ZERO DISSIPATION LIMIT TO A RIEMANN SOLUTION FOR THE COMPRESSIBLE NAVIER-STOKES SYSTEM OF GENERAL GAS

For the general gas including ideal polytropic gas, we study the zero dissipation limit problem of the full 1-D compressible Navier-Stokes equations toward the superposition of contact discontinuity and two rarefaction waves. In the case of both smooth and Riemann initial data, we show that if the solutions to the corresponding Euler system consist of the composite wave of two rarefaction wave and contact discontinuity, then there exist solutions to Navier-Stokes equations which converge to the Riemman solutions away from the initial layer with a decay rate in any fixed time interval as the viscosity and the heat-conductivity coefficients tend to zero. The proof is based on scaling arguments, the construction of the approximate profiles and delicate energy estimates. Notice that we have no need to restrict the strengths of the contact discontinuity and rarefaction waves to be small.     

ZERO DISSIPATION LIMIT OF THE COMPRESSIBLE HEAT-CONDUCTING NAVIER-STOKES EQUATIONS IN HE PRESENCE OF THE SHOCK

The zero dissipation limit of the compressible heat-conducting Navier-Stokes equations in the presence of the shock is investigated. It is shown that when the heat ε→ 0 (see (1.3)), if the solution of the corresponding Euler equations is piecewise smooth with shock wave satisfying the Lax entropy condition, then there exists a smooth solution to the Navier-Stokes equations, which converges to the piecewise smooth shock solution of the Euler equations away from the shock discontinuity at a rate of ε. The proof is given by a combination of the energy estimates and the matched asymptotic analysis introduced in [3].     

Inviscid Limit for Scalar Viscous Conservation Laws in Presence of Strong Shocks and Boundary Layers

In this paper, we study the inviscid limit problem for the scalar viscous conservation laws on half plane. We prove that if the solution of the corresponding inviscid equation on half plane is piecewise smooth with a single shock satisfying the entropy condition, then there exist solutions to the viscous conservation laws which converge to the inviscid solution away fromthe shock discontinuity and the boundary at a rate of ε^1 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.     

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.     

Stability of supersonic contact discontinuity for two-dimensional steady compressible Euler flows in a finite nozzle

In this paper, we study the stability of supersonic contact discontinuity for the two-dimensional steady compressible Euler flows in a finitely long nozzle of varying cross-sections. We formulate the problem as an initial–boundary value problem with the contact discontinuity as a free boundary. To deal with the free boundary value problem, we employ the Lagrangian transformation to straighten the contact discontinuity and then the free boundary value problem becomes a fixed boundary value problem. We develop an iteration scheme and establish some novel estimates of solutions for the first order of hyperbolic equations on a cornered domain. Finally, by using the inverse Lagrangian transformation and under the assumption that the incoming flows and the nozzle walls are smooth perturbations of the background state, we prove that the original free boundary problem admits a unique weak solution which is a small perturbation of the background state and the solution consists of two smooth supersonic flows separated by a smooth contact discontinuity.     

Gradient bounds for minimizers of free discontinuity problems related to cohesive zone models in fracture mechanics

In this note we consider a free discontinuity problem for a scalar function, whose energy depends also on the size of the jump. We prove that the gradient of every smooth local minimizer never exceeds a constant, determined only by the data of the problem.     

Asymptotic stability of contact waves for the compressible Navier–Stokes equations

In this article, we study the large-time asymptotic behaviour of contact wave for the Cauchy problem of one-dimensional compressible Navier–Stokes equations with zero viscosity. When the Riemann problem for the Euler system admits a contact discontinuity solution, we can construct a contact wave, which approximates the contact discontinuity on any finite-time interval for small heat conduction and then runs away from it for large time, and prove that it is nonlinearly stable provided that the strength of contact discontinuity and the perturbation of the initial data are suitably small.     

Detecting discontinuity points from spectral data with the quotient-difference (qd) algorithm

This paper introduces a new technique for the localization of discontinuity points from spectral data. Through this work, we will be able to detect discontinuity points of a 2π-periodic piecewise smooth function from its Fourier coefficients. This could be useful in detecting edges and reducing the effects of the Gibbs phenomenon which appears near discontinuities and affects signal restitution. Our approach consists in moving from a discontinuity point detection problem to a pole detection problem, then adapting the quotient-difference (qd) algorithm in order to detect those discontinuity points.     

A hybrid convex variational model for image restoration

We propose a new hybrid model for variational image restoration using an alternative diffusion switching non-quadratic function with a parameter. The parameter is chosen adaptively so as to minimize the smoothing near the edges and allow the diffusion to smooth away from the edges. This model belongs to a class of edge-preserving regularization methods proposed in the past, the ?-function formulation. This involves a minimizer to the associated energy functional. We study the existence and uniqueness of the energy functional of the model. Using real and synthetic images we show that the model is effective in image restoration.     

Convergence to the superposition of rarefaction waves and contact discontinuity for the 1-D compressible Navier–Stokes–Korteweg system

This paper is concerned with the vanishing capillarity–viscosity limit for the one-dimensional compressible Navier–Stokes–Korteweg system to the Riemann solution of the Euler system that consists of the supposition of two rarefaction waves and a contact discontinuity. It is shown that there exists a family of smooth solutions to the compressible Navier–Stokes–Korteweg system which converge to the Riemann solution away from the initial time t=0$t=0$ and the contact discontinuity located at x=0$x=0$, as the coefficients of capillarity, viscosity and heat conductivity tend to zero. Moreover, a uniform convergence rate in terms of the above physical parameters is also obtained. Here, the strengths of both the rarefaction waves and the contact discontinuity are not required to be small.     

Rotationally symmetric 1-harmonic maps from D 2 to S 2

We consider rotationally symmetric 1-harmonic maps from D ² to S ² subject to Dirichlet boundary conditions. We prove that the corresponding energy—a degenerate non-convex functional with linear growth—admits a unique minimizer, and that the minimizer is smooth in the bulk and continuously differentiable up to the boundary. We also show that, in contrast with 2-harmonic maps, a range of boundary data exists such that the energy admits more than one smooth critical point: more precisely, we prove that the corresponding Euler–Lagrange equation admits a unique (up to scaling and symmetries) global solution, which turns out to be oscillating, and we characterize the minimizer and the smooth critical points of the energy as the monotone, respectively non-monotone, branches of such solution. R. Dal Passo passed away on 8th August 2007. Endowed with great strength, creativity and humanity, Roberta has been an outstanding mathematician, an extraordinary teacher and a wonderful friend. Farewell, Roberta.     

MECHANISM FOR THE TRANSITION FROM A REGULAR REFLECTION TO A MACH REFLECTION OR A VON NEUMANN REFLECTION

In this paper,by taking into account the thickness of the incident shock as well as the influence of the boundary layer,we point out that even in a regular reflection there should be present a contact discontinuity.By using the smallest energy criterion,the inclined angle of this contact discontinuity can be determined for differen incident angle.Then,with this inclined contact discontinuity,together with the law of conservation of mass,the mechanism for the transition from a regular reflection to a Mach reflection or a von Neumann reflection becomes clear.The important roles played by the leftest point in the reflected shock polar are identified.     

Discontinuous Solutions of Linearized, Steady-State, Viscous, Compressible Flows

In thIs paper we prove the existence and uniqueness of the solution to a linearized, two dimensional, steady state, viscous, compressible Navier-Stokes equations in a strip. We obtain a regularity result and a new a priori estimate of the solution. We establish the discontinuity of the solution when the boundary data of pressure have a jump discontinuity. We also derive a formula for the discontinuous part and show that the remainder is smooth in the strip.     

Filtering in Legendre spectral methods

We discuss the impact of modal filtering in Legendre spectral methods, both on accuracy and stability. For the former, we derive sufficient conditions on the filter to recover high order accuracy away from points of discontinuity. Computational results confirm that less strict necessary conditions appear to be adequate. We proceed to discuss a instability mechanism in polynomial spectral methods and prove that filtering suffices to ensure stability. The results are illustrated by computational experiments.

     

Development of a two‐dimensional finite element model for pure advective equation

This article deals with a six‐parameter flux corrected transport (FCT) Taylor Galerkin finite element model for solving the pure convection equation that admits discontinuities. Modified equation analysis is conducted to optimize the scheme accuracy in the smooth flow. In the presence of discontinuity, the introduced free parameters are further controlled to yield a slower group velocity ahead of the discontinuity using the group velocity theory. In contrast, free parameters that result in a faster group velocity than the analytic phase velocity are chosen to reduce postdiscontinuity oscillations. The success in employing the FCT technique of Zalesak is to obtain a monotone solution and we apply the M‐matrix theory to achieve the goal. To validate the proposed finite element model, analytic tests, which are amenable to smooth as well as sharply varying solutions, are conducted. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 302–326, 2004     

Generic bifurcation of refracted systems

In this article we discuss some qualitative and geometric aspects of non-smooth dynamical systems theory. Our goal is to study the diagram bifurcation of typical singularities that occur generically in one parameter families of certain piecewise smooth vector fields named Refracted Systems. Such systems has a codimension-one submanifold as its discontinuity set.     

Partial compactness for Landau-Lifshitz maxwell equation in two-dimension

We study the partial regularity of weak solutions to the 2-dimensional LandauLifshitz equations coupled with time dependent Maxwell equations by Ginzburg-Landau type approximation. Outside an energy concentration set of locally finite 2-dimensional parabolic Hausdorff measure, we prove the uniform local C ∞ bounds for the approaching solutions and then extract a subsequence converging to a global weak solution of the Landau-Lifshitz-Maxwell equations which are smooth away from finitely many points.     

Iterative methods based on spline approximations to detect discontinuities from Fourier data

Recently, spline approximations have been proposed for the reconstruction of piecewise smooth functions from Fourier data. That approach makes possible to retrieve the functions from their Fourier coefficients for any given degree of accuracy when the discontinuity points are known. In this paper we present iterative methods based on those spline approximations, for several degrees, to find locations and amplitudes of the jumps of a piecewise smooth function, given its Fourier coefficients. We also present numerical experiments comparing with different previous approaches.     

Zero dissipation limit to rarefaction waves for the one-dimensional navier-stokes equations of compressible isentropic gases

We study the zero dissipation limit problem for the one-dimensional Navier-Stokes equations of compressible, isentropic gases in the case that the corresponding Euler equations have rarefaction wave solutions. We prove that the solutions of the Navier-Stokes equations with centered rarefaction wave data exist for all time, and converge to the centered rarefaction waves as the viscosity vanishes, uniformly away from the initial discontinuities. In the case that either the effects of initial layers are ignored or the rarefaction waves are smooth, we then obtain a rate of convergence which is valid uniformly for all time. Our method of proof consists of a scaling argument and elementary energy analysis, based on the underlying wave structure. © 1993 John Wiley & Sons, Inc.