Global Hölder continuity of weak solutions to quasilinear divergence form elliptic equations

We derive global Hölder regularity for the -weak solutions to the quasilinear, uniformly elliptic equation

div(aij(x,u)Dju+ai(x,u))+a(x,u,Du)=0     

A priori estimates for solutions of anisotropic elliptic equations

We prove universal, pointwise, a priori estimates for nonnegative solutions of anisotropic nonlinear elliptic equations.     

A priori estimates and existence of positive solutions for strongly nonlinear problems

In this paper we study the existence of positive solutions for a nonlinear Dirichlet problem involving the m-Laplacian. The nonlinearity considered depends on the first derivatives; in such case, variational methods cannot be applied. So, we make use of topological methods to prove the existence of solutions. We combine a blow-up argument and a Liouville-type theorem to obtain a priori estimates. Some Harnack-type inequalities which are needed in our reasonings are also proved.     

On the generalized self-similar singularities for the Euler and the Navier-Stokes equations

In this paper we study blow-up rates and the blow-up profiles of possible asymptotically self-similar singularities of the Euler and the Navier-Stokes equations, where the sense of convergence and self-similarity are considered in various generalized senses. We improve substantially, in particular, the previous nonexistence results of self-similar/asymptotically self-similar singularities. Generalization of the self-similar transforms is also considered, and by appropriate choice of the parameterized transform we obtain new a priori estimates for the Euler and the Navier-Stokes equations depending on a free parameter.     

Global existence for weak solutions of the Cauchy problem in a model of radiation hydrodynamics

We consider a model based on the Navier–Stokes–Fourier system coupled to a transport equation, recently proposed in order to describe the thermal effects in low Mach number radiative flows. We establish global-in-time existence in weighted spaces for the associated Cauchy problem in the framework of weak solutions.     

“A priori” estimates, uniqueness and existence of positive solutions of Yamabe type equations on complete manifolds

We study the asymptotic behaviour of non-negative solutions of Yamabe type equations on a complete Riemannian manifold. Then we provide a comparison result, based on a form of the weak maximum principle at infinity, which together with the “a priori” estimates previously obtained, yields uniqueness under very general Ricci assumptions. The paper ends with an existence result and an application to the non-compact Yamabe problem.     

The existence of solutions to second-order singular boundary value problems

This article analyzes qualitative properties of solutions to two-point boundary value problems for singular ordinary differential equations. In particular, we form new approaches that ensure that all possible solutions satisfy certain a priori bounds. The methods involve differential inequalities.     

Global existence and uniqueness of weak solutions in critical spaces for a mathematical model in superfluidity

We prove the global‐in‐time existence and uniqueness of weak solutions in critical spaces for a mathematical model in superfluidity, with initial data ψ₀,A₀ ∈ L³,u₀ ∈ L^{3 ∕ 2},u₀ ≥ 0 in three dimension and in two dimension. Copyright © 2014 John Wiley & Sons, Ltd.     

On uniqueness and existence of entropy solutions for a nonlinear parabolic problem with absorption

The aim of this paper is to study, for L ¹-data, the absorption problem of parabolic type : with Dirichlet boundary conditions and initial conditions. Here a satisfies the classical Leray-Lions hypotheses and β(x, ·) is the subdifferential ∂j(x, ·), where j is a convex function such that j(·, 0) = 0. Existence and uniqueness of an entropy solution is established.      

Global existence of smooth solutions to two-dimensional compressible isentropic Euler equations for Chaplygin gases

In this paper we investigate the two-dimensional compressible isentropic Euler equations for Chaplygin gases. Under the assumption that the initial data is close to a constant state and the vorticity of the initial velocity vanishes, we prove the global existence of the smooth solution to the Cauchy problem for twodimensional flow of Chaplygin gases.     

Global existence of classical solutions of Goursat problem for quasilinear hyperbolic systems with BV data

We investigate the existence of a global classical solution to the Goursat problem for linearly degenerate quasilinear hyperbolic systems. As the result in [A. Bressan, Contractive metrics for nonlinear hyperbolic systems, Indiana Univ. Math. J. 37 (1988) 409–421] suggests that one may achieve global smoothness even if the C¹ norm of the initial data is large, we prove that, if the C¹ norm of the boundary data is bounded but possibly large, and the BV norm of the boundary data is sufficiently small, then the solution remains C¹ globally in time. Applications include the equation of time‐like extremal surfaces in Minkowski space R^{1 + (1 + n)} and the one‐dimensional Chaplygin gas equations. Copyright © 2013 John Wiley & Sons, Ltd.     

Global existence of weak discontinuous solutions to the Cauchy problem with small BV initial data for quasilinear hyperbolic systems

In this paper, we study the Cauchy problem for quasilinear hyperbolic system with a kind of non‐smooth initial data. Under the assumption that the initial data possess a suitably small bounded variation norm, a necessary and sufficient condition is obtained to guarantee the existence and uniqueness of global weak discontinuous solution. Copyright © 2014 John Wiley & Sons, Ltd.     

A class of pseudo‐parabolic equations: existence,uniqueness of weak solutions,and error estimates for the Euler‐implicit discretization

In this paper, we investigate a class of pseudo‐parabolic equations. Such equations model two‐phase flow in porous media where dynamic effects are included in the capillary pressure. The existence and uniqueness of a weak solution are proved, and error estimates for an Euler implicit time discretization are obtained. Copyright © 2011 John Wiley & Sons, Ltd.     

Global existence and nonexistence of solution for Cauchy problem of multidimensional double dispersion equations

In this paper we consider the Cauchy problem of multidimensional generalized double dispersion equations utt−Δu−Δutt+Δ²u=Δf(u), where f(u)=ap|u|. By potential well method we prove the existence and nonexistence of global weak solution without establishing the local existence theory. And we derive some sharp conditions for global existence and lack of global existence solution.     

Global existence and blow-up of solutions to an evolution p-Laplace system coupled via nonlocal sources

The aim of this paper is to investigate the behavior of positive solutions to the following system of evolution p-Laplace equations coupled via nonlocal sources:

     

非线性退缩抛物组初边值问题解的存在性和唯一性

本文讨论一类非线性退缩抛物型方程组初边值问题广义问题广义解的存在性和唯一性。     

Algorithms for the universal and a priori TSP

We present two simple results for generalizations of the traveling salesman problem (TSP): for the universal TSP, we show that one can compute a tour that is universally optimal whenever the input is a tree metric. A (randomized) O(logn)-approximation algorithm for the a priori TSP follows as a corollary.