Vanishing viscosity and Debye-length limit to rarefaction wave with vacuum for the 1D bipolar Navier–Stokes–Poisson equation

In this paper, we consider the one-dimensional (1D) compressible bipolar Navier–Stokes–Poisson equations. We know that when the viscosity coefficient and Debye length are zero in the compressible bipolar Navier–Stokes–Poisson equations, we have the compressible Euler equations. Under the case that the compressible Euler equations have a rarefaction wave with one-side vacuum state, we can construct a sequence of the approximation solution to the one-dimensional bipolar Navier–Stokes–Poisson equations with well-prepared initial data, which converges to the above rarefaction wave with vacuum as the viscosity and the Debye length tend to zero. Moreover, we also obtain the uniform convergence rate. The results are proved by a scaling argument and elaborate energy estimate.     

Convergence of the quantum Navier–Stokes–Poisson equations to the incompressible Euler equations for general initial data

In this paper, we are concerned with the rigorous proof of the convergence of the quantum Navier–Stokes-Poisson system to the incompressible Euler equations via the combined quasi-neutral, vanishing damping coefficient and inviscid limits in the three-dimensional torus for general initial data. Furthermore, the convergence rates are obtained.     

Blowup criteria for strong solutions to the compressible Navier–Stokes equations with variable viscosity

For the compressible Navier–Stokes equations with viscosity and heat conductivity coefficients possibly depending on the density or temperature, several blowup criteria are given to the local-in-time strong solutions. The proof is based on energy methods together with elliptic and parabolic estimates adopted to the present situation.     

Inviscid limit for Navier–Stokes equations in domains with permeable boundaries

This work is concerned with 2D-Navier Stokes equations in a multiply-connected bounded domain with permeable walls. The permeability is described by a Navier type condition. Our aim is to show that the inviscid limit is a solution of the Euler equations, satisfying the Navier type condition on the inflow zone of the walls.     

Incompressible limit of the degenerate quantum compressible Navier–Stokes equations with general initial data

In this paper we study the incompressible limit of the degenerate quantum compressible Navier–Stokes equations in a periodic domain ${\mathbb{T}}^3$ and the whole space ${\mathbb{R}}^3$ with general initial data. In the periodic case, by applying the refined relative entropy method and carrying out the detailed analysis on the oscillations of velocity, we prove rigorously that the gradient part of the weak solutions (velocity) of the degenerate quantum compressible Navier–Stokes equations converge to the strong solution of the incompressible Navier–Stokes equations. Our results improve considerably the ones obtained by Yang, Ju and Yang [25] where only the well-prepared initial data case is considered. While for the whole space case, thanks to the Strichartz's estimates of linear wave equations, we can obtain the convergence of the weak solutions of the degenerate quantum compressible Navier–Stokes equations to the strong solution of the incompressible Navier–Stokes/Euler equations with a linear damping term. Moreover, the convergence rates are also given.     

Blowup of solutions for the compressible Navier–Stokes equations with density-dependent viscosity coefficients

We study the blowup phenomena of solutions to the compressible Navier–Stokes equations with density-dependent viscosity coefficients in arbitrary dimensions. By constructing a family of self-similar analytical solutions with spherical symmetry, some interesting information including the blowup and expanding properties are shown. In addition, the case of constant viscosity coefficients is also considered. The approach is based on the phase plane method.     

Steady solution and its stability for Navier–Stokes equations with general Navier slip boundary condition

The steady solution and the asymptotic behavior of the corresponding nonsteady solution are studied for Navier–Stokes equations under the general Navier slip boundary condition. The existence of a unique stationary solution is established. It is also proved that this solution is asymptotically stable under some restrictions on the data. Bibliography: 16 titles. Dedicated to Vsevolod Alekseevich Solonnikov on the occasion of his jubilee Published in Zapiski Nauchnykh Seminarov POMI, Vol. 362, 2008, pp. 153–175.     

Boundary layer problem: Navier–Stokes equations and Euler equations

This work is concerned with the boundary layer turbulence, which is an outstanding problem in fluid mechanics. We consider an incompressible viscous fluid in 2D domains with permeable walls. The permeability is described by the Yudovich condition. The goal of the article is to study the fluid behavior at vanishing viscosity (large Reynold’s numbers). We show that the vanishing viscous limit is a solution of the Euler equations with the Yudovich condition on the inflow region of the boundary.     

On strong convergence of attractors of Navier–Stokes equations in the limit of vanishing viscosity

Mathematical Notes -     

Euler implicit/explicit iterative scheme for the stationary Navier–Stokes equations

In this paper, a new uniqueness assumption (A2) of the solution for the stationary Navier–Stokes equations is presented. Under assumption (A2), the exponential stability of the solution $(\bar{u},\bar{p})$ for the stationary Navier–Stokes equations is proven. Moreover, the Euler implicit/explicit scheme based on the mixed finite element is applied to solve the stationary Navier–Stokes equations. Finally, the almost unconditionally stability is proven and the optimal error estimates uniform in time are provided for the scheme.     

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.     

Components reduction regularity results for the Navier–Stokes equations in general dimensions

We consider the Cauchy problem of the Navier–Stokes equations in arbitrary dimensions, and establish several new components reduction regularity criteria.     

Vacuum problem for 1D compressible Navier–Stokes equations with gravity and general pressure law

In this paper, we study the free boundary problem for the one-dimensional isentropic Navier–Stokes equations with gravity and vacuum for the general pressure P = P(ρ). We mainly obtain global existence, the uniqueness and asymptotic behavior of the weak solution. In particular, we get the result of Theorem 4.7, which shows that the time-asymptotic state corresponds to the hydrostatic pressure law.