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.     

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.     

Existence of global L∞ solutions to a generalized n × n hyperbolic system of LeRoux type

In this article, we give the existence of global L^∞ bounded entropy solutions to the Cauchy problem of a generalized n × n hyperbolic system of LeRoux type. The main difficulty lies in establishing some compactness estimates of the viscosity solutions because the system has been generalized from 2 × 2 to n × n and more linearly degenerate characteristic fields emerged, and the emergence of singularity in the region {v₁=0} is another difficulty. We obtain the existence of the global weak solutions using the compensated compactness method coupled with the construction of entropy-entropy flux and BV estimates on viscous solutions.     

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.     

The convergence of non-Newtonian fluids to Navier-Stokes equations

In this paper, the convergence of solutions for incompressible dipolar viscous non-Newtonian fluids is investigated. We obtain the conclusion that the solutions of non-Newtonian fluids converge to the solutions of Navier-Stokes equations in the sense of L²-norm (resp. H¹-norm), as the viscosities tend to zero and the initial data belong to H¹(Ω) (resp. H²(Ω)). Moreover, we obtain L^∞-norm convergence of solutions if the initial data belong to H²(Ω).     

Global existence of solutions for one-dimensional compressible navier-stokes equations in the half space

We prove the existence of global solutions to the initial-boundary-value problem on the half space R＋ for a one-dimensional viscous ideal polytropic gas. Some suitable assumptions are made to guarantee the existence of smooth solutions. Employing the L2- energy estimate, we prove that the impermeable problem has a unique global solutionis.     

Multidimensional viscous shocks II: The small viscosity limit

In this paper we prove the existence of curved, multidimensional viscous shocks and also justify the small‐viscosity limit. Starting with a curved, multidimensional (inviscid) shock solution to a system of hyperbolic conservation laws, we show that the shock can be obtained as a small‐viscosity limit of solutions to an associated parabolic problem (viscous shocks). The two main hypotheses are a natural Evans function assumption on the viscous profile, together with a restriction on how much the shock can deviate from flatness. The main tools are a conjugation lemma that removes x_N/? dependence from the linearization of the parabolic problem about the viscous profile, new degenerate Kreiss‐type symmetrizers used to prove an L² estimate for the linearized problem, and a finite‐regularity calculus of semiclassical and mixed type (classical‐semiclassical) pseudodifferential operators. © 2003 Wiley Periodicals, Inc.     

Smooth interface in a two-component Stokes flow

The problem considered is that of evolution of the free boundary Γ(t) separating two immiscible viscous fluids with different constant densities and viscosities. The motion is described by the Stokes equations driven by the gravity force. We prove the existence of classical solutions for small timet and establish that the free boundary Γ(t)∈C ^l+2 (l>0 is an arbitrary non-integer number)     

On a viscous Hamilton-Jacobi equation with an unbounded potential term

We present comparison, uniqueness and existence results for unbounded solutions of a viscous Hamilton-Jacobi or eikonal equation. The equation includes an unbounded potential term V(x) subject to a quadratic upper bound. The results are obtained through a tailor-made change of variables in combination with the Hopf-Cole transformation. An integral representation formula for the solution of the Cauchy problem is derived in the case where V(x)=ω²|x|²/2.     

Fujita phenomena in nonlinear pseudo-parabolic system

This paper deals with the Cauchy problem to the nonlinear pseudo-parabolic system ut-△u-αut=vp,vt-△v-αvt=uqwith p,q 1 and pq1,where the viscous terms of third order are included.We first find the critical Fujita exponent,and then determine the second critical exponent to characterize the critical space-decay rate of initial data in the co-existence region of global and non-global solutions.Moreover,time-decay profiles are obtained for the global solutions.It can be found that,diferent from those for the situations of general semilinear heat systems,we have to use distinctive techniques to treat the influence from the viscous terms of the highest order.To fix the non-global solutions,we exploit the test function method,instead of the general Kaplan method for heat systems.To obtain the global solutions,we apply the Lp-Lq technique to establish some uniform Lmtime-decay estimates.In particular,under a suitable classification for the nonlinear parameters and the initial data,various Lmtime-decay estimates in the procedure enable us to arrive at the time-decay profiles of solutions to the system.It is mentioned that the general scaling method for parabolic problems relies heavily on regularizing efect to establish the compactness of approximating solutions,which cannot be directly realized here due to absence of the smooth efect in the pseudo-parabolic system.     

A singular Monge-Ampère equation on unbounded domains

In this paper, we study the Dirichlet problem for a singular Monge-Amp`ere type equation on unbounded domains. For a few special kinds of unbounded convex domains, we find the explicit formulas of the solutions to the problem. For general unbounded convex domain ?, we prove the existence for solutions to the problem in the space C∞(?) ∩ C(?). We also obtain the local C1/2-estimate up to the ?? and the estimate for the lower bound of the solutions.     

The asymptotic behavior of globally smooth solutions of the multidimensional isentropic hydrodynamic model for semiconductors

In this paper we study the asymptotic behavior of globally smooth solutions of the Cauchy problem for the multidimensional isentropic hydrodynamic model for semiconductors in R^d. We prove that smooth solutions (close to equilibrium) of the problem converge to a stationary solution exponentially fast as t→+∞.     

Construction of continuous solutions and stability for the polynomial-like iterative equation

Most of known results such as existence, uniqueness and stability for polynomial-like iterative equations were given under the assumption that the coefficient of the first order iteration term does not vanish. The existence with a non-zero leading coefficient was therefore raised as an open problem. It was positively answered for local C¹ solutions later. In this paper this problem is answered further by constructing C⁰ solutions. Moreover, we discuss the stability of those C⁰ solutions, which consequently implies a result of the stability for iterative roots.     

Global solutions and self-similar solutions of semilinear parabolic equations with nonlinear gradient terms

In this paper we consider the Cauchy problem of semilinear parabolic equations with nonlinear gradient terms a(x)|u|^q−1u|∇u|^p. We prove the existence of global solutions and self-similar solutions for small initial data. Moreover, for a class of initial data we show that the global solutions behave asymptotically like self-similar solutions as t→∞.     

Nano boundary layers over stretching surfaces

In this paper, we present similarity solutions for the nano boundary layer flows with Navier boundary condition. We consider viscous flows over a two-dimensional stretching surface and an axisymmetric stretching surface. The resulting nonlinear ordinary differential equations are solved analytically by the Homotopy Analysis Method. Numerical solutions are obtained by using a boundary value problem solver, and are shown to agree well with the analytical solutions. The effects of the slip parameter K and the suction parameter s on the fluid velocity and on the tangential stress are investigated and discussed. As expected, we find that for such fluid flows at nano scales, the shear stress at the wall decreases (in an absolute sense) with an increase in the slip parameter K.     

On the Navier–Stokes flows for heat-conducting fluids with mixed boundary conditions

We consider a coupled model for steady flows of viscous incompressible heat-conducting fluids with temperature dependent material coefficients in a fixed three-dimensional open cylindrical channel. We introduce the Banach spaces X and Y to be the space of possible solutions of this problem and the space of its data, respectively. We show that the corresponding operator of the problem acting between X and Y is Fréchet differentiable. Applying the local diffeomorphism theorem we get the local solvability results for a variational formulation.     

Global solutions of the Navier-Stokes equations for compressible flow with density-dependent viscosity and discontinuous initial data

In this paper, we study the evolutions of the interfaces between the gas and the vacuum for viscous one-dimensional isentropic gas motions. We prove the global existence and uniqueness for discontinuous solutions of the Navier-Stokes equations for compressible flow with density-dependent viscosity coefficient. Precisely, the viscosity coefficient μ is proportional to ρ^θ with 0<θ<1. Specifically, we require that the initial density be piecewise smooth with arbitrarily large jump discontinuities, bounded above and below away from zero, in the interior of gas. We show that the discontinuities in the density persist for all time, and give a decay result for the density as t→+∞.     

Uniqueness and structure of solutions to the Dirichlet problem for an elliptic system

In this paper, we consider the Dirichlet problem for an elliptic system on a ball in R². By investigating the properties for the corresponding linearized equations of solutions, and adopting the Pohozaev identity and Implicit Function Theorem, we show the uniqueness and the structure of solutions.