Existence in the Large for Pressure-Gradient System

In this paper, the authors use Glimm scheme to study the global existence of BV solutions to Cauchy problem of the pressure-gradient system with large initial data. To this end, some important properties of the shock curves of the pressure-gradient system in the Riemann invariant coordinate system and verify that the shock curves satisfy Diperna’s conditions (see [Diperna, R. J., Existence in the large for quasilinear hyperbolic conservation laws, Arch. Ration. Mech. Anal., 52(3), 1973, 244–257]) are studied. Then they construct the approximate solution sequence through Glimm scheme. By establishing accurate local interaction estimates, they prove the boundedness of the approximate solution sequence and its total variation.     

Finding all solutions of nonlinearly constrained systems of equations

A new approach is proposed for finding all-feasible solutions for certain classes of nonlinearly constrained systems of equations. By introducing slack variables, the initial problem is transformed into a global optimization problem (P) whose multiple global minimum solutions with a zero objective value (if any) correspond to all solutions of the initial constrained system of equalities. All-globally optimal points of (P) are then localized within a set of arbitrarily small disjoint rectangles. This is based on a branch and bound type global optimization algorithm which attains finite-convergence to each of the multiple global minima of (P) through the successive refinement of a convex relaxation of the feasible region and the subsequent solution of a series of nonlinear convex optimization problems. Based on the form of the participating functions, a number of techniques for constructing this convex relaxation are proposed. By taking advantage of the properties of products of univariate functions, customized convex lower bounding functions are introduced for a large number of expressions that are or can be transformed into products of univariate functions. Alternative convex relaxation procedures involve either the difference of two convex functions employed in BB [23] or the exponential variable transformation based underestimators employed for generalized geometric programming problems [24]. The proposed approach is illustrated with several test problems. For some of these problems additional solutions are identified that existing methods failed to locate.     

Qualitative Analysis and Travelling Wave Solutions for the Generalized Pochhammer–Chree Equation with a Dissipation Term

The purpose of this paper is to reveal the influence of dissipation on travelling wave solutions of the generalized Pochhammer–Chree equation with a dissipation term, and provides travelling wave solutions for this equation. Applying the theory of planar dynamical systems, we obtain ten global phase portraits of the dynamic system corresponding to this equation under various parameter conditions. Moreover, we present the relations between the properties of travelling wave solutions and the dissipation coefficient r of this equation. We find that a bounded travelling wave solution appears as a bell profile solitary wave solution or a periodic travelling wave solution when r= 0; a bounded travelling wave solution appears as a kink profile solitary wave solution when |r| > 0 is large; a bounded travelling wave solution appears as a damped oscillatory solution when |r| > 0 is small. Further, by using undetermined coefficient method, we get all possible bell profile solitary wave solutions and approximate damped oscillatory solutions for this equation. Error estimates indicate that the approximate solutions are meaningful.     

Applications of the Differential Calculus to Nonlinear Elliptic Operators with Discontinuous Coefficients

The paper concerns Dirichlet’s problem for second order quasilinear non-divergence form elliptic equations with discontinuous coefficients. We start with suitable structure, growth, and regularity conditions ensuring solvability of the problem under consideration. Fixing then a solution u ₀ such that the linearized at u ₀ problem is non-degenerate, we apply the Implicit Function Theorem. As a result we get that for all small perturbations of the coefficients there exists exactly one solution u ≈ u ₀ which depends smoothly (in W ^2,p with p larger than the space dimension) on the data. For that, no structure and growth conditions are needed and the perturbations of the coefficients can be general L ^∞-functions of the space variable x. Moreover, we show that the Newton Iteration Procedure can be applied in order to obtain a sequence of approximate (in W ^2,p ) solutions for u ₀.     

On theLU factorization ofM-matrices

Summary In this paper, we give in Theorem 1 a characterization, based on graph theory, of when anM-matrixA admits anLU factorization intoM-matrices, whereL is a nonsingular lower triangularM-matrix andU is an upper triangularM-matrix. This result generalizes earlier factorization results of Fiedler and Pták (1962) and Kuo (1977). As a consequence of Theorem 1, we show in Theorem 3 that the conditionx ^T A0 ^T for somex>0, for anM-matrixA, is both necessary and sufficient forPAP ^T to admit such anLU factorization for everyn×n permutation matrixP. This latter result extends recent work of Funderlic and Plemmons (1981). Finally, Theorem 1 is extended in Theorem 5 to give a characterization, similarly based on graph theory, of when anM-matrixA admits anLU factorization intoM-matrices.Dedicated to Professor Ky Fan on his sixty-seventh birthday, September 19, 1981.Research supported in part by the Air Force Office of Scientific Research, and by the Department of Energy     

A global Inverse Map Theorem and biLipschitz maps in the Heisenberg group

Abstract We prove a global Inverse Map Theorem for a map f from the Heisenberg group into itself, provided the Pansu differential of f is continuous, non singular and satisfies some growth conditions at infinity. An estimate for the Lipschitz constant (with respect to the Carnot–Carathéodory distance in ) of a continuously Pansu differentiable map is included. This gives a characterization of (continuously Pansu differentiable) globally biLipscitz deformations of in term of a pointwise estimate of their differential. Keywords: Inverse problem, Heisenberg group     

Periodic and almost periodic solutions of conservation laws: Global existence and decay

In this paper we survey recent results on the decay of periodic and almost periodic solutions of conservation laws. We also recall some recent results on the global existence of periodic solutions of conservation laws systems which lie inBV _loc and are constructed through Glimm scheme. The latter motivates a discussion on a possible strategy for solving the open problem of the global existence of periodic solutions of the Euler equations for nonisentropic gas dynamics. We base our decay analysis on a general result about space-time functions which are almost periodic in the space variable, established here for the first time. This result is an abstract version of Theorem 2.1 in [31], which in turn is an extention of the combined result given by Theorems 3.1–3.2 in [9].     

On the initial value problem for the Vlasov-Poisson-Fokker-Planck system with initial data in Lp spaces

In this paper the global existence of weak solutions for the Vlasov-Poisson-Fokker-Planck equations in three dimensions is proved with an L¹ ∩ L^p initial data. Also, the global existence of weak solutions in four dimensions with small initial data is studied. A convergence of the solutions is obtained to those built by E. Horst and R. Hunze when the Fokker-Planck term vanishes. In order to obtain the a priori necessary estimates a sequence of approximate problems is introduced. This sequence is obtained starting from a non-linear regulation of the problem together with a linearization via a time retarded mollification of the non-linear term. The a priori bounds are reached by means of the control of the kinetic energy in the approximate sequence of problems. Then, the proof is completed obtaining the equicontinuity properties which allow to pass to the limit.     

Qualitative analysis and solutions of bounded traveling wave for B-BBM equation

In this paper, we apply the theory of planar dynamical systems to carry out qualitative analysis for the dynamical system corresponding to B-BBM equation, and obtain global phase portraits under various parameter conditions. Then, the relations between the behaviors of bounded traveling wave solutions and the dissipation coeffiicient μ are investigated. We find that a bounded traveling wave solution appears as a kink profile solitary wave solution when μ is more than the critical value obtained in this paper, while a bounded traveling wave solution appears as a damped oscillatory solution when μ is less than it. Furthermore, we explain the solitary wave solutions obtained in previous literature, and point out their positions in global phase portraits. In the meantime, approximate damped oscillatory solutions are given by means of undetermined coefficients method. Finally, based on integral equations that reflect the relations between the approximate damped oscillatory solutions and the implicit exact damped oscillatory solutions, error estimates for the approximate solutions are presented.     

First Order Differential Operators Associated to the Cauchy—Riemann Operator in the Plane

The article deals with initial value problems of type δw/δt = Fw, w(0, ·) = φ where t is the time and F is a linear first order operator acting in the z = x ? iy-plane. In view of the classical Cauchy-Kovalevkaya Theorem, the initial value problem is solvable provided F has holomorphic coefficients and the initial function is holomorphic. On the other hand, the Lewy example [H. Lewy (1957). An example of a smooth linear partial differential equation without solution. Ann. of Math., 66, 155–158.] shows that there are equations of the above form with infinitely differentiable coefficients not having any solutions. The article in hand constructs, conversely, all linear operators F for which the initial value problem with an arbitrary holomorphic initial function is always solvable. In particular, we shall see that there are equations of that type whose coefficients are only continuous.     

Infinitely many solutions of dirichlet problem for <Emphasis Type="Italic">p</Emphasis>-mean curvature operator

The existence of infinitely many solutions of the following Dirichlet problem for p-mean curvature operator: is considered, where Θ is a bounded domain in R ⁿ (n>p>1) with smooth boundary ∂Θ. Under some natural conditions together with some conditions weaker than (AR) condition, we prove that the above problem has infinitely many solutions by a symmetric version of the Mountain Pass Theorem if . Supported by the National Natural Science Foundation of China (10171032) and the Guangdong Provincial Natural Science Foundation (011606).     

On the periodic solutions of certain class of seventh-order differential equations

The aim of this paper is to investigate sufficient conditions (Theorem 1) for the nonexistence of nontrivial periodic solutions of equation (1.1) withp ≡ 0 and (Theorem 2) for the existence of periodic solutions of equation (1.1).     

Approximate damped oscillatory solutions for compound KdV-Burgers equation and their error estimates

In this paper,we focus on studying approximate solutions of damped oscillatory solutions of the compound KdV-Burgers equation and their error estimates.We employ the theory of planar dynamical systems to study traveling wave solutions of the compound KdV-Burgers equation.We obtain some global phase portraits under different parameter conditions as well as the existence of bounded traveling wave solutions.Furthermore,we investigate the relations between the behavior of bounded traveling wave solutions and the dissipation coefficient r of the equation.We obtain two critical values of r,and find that a bounded traveling wave appears as a kink profile solitary wave if |r| is greater than or equal to some critical value,while it appears as a damped oscillatory wave if |r| is less than some critical value.By means of analysis and the undetermined coefficients method,we find that the compound KdV-Burgers equation only has three kinds of bell profile solitary wave solutions without dissipation.Based on the above discussions and according to the evolution relations of orbits in the global phase portraits,we obtain all approximate damped oscillatory solutions by using the undetermined coefficients method.Finally,using the homogenization principle,we establish the integral equations reflecting the relations between exact solutions and approximate solutions of damped oscillatory solutions.Moreover,we also give the error estimates for these approximate solutions.     

On the global weak solutions for a modified two‐component Camassa‐Holm equation

In this paper, we investigate the existence of global weak solutions to the Cauchy problem of a modified two‐component Camassa‐Holm equation with the initial data satisfying lim_{x → ±∞}u₀(x) = u_±. By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution for the system under the assumption u_? ≤ u₊. The global weak solution is obtained as a limit of approximation solutions. The key elements in our analysis are the Helly theorem and the estimation of energy for approximation solutions in $H^1(\mathbb {R})\times H^1(\mathbb {R})In this paper, we investigate the existence of global weak solutions to the Cauchy problem of a modified two‐component Camassa‐Holm equation with the initial data satisfying lim_{x → ±∞}u₀(x) = u_±. By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution for the system under the assumption u_? ≤ u₊. The global weak solution is obtained as a limit of approximation solutions. The key elements in our analysis are the Helly theorem and the estimation of energy for approximation solutions in $H^1(\mathbb {R})\times H^1(\mathbb {R})$ and some a priori estimates on the first‐order derivatives of approximation solutions.     

Nonexistence of Global Solutions to Nonlinear Stochastic Wave Equations in Mean L p -Norm

This article is concerned with explosive solutions of the initial-boundary problem for a class of nonlinear stochastic wave equations in a domain 𝒟 ? ? ^d . Under appropriate conditions on the initial data, the nonlinear term and the noise intensity, it is proved in Theorem 3.4 that there cannot exist a global solution and the local solution will blow up at a finite time in the mean L ^p  ? norm for p ≥ 1. An example is given to show the application of this theorem.     

Initial-boundary problem for the 1-D Euler-Boltzmann equations in radiation hydrodynamics

We study the initial-boundary value problem for the one dimensional EulerBoltzmann equation with reflection boundary condition. For initial data with small total variation, we use a modified Glimm scheme to construct the global approximate solutions(U_(△t,d), I_(△t,d)) and prove that there is a subsequence of the approximate solutions which is convergent to the global solution.     

Global weak solutions for a periodic two‐component μ‐Camassa–Holm system

In this paper, we consider the global existence of weak solutions for a two‐component μ‐Camassa–Holm system in the periodic setting. Global existence for strong solutions to the system with smooth approximate initial value is derived. Then, we show that the limit of approximate solutions is a global‐in‐time weak solution of the two‐component μ‐Camassa–Holm system. Copyright © 2013 John Wiley & Sons, Ltd.     

Approximation of infinite matrices by matricial Haar polynomials

The main goal of this paper is to extend the approximation theorem of contiuous functions by Haar polynomials (see Theorem A) to infinite matrices (see Theorem C). The extension to the matricial framework will be based on the one hand on the remark that periodic functions which belong toL ^∞ (T) may be one-to-one identified with Toeplitz matrices fromB(l ₂) (see Theorem 0) and on the other hand on some notions given in the paper. We mention for instance:ms—a unital commutative subalgebra ofl ^∞,C(l ₂) the matricial analogue of the space of all continuous periodic functionsC(T), the matricial Haar polynomials, etc. In Section 1 we present some results concerning the spacems—a concept important for this generalization, the proof of the main theorem being given in the second section. Partially supported by EUROMMAT ICA1-CT-2000-70022. Partially supported by V-Stabi-RUM/1022123. Partially supported by EUROMMAT ICA1-CT-2000-70022 and V-Stabi-RUM/1022123.     

An existence theorem for the unmodified Vlasov Equation

The initial value problem for the modified Vlasov equation with a mollification parameter δ > 0, as introduced by Batt, has a unique global solution in the weak sense whenever f₀ ε L¹ and f₀ ≧ 0 λ-a.e. Assuming boundedness of f₀ and boundedness of the kinetic energy, it is shown that, as δ → 0, there are subsequences δ_n → 0 such that the corresponding solutions converge weakly in the measure-theoretical sense. The limits are shown to be global weak solutions of the initial value problem for Vlasov's equation, and these solutions are seen to be weakly continuous with respect to t. For the plasma physical case, boundedness of the kinetic energy is a consequence of energy conservation.