The （G＇/G, 1/G）-expansion method and its application to travelling wave solutions of the Zakharov equations

The （G＇/G, 1/G）-expansion method for finding exact travelling wave solutions of nonlinear evolution equations, which can be thought of as an extension of the （G＇/G）-expansion method proposed recently, is presented. By using this method abundant travelling wave so- lutions with arbitrary parameters of the Zakharov equations are successfully obtained. When the parameters are replaced by special values, the well-known solitary wave solutions of the equations are rediscovered from the travelling waves.     

Vanishing viscosity limit of the Navier‐Stokes equations to the euler equations for compressible fluid flow

We establish the vanishing viscosity limit of the Navier‐Stokes equations to the isentropic Euler equations for one‐dimensional compressible fluid flow. For the Navier‐Stokes equations, there exist no natural invariant regions for the equations with the real physical viscosity term so that the uniform sup‐norm of solutions with respect to the physical viscosity coefficient may not be directly controllable. Furthermore, convex entropy‐entropy flux pairs may not produce signed entropy dissipation measures. To overcome these difficulties, we first develop uniform energy‐type estimates with respect to the viscosity coefficient for solutions of the Navier‐Stokes equations and establish the existence of measure‐valued solutions of the isentropic Euler equations generated by the Navier‐Stokes equations. Based on the uniform energy‐type estimates and the features of the isentropic Euler equations, we establish that the entropy dissipation measures of the solutions of the Navier‐Stokes equations for weak entropy‐entropy flux pairs, generated by compactly supported C² test functions, are confined in a compact set in H^?1, which leads to the existence of measure‐valued solutions that are confined by the Tartar‐Murat commutator relation. A careful characterization of the unbounded support of the measure‐valued solution confined by the commutator relation yields the reduction of the measurevalued solution to a Dirac mass, which leads to the convergence of solutions of the Navier‐Stokes equations to a finite‐energy entropy solution of the isentropic Euler equations with finite‐energy initial data, relative to the different end‐states at infinity. © 2010 Wiley Periodicals, Inc.     

Mild solutions for abstract fractional differential equations

We propose a unified functional analytic approach to derive a variation of constants formula for a wide class of fractional differential equations using results on (a,?k)-regularized families of bounded and linear operators, which covers as particular cases the theories of C ₀-semigroups and cosine families. Using this approach we study the existence of mild solutions to fractional differential equation with nonlocal conditions. We also investigate the asymptotic behaviour of mild solutions to abstract composite fractional relaxation equations. We include in our analysis the Basset and Bagley–Torvik equations.     

Convergence of the Lax–Friedrichs scheme for isothermal gas dynamics with semiconductor devices

In this paper we study the existence of global solutions to the Euler equations of compressible isothermal gas dynamics with semiconductor devices. We construct the approximate solutions by Lax–Friedrichs scheme. The convergence and consistency are obtained by using the compensated compactness framework for γ = 1. The global entropy solutions in L^∞ are obtained. We deal with the initial data containing unbounded velocity which is different from the isentropic case. Received: November 18, 2003     

Stokes and Navier–Stokes equations with nonhomogeneous boundary conditions

In this paper, we study the existence and regularity of solutions to the Stokes and Oseen equations with nonhomogeneous Dirichlet boundary conditions with low regularity. We consider boundary conditions for which the normal component is not equal to zero. We rewrite the Stokes and the Oseen equations in the form of a system of two equations. The first one is an evolution equation satisfied by Pu, the projection of the solution on the Stokes space – the space of divergence free vector fields with a normal trace equal to zero – and the second one is a quasi-stationary elliptic equation satisfied by (I−P)u, the projection of the solution on the orthogonal complement of the Stokes space. We establish optimal regularity results for Pu and (I−P)u. We also study the existence of weak solutions to the three-dimensional instationary Navier–Stokes equations for more regular data, but without any smallness assumption on the initial and boundary conditions.     

The generalized Riemann problem for a scalar Chapman–Jouguet combustion model

The generalized Riemann problem for a scalar Chapman–Jouguet combustion model in a neighborhood of the origin (t > 0) on the (x, t) plane is studied. Under the entropy conditions, we obtain the solutions constructively. It is found that, for some cases, the perturbed Riemann solutions are essentially different from the corresponding Riemann solutions. The perturbation may transform a combustion wave CJDT into SDT in the neighborhood of the origin. Especially, it can be observed that burning happens although the corresponding Riemann solution doesn’t contain combustion waves, which exhibits the instability for unburnt states. This work is supported by NSFC 10671120     

The dressing method and nonlocal Riemann-Hilbert problems

Summary We consider equations in 2+1 solvable in terms of a nonlocal Riemann-Hilbert problem and show that for such an equation there exists a unified dressing method which yields: (i) a Lax pair suitable for obtaining solutions that are perturbations of an arbitrary exact solution of the given equation; (ii) certain integrable generalizations of the given equation. Using this generalized dressing method large classes of solutions of these equations, including dromions and line dromions, can be obtained. The method is illustrated by using theN-wave interactions, the Davey-Stewartson I, and the Kadomtsev-Petviashvili I equations. We also show that a careful application of the usual dressing method yields a certain generalization of theN-wave interactions.     

On the theory of divergence-measure fields and its applications

Divergence-measure fields are extended vector fields, including vector fields inL ^p and vector-valued Radon measures, whose divergences are Radon measures. Such fields arise naturally in the study of entropy solutions of nonlinear conservation laws and other areas. In this paper, a theory of divergence-measure fields is presented and analyzed, in which normal traces, a generalized Gauss-Green theorem, and product rules, among others, are established. Some applications of this theory to several nonlinear problems in conservation laws and related areas are discussed. In particular, with the aid of this theory, we prove the stability of Riemann solutions, which may contain rarefaction waves, contact discontinuities, and/or vacuum states, in the class of entropy solutions of the Euler equations for gas dynamics.Dedicated to Constantine Dafermos on his 60^th birthday     

Thermodynamics from the differential geometry standpoint

We study the differential-geometric structure of the space of thermodynamic states in equilibrium thermodynamics. We demonstrate that this space is a foliation of codimension two and find variables in which the foliation fibers are flat. We show that we can introduce a symplectic structure on this space: the external derivative of the 1-form of the heat source, which has the form of the skew-symmetric product dT 蝃 dS in the found variables. The entropy S then plays the role of the Lagrange function (or Hamiltonian) in mechanics, completely determining the thermodynamic system. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 1, pp. 141–148, October, 2008.     

On solutions of stochastic differential equations with drift

Summary We study stochastic differential equations of the formdX _t=(X _t)dM_t+b(X_t)d_t whereM is a continuous local martingale and <M> stands for its quadratic variation process. The conditions introduced by Engelbert and Schmidt, which ensure the existence and uniqueness in law of solutions of SDE's driven by the Wiener process without drift (or with generalized drift) are shown to be no longer valid.     

Discrete approximations of BV solutions to doubly nonlinear degenerate parabolic equations

Summary. In this paper we present and analyse certain discrete approximations of solutions to scalar, doubly nonlinear degenerate, parabolic problems of the form under the very general structural condition . To mention only a few examples: the heat equation, the porous medium equation, the two-phase flow equation, hyperbolic conservation laws and equations arising from the theory of non-Newtonian fluids are all special cases of (P). Since the diffusion terms a(s) and b(s) are allowed to degenerate on intervals, shock waves will in general appear in the solutions of (P). Furthermore, weak solutions are not uniquely determined by their data. For these reasons we work within the framework of weak solutions that are of bounded variation (in space and time) and, in addition, satisfy an entropy condition. The well-posedness of the Cauchy problem (P) in this class of so-called BV entropy weak solutions follows from a work of Yin [18]. The discrete approximations are shown to converge to the unique BV entropy weak solution of (P). Received November 10, 1998 / Revised version received June 10, 1999 / Published online June 8, 2000     

Entropy Dissipation Methods for Degenerate ParabolicProblems and Generalized Sobolev Inequalities

We analyse the large-time asymptotics of quasilinear (possibly) degenerate parabolic systems in three cases: 1) scalar problems with confinement by a uniformly convex potential, 2) unconfined scalar equations and 3) unconfined systems. In particular we are interested in the rate of decay to equilibrium or self-similar solutions. The main analytical tool is based on the analysis of the entropy dissipation. In the scalar case this is done by proving decay of the entropy dissipation rate and bootstrapping back to show convergence of the relative entropy to zero. As by-product, this approach gives generalized Sobolev-inequalities, which interpolate between the Gross logarithmic Sobolev inequality and the classical Sobolev inequality. The time decay of the solutions of the degenerate systems is analyzed by means of a generalisation of the Nash inequality. Porous media, fast diffusion, p-Laplace and energy transport systems are included in the considered class of problems. A generalized Csiszár–Kullback inequality allows for an estimation of the decay to equilibrium in terms of the relative entropy.     

Studies on Nonhomogeneous Multi-point BVPs of Difference Equations with One-Dimensional p-Laplacian

This article deals with a discrete type multi-point BVP of difference equations. The sufficient conditions to guarantee the existence of at least three positive solutions are established. An example is presented to illustrate the main results. It is the purpose of this paper to show that the approach to get positive solutions of BVPs by using multifixed-point theorems can be extended to treat nonhomogeneous BVPs. The emphasis is put on the nonlinear term f involved with the first order delta operators Δx(n) and Δx(n + 1). The difference concerned is a implicit difference equation.     

A steady-state system in non-equilibrium thermodynamics including thermal and electrical effects

The steady-state equations for a charged gas or fluid consisting of several components, exposed to an electric field, are considered. These equations form a system of strongly coupled, quasilinear elliptic equations which in some situations can be derived from the Boltzmann equation. The model uses the duality between the thermodynamic fluxes and the thermodynamic forces. Physically motivated mixed Dirichlet–Neumann boundary conditions are prescribed. The existence of generalized solutions is proven. The key of the proof is a transformation of the problem by using the entropic variables, or electro-chemical potentials, which symmetrize the equations. The uniqueness of weak solutions is shown under the assumption that the boundary data are not far from the thermal equilibrium. A general uniqueness result cannot be expected for physical reasons. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Algebraic stability and the existence of solutions of implicit Runge-Kutta equations

An implicit Runge-Kutta method, applied to an initial value problem, gives systems of algebraic equations. Under natural assumptions concerning the differential system, there are known conditions on the method which guarantee that the algebraic equations have unique solutions. It is shown that these conditions are closely related to the requirement that the method be (k(l),l)-algebraically stable on an interval [0,).     

Approximation of entropy on hyperbolic sets for one-dimensional maps and their multidimensional perturbations

We consider piecewise monotone (not necessarily, strictly) piecewise C ² maps on the interval with positive topological entropy. For such a map f we prove that its topological entropy h _top(f) can be approximated (with any required accuracy) by restriction on a compact strictly f-invariant hyperbolic set disjoint from some neighborhood of prescribed set consisting of periodic attractors, nonhyperbolic intervals and endpoints of monotonicity intervals. By using this result we are able to generalize main theorem from [1] on chaotic behavior of multidimensional perturbations of solutions for difference equations which depend on two variables at nonperturbed value of parameter.     

Iterative orthogonal direction methods for Hermitian minimum norm solutions of two consistent matrix equations

The consistent conditions and the general expressions about the Hermitian solutions of the linear matrix equations AXB=C and (AX, XB)=(C, D) are studied in depth, where A, B, C and D are given matrices of suitable sizes. The Hermitian minimum F‐norm solutions are obtained for the matrix equations AXB=C and (AX, XB)=(C, D) by Moore–Penrose generalized inverse, respectively. For both matrix equations, we design iterative methods according to the fundamental idea of the classical conjugate direction method for the standard system of linear equations. Numerical results show that these iterative methods are feasible and effective in actual computations of the solutions of the above‐mentioned two matrix equations. Copyright © 2006 John Wiley & Sons, Ltd.     

Entropy Dissipation Methods for Degenerate ParabolicProblems and Generalized Sobolev Inequalities

We analyse the large-time asymptotics of quasilinear (possibly) degenerate parabolic systems in three cases: 1) scalar problems with confinement by a uniformly convex potential, 2) unconfined scalar equations and 3) unconfined systems. In particular we are interested in the rate of decay to equilibrium or self-similar solutions. The main analytical tool is based on the analysis of the entropy dissipation. In the scalar case this is done by proving decay of the entropy dissipation rate and bootstrapping back to show convergence of the relative entropy to zero. As by-product, this approach gives generalized Sobolev-inequalities, which interpolate between the Gross logarithmic Sobolev inequality and the classical Sobolev inequality. The time decay of the solutions of the degenerate systems is analyzed by means of a generalisation of the Nash inequality. Porous media, fast diffusion, p-Laplace and energy transport systems are included in the considered class of problems. A generalized Csiszár–Kullback inequality allows for an estimation of the decay to equilibrium in terms of the relative entropy. (Received 11 October 2000; in revised form 13 March 2001)     

Construction of Solutions and L^1-error Estimates of Viscous Methods for Scalar Conservation Laws with Boundary

This paper is concerned with an initial boundary value problem for strictly convex conservation laws whose weak entropy solution is in the piecewise smooth solution class consisting of finitely many discontinuities. By the structure of the weak entropy solution of the corresponding initial value problem and the boundary entropy condition developed by Bardos-Leroux Nedelec, we give a construction method to the weak entropy solution of the initial boundary value problem. Compared with the initial value problem, the weak entropy solution of the initial boundary value problem includes the following new interaction type： an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary. According to the structure and some global estimates of the weak entropy solution, we derive the global L^1-error estimate for viscous methods to this initial boundary value problem by using the matching travelling wave solutions method. If the inviscid solution includes the interaction that an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary, or the inviscid solution includes some shock wave which is tangent to the boundary, then the error of the viscosity solution to the inviscid solution is bounded by O（ε^1/2） in L^1-norm; otherwise, as in the initial value problem, the L^1-error bound is O（ε｜ In ε｜）.     

Solutions of Semilinear Differential Equations Related to Harmonic Functions

This is an attempt to establish a link between positive solutions of semilinear equations Lu=−ψ(u) and Lv=ψ(v) where L is a second order elliptic differential operator and ψ is a positive function. The equations were investigated separately by a number of authors. We try to link them via positive solutions of a linear equation Lu=0 (we call them L-harmonic functions). Let D be an arbitrary open subset of ^d and let (D), (D) and (D) stand for the sets of all positive solutions in D for three equations mentioned above. We establish a 1–1 correspondence between certain subclasses of these classes. Similar results are obtained also for the corresponding parabolic equations. A probabilistic interpretation in terms of a superdiffusion is given in [1].