The inviscid limit for an inflow problem of compressible viscous gas in presence of both shocks and boundary layers

In this paper, we study the inviscid limit problem for the Navier-Stokes equations of one-dimensional compressible viscous gas on half plane. We prove that if the solution of the inviscid Euler system on half plane is piecewise smooth with a single shock satisfying the entropy condition, then there exist solutions to Navier-Stokes equations which converge to the inviscid solution away from the shock discontinuity and the boundary at an optimal rate of ε¹ as the viscosity ε tends to zero.     

Zero dissipation limit to strong contact discontinuity for the 1-D compressible Navier-Stokes equations

In this paper, we study the zero dissipation limit problem for the one-dimensional compressible Navier-Stokes equations. We prove that if the solution of the inviscid Euler equations is piecewise constants with a contact discontinuity, then there exist smooth solutions to the Navier-Stokes equations which converge to the inviscid solution away from the contact discontinuity at a rate of as the heat-conductivity coefficient κ tends to zero, provided that the viscosity μ is of higher order than the heat-conductivity κ. Without loss of generality, we set μ≡0. Here we have no need to restrict the strength of the contact discontinuity to be small.     

Local well-posedness for the homogeneous Euler equations

We introduce Triebel-Lizorkin-Lorentz function spaces, based on the Lorentz L^p,q-spaces instead of the standard L^p-spaces, and prove a local-in-time unique existence and a blow-up criterion of solutions in those spaces for the Euler equations of inviscid incompressible fluid in Rⁿ,n≥2. As a corollary we obtain global existence of solutions to the 2D Euler equations in the Triebel-Lizorkin-Lorentz space. For the proof, we establish the Beale-Kato-Majda type logarithmic inequality and commutator estimates in our spaces. The key methods of proof used are the Littlewood-Paley decomposition and the paradifferential calculus by J.M. Bony.     

Navier边界条件下湖方程的边界层问题

考虑光滑区域上二维粘性湖方程在Navier边界条件下的无粘极限问题,证明了具有Navier边界条件粘性湖方程的边界层在Sobolev空间中是非线性稳定的,验证了具有较弱强度的边界层的渐近展开的合理性.     

Shallow water equations: viscous solutions and inviscid limit

We establish the inviscid limit of the viscous shallow water equations to the Saint-Venant system. For the viscous equations, the viscosity terms are more degenerate when the shallow water is close to the bottom, in comparison with the classical Navier-Stokes equations for barotropic gases; thus, the analysis in our earlier work for the classical Navier-Stokes equations does not apply directly, which require new estimates to deal with the additional degeneracy. We first introduce a notion of entropy solutions to the viscous shallow water equations and develop an approach to establish the global existence of such solutions and their uniform energy-type estimates with respect to the viscosity coefficient. These uniform estimates yield the existence of measure-valued solutions to the Saint-Venant system generated by the viscous solutions. Based on the uniform energy-type estimates and the features of the Saint-Venant system, we further establish that the entropy dissipation measures of the viscous solutions for weak entropy-entropy flux pairs, generated by compactly supported C ² test-functions, are confined in a compact set in H ^?1, which yields that the measure-valued solutions are confined by the Tartar-Murat commutator relation. Then, the reduction theorem established in Chen and Perepelitsa [5] for the measure-valued solutions with unbounded support leads to the convergence of the viscous solutions to a finite-energy entropy solution of the Saint-Venant system with finite-energy initial data, which is relative with respect to the different end-states of the bottom topography of the shallow water at infinity. The analysis also applies to the inviscid limit problem for the Saint-Venant system in the presence of friction.     

From the dynamics of gaseous stars to the incompressible Euler equations

A model for the dynamics of gaseous stars is introduced and formulated by the Navier-Stokes-Poisson system for compressible, reacting gases. The combined quasineutral and inviscid limit of the Navier-Stokes-Poisson system in the torus Tn is investigated. The convergence of the Navier-Stokes-Poisson system to the incompressible Euler equations is proven for the global weak solution and for the case of general initial data.     

Navier-Stokes regularization of multidimensional Euler shocks

We establish existence and stability of multidimensional shock fronts in the vanishing viscosity limit for a general class of conservation laws with “real”, or partially parabolic, viscosity including the Navier-Stokes equations of compressible gas dynamics with standard or van der Waals-type equation of state. More precisely, given a curved Lax shock solution u⁰ of the corresponding inviscid equations for which (i) each of the associated planar shocks tangent to the shock front possesses a smooth viscous profile and (ii) each of these viscous profiles satisfies a uniform spectral stability condition expressed in terms of an Evans function, we construct nearby smooth viscous shock solutions uε of the viscous equations converging to u⁰ as viscosity ε→0, and establish for these sharp linearized stability estimates generalizing those of Majda in the inviscid case. Conditions (i)-(ii) hold always for shock waves of sufficiently small amplitude, but in general may fail for large amplitudes.We treat the viscous shock problem considered here as a representative of a larger class of multidimensional boundary problems arising in the study of viscous fluids, characterized by sharp spectral conditions rather than symmetry hypotheses, which can be analyzed by Kreiss-type symmetrizers.Compared to the strictly parabolic (artificial viscosity) case, the main new features of the analysis appear in the high frequency estimates for the linearized problem. In that regime we use frequency-dependent conjugators to decouple parabolic components that are smoothed from hyperbolic components (like density in Navier-Stokes) that are not. The construction of the conjugators and the subsequent estimates depend on a careful spectral analysis of the linearized operator.     

Radial equivalence and study of self-similarity for two very fast diffusion equations

In this paper we continue the study of the radial equivalence between the porous medium equation and the evolution p-Laplacian equation, begun in a previous work. We treat the cases m<0 and p<1. We perform an exhaustive study of self-similar solutions for both equations, based on a phase-plane analysis and the correspondences we discover. We also obtain special correspondence relations and self-maps for the limit case m=−1, p=0, which is particularly important in applications in image processing. We also find self-similar solutions for the very fast p-Laplacian equation that have finite mass and, in particular, some of them that conserve mass, while this phenomenon is not true for the very fast diffusion equation.     

The critical exponents for the quasi-linear parabolic equations with inhomogeneous terms

This paper deals with the critical exponents for the quasi-linear parabolic equations in Rn and with an inhomogeneous source, or in exterior domains and with inhomogeneous boundary conditions. For n?3, σ>−2/n and p>max{1,1+σ}, we obtain that pc=n(1+σ)/(n−2) is the critical exponent of these equations. Furthermore, we prove that if max{1,1+σ}<p?pc, then every positive solution of these equations blows up in finite time; whereas these equations admit the global positive solutions for some f(x) and some initial data u₀(x) if p>pc. Meantime, we also demonstrate that every positive solution of these equations blows up in finite time provided n=1,2, σ>−1 and p>max{1,1+σ}.     

Convergence of euler-stokes splitting of the navier-stokes equations

We consider approximation by partial time steps of a smooth solution of the Navier-Stokes equations in a smooth domain in two or three space dimensions with no-slip boundary condition. For small k > 0, we alternate the solution for time k of the inviscid Euler equations, with tangential boundary condition, and the solution of the linear Stokes equations for time k, with the no-slip condition imposed. We show that this approximation remains bounded in H^2,p and is accurate to order k in L^p for p > ∞. The principal difficulty is that the initial state for each Stokes step has tangential velocity at the boundary generated during the Euler step, and thus does not satisfy the boundary condition for the Stokes step. The validity of such a fractional step method or splitting is an underlying principle for some computational methods. © 1994 John Wiley & Sons, Inc.     

The Convergence of the Navier–Stokes–Poisson System to the Incompressible Euler Equations

ABSTRACT

The combining quasineutral and inviscid limit of the Navier–Stokes–Poisson system in the torus 𝕋 ^d , d ≥ 1 is studied. The convergence of the Navier–Stokes–Poisson system to the incompressible Euler equations is proven for the global weak solution and for the case of general initial data.     

The homogeneous dirichlet problem for quasilinear anisotropic degenerate parabolic-hyperbolic equation with L initial value

The aim of this paper is to prove the well-posedness (existence and uniqueness) of the L^p entropy solution to the homogeneous Dirichlet problems for the anisotropic degenerate parabolic-hyperbolic equations with L^p initial value. We use the device of doubling variables and some technical analysis to prove the uniqueness result. Moreover we can prove that the L^p entropy solution can be obtained as the limit of solutions of the corresponding regularized equations of nondegenerate parabolic type.     

Convergence of rank-type equations

Convergence results are presented for rank-type difference equations, whose evolution rule is defined at each step as the kth largest of p univariate difference equations. If the univariate equations are individually contractive, then the equation converges to a fixed point equal to the kth largest of the individual fixed points of the univariate equations. Examples are max-type equations for k = 1, and the median of an odd number p of equations, for k = (p + 1)/2. In the non-hyperbolic case, conjectures are stated about the eventual periodicity of the equations, generalizing long-standing conjectures of G. Ladas.     

Homogenization and correctors for monotone problems in cylinders of small diameter

In this paper we study the homogenization of monotone diffusion equations posed in an N  -dimensional cylinder which converges to a (one-dimensional) segment line. In other terms, we pass to the limit in diffusion monotone equations posed in a cylinder whose diameter tends to zero, when simultaneously the coefficients of the equations (which are not necessarily periodic) are also varying. We obtain a limit system in both the macroscopic (one-dimensional) variable and the microscopic variable. This system is nonlocal. From this system we obtain by elimination an equation in the macroscopic variable which is local, but in contrast with usual results, the operator depends on the right-hand side of the equations. We also obtain a corrector result, i.e. an approximation of the gradients of the solutions in the strong topology of the space L^p$L^p$ in which the monotone operators are defined.     

Multiplicity of solutions for a class of elliptic systems with critical Sobolev exponent

In this paper we consider systems of p-q-Laplacian elliptic equations with critical Sobolev exponent. The existence and multiplicity results of solutions are obtained by a limit index method.     

On the structural stability of the Euler-Voigt and Navier-Stokes-Voigt models

We consider the Euler-Voigt equations and the Navier-Stokes-Voigt equations, which are obtained by an inviscid α-regularization from the corresponding equations. The main result we show is the structural stability of the system in terms of the variations of both viscosity and regularization parameters.     

Convergence rate of numerical solutions to SFDEs with jumps

In this paper, we are interested in numerical solutions of stochastic functional differential equations with jumps. Under a global Lipschitz condition, we show that the pth-moment convergence of Euler-Maruyama numerical solutions to stochastic functional differential equations with jumps has order 1/p for any p≥2. This is significantly different from the case of stochastic functional differential equations without jumps, where the order is 1/2 for any p≥2. It is therefore best to use the mean-square convergence for stochastic functional differential equations with jumps. Moreover, under a local Lipschitz condition, we reveal that the order of mean-square convergence is close to 1/2, provided that local Lipschitz constants, valid on balls of radius j, do not grow faster than logj.     

Viscous limits for piecewise smooth solutions of the p-system

We study the zero-dissipation problem for a one-dimensional model system for the isentropic flow of a compressible viscous gas, the so-called p-system with viscosity. When the solution of the inviscid problem is piecewise smooth and having finitely many noninteracting shocks satisfying the entropy condition, there exists unique solution to the viscous problem which converges to the given inviscid solution away from shock discontinuities at a rate of order ε as the viscosity coefficient ε goes to zero. The proof is given by a matched asymptotic analysis and an elementary energy method. And we do not need the smallness condition on the shock strength.     

On the technique for passing to the limit in nonlinear elliptic equations

We consider the problem of passing to the limit in a sequence of nonlinear elliptic problems. The “limit” equation is known in advance, but it has a nonclassical structure; namely, it contains the p-Laplacian with variable exponent p = p(x). Such equations typically exhibit a special kind of nonuniqueness, known as the Lavrent’ev effect, and this is what makes passing to the limit nontrivial. Equations involving the p(x)-Laplacian occur in many problems of mathematical physics. Some applications are included in the present paper. In particular, we suggest an approach to the solvability analysis of a well-known coupled system in non-Newtonian hydrodynamics (“stationary thermo-rheological viscous flows”) without resorting to any smallness conditions.     

Optimal spectral theory of the linear Boltzmann equation

We give optimal compactness results in L^p spaces ( 1<p<∞) related to spectral theory of general neutron transport equations on spatial domains with finite Lebesgue measure.