals

We prove existence and asymptotic stability of the stationary solution for the compressible Navier-Stokes equations for isentropic gas dynamics with a density dependent diffusion in a bounded interval. We present the necessary conditions to be imposed on the boundary data which ensure existence and uniqueness of the steady state, and we subsequent investigate its stability properties by means of the construction of a suitable Lyapunov functional for the system. The Saint-Venant system, modeling the dynamics of a shallow compressible fluid, fits into this general framework.

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 &amp;gt; 1 remains a challenging problem. The only available results under γ &amp;gt; 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 &amp;gt; 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 &amp;gt; 3.

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

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

Asymptotic analysis for a Vlasov–Fokker–Planck/Navier–Stokes system in a bounded domain

We study an asymptotic analysis of a coupled system of kinetic and fluid equations. More precisely, we deal with the nonlinear Vlasov–Fokker–Planck equation coupled with the compressible isentropic Navier–Stokes system through a drag force in a bounded domain with the specular reflection boundary condition for the kinetic equation and homogeneous Dirichlet boundary condition for the fluid system. We establish a rigorous hydrodynamic limit corresponding to strong noise and local alignment force. The limiting system is a type of two-phase fluid model consisting of the isothermal Euler system and the compressible Navier–Stokes system. Our main strategy relies on the relative entropy argument based on the weak–strong uniqueness principle. For this, we provide a global-in-time existence of weak solutions for the coupled kinetic-fluid system. We also show the existence and uniqueness of strong solutions to the limiting system in a bounded domain with the kinematic boundary condition for the Euler system and Dirichlet boundary condition for the Navier–Stokes system.

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.