Viscosity methods for zeroes of accretive operators

This paper deals with the general iteration method , for calculating a particular zero of A, an m-accretive operator in a Banach space X, T_n being a sequence of nonexpansive self-mappings in X. Under suitable conditions on the parameters and X, we state strong and weak convergence results of (x_n). We also show how to compute a common zero of two m-accretive operators in X.     

Weak and strong solutions for the incompressible Navier-Stokes equations with damping

In this paper, we show that the Cauchy problem of the Navier-Stokes equations with damping α|u|^β−1u(α>0) has global weak solutions for any β?1, global strong solution for any β?7/2 and that the strong solution is unique for any 7/2?β?5.     

Existence of periodic solutions for nonlinear evolution equations with pseudo monotone operators

In this paper, we study the existence of -periodic solutions for the problem

where is a -periodic, pseudo monotone mapping from a reflexive Banach space into its dual.

     

Weak solutions for stochastic differential equations with additive fractional noise

In this paper, we show the existence of a weak solution for a stochastic differential equation driven by an additive fractional Brownian motion with Hurst parameter , and a discontinuous drift. The proof of this result is based on the Girsanov theorem for the fractional Brownian motion.     

On the definition of viscosity solutions for parabolic equations

In this short note we suggest a refinement for the definition of viscosity solutions for parabolic equations. The new version of the definition is equivalent to the usual one and it better adapts to the properties of parabolic equations. The basic idea is to determine the admissibility of a test function based on its behavior prior to the given moment of time and ignore what happens at times after that.

     

Weak solutions to the stochastic porous media equation via Kolmogorov equations: The degenerate case

A stochastic version of the porous medium equation with coloured noise is studied. The corresponding Kolmogorov equation is solved in the space L²(H,ν) where ν is an infinitesimally excessive measure. Then a weak solution is constructed.     

Flow invariance for solutions to nonlinear nonautonomous partial differential delay equations

We investigate the problem of existence and flow invariance of mild solutions to nonautonomous partial differential delay equations , t?s, us=φ, where B(t) is a family of nonlinear multivalued, α-accretive operators with D(B(t)) possibly depending on t, and the operators F(t,.) being defined—and Lipschitz continuous—possibly only on “thin” subsets of the initial history space E. The results are applied to population dynamics models. We also study the asymptotic behavior of solutions to this equation. Our analysis will be based on the evolution operator associated to the equation in the initial history space E.     

Weak and smooth solutions to incompressible Navier-Stokes-Landau-Lifshitz-Maxwell equations

Considering the Navier-Stokes-Landau-Lifshitz-Maxwell equations, in dimensions two and three, we use Galerkin method to prove the existence of weak solution. Then combine the a priori estimates and induction technique, we obtain the existence of smooth solution.     

A New Proof for the Equivalence of Weak and Viscosity Solutions for the p-Laplace Equation

In this paper, we give a new proof for the fact that the distributional weak solutions and the viscosity solutions of the p-Laplace equation ?div(|Du| ^p?2 Du) = 0 coincide. Our proof is more direct and transparent than the original proof of Juutinen et al. [8 Juutinen , P. , Lindqvist , P. , Manfredi , J.J. ( 2001 ). On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation . SIAM J. Math. Anal. 33 : 699 – 717 .[Crossref], [Web of Science ®] , [Google Scholar]], which relied on the full uniqueness machinery of the theory of viscosity solutions. We establish a similar result also for the solutions of the non-homogeneous version of the p-Laplace equation.     

On Weak Solutions of Stochastic Differential Equations II

In the first part of this article a new method of proving existence of weak solutions to stochastic differential equations with continuous coefficients having at most linear growth was developed. In this second part, we show that the same method may be used even if the linear growth hypothesis is replaced with a suitable Lyapunov condition.     

Symmetry properties for solutions of nonlocal equations involving nonlinear operators

We pursue the study of one-dimensional symmetry of solutions to nonlinear equations involving nonlocal operators. We consider a vast class of nonlinear operators and in a particular case it covers the fractional p-Laplacian operator. Just like the classical De Giorgi's conjecture, we establish a Poincaré inequality and a linear Liouville theorem to provide two different proofs of the one-dimensional symmetry results in two dimensions. Both approaches are of independent interests. In addition, we provide certain energy estimates for layer solutions and Liouville theorems for stable solutions. Most of the methods and ideas applied in the current article are applicable to nonlocal operators with general kernels where the famous extension problem, given by Caffarelli and Silvestre, is not necessarily known.     

Existence of partially regular weak solutions to Landau-Lifshitz-Maxwell equations

In this paper, the authors establish the existence of partially regular weak solutions to the Landau-Lifshitz equations coupling with static Maxwell systems in 3 dimensions by Ginzburg-Landau approximation. It is proved that the Hausdorff measure of the singular set is locally finite. This extends the similar results of Ding and Guo [S. Ding, B. Guo, Hausdorff measure of the singular set of Landau-Lifshitz equations with a nonlocal term, Comm. Math. Phys. 250 (1) (2004) 95-117] from the stationary solutions to weak solutions and the results of Wang [C. Wang, On Landau-Lifshitz equations in dimensions at most four, Indiana Univ. Math. J. 55 (5) (2006) 1615-1644] from Landau-Lifshitz equations to Landau-Lifshitz-Maxwell equations.     

Boundary value problems for difference equations with causal operators

By using the monotone iterative method, some new results are established for nonlinear boundary conditions of difference problems with causal operators. We formulate sufficient conditions under which such problems have extremal solutions. Difference inequalities with causal operators are also discussed. Two examples are added to illustrate the results.     

Computing periodic solutions of linear differential-algebraic equations by waveform relaxation

We propose an algorithm, which is based on the waveform relaxation (WR) approach, to compute the periodic solutions of a linear system described by differential-algebraic equations. For this kind of two-point boundary problems, we derive an analytic expression of the spectral set for the periodic WR operator. We show that the periodic WR algorithm is convergent if the supremum value of the spectral radii for a series of matrices derived from the system is less than 1. Numerical examples, where discrete waveforms are computed with a backward-difference formula, further illustrate the correctness of the theoretical work in this paper.

     

Singular solutions of Hessian fully nonlinear elliptic equations

We study Hessian fully nonlinear uniformly elliptic equations and show that the second derivatives of viscosity solutions of those equations (in 12 or more dimensions) can blow up in an interior point of the domain. We prove that the optimal interior regularity of such solutions is no more than C1+?, showing the optimality of the known interior regularity result. The same is proven for Isaacs equations. We prove the existence of non-smooth solutions to fully nonlinear Hessian uniformly elliptic equations in 11 dimensions. We study also the possible singularity of solutions of Hessian equations defined in a neighborhood of a point and prove that a homogeneous order 0<α<1 solution of a Hessian uniformly elliptic equation in a punctured ball should be radial.     

A note on global existence of weak solutions to the compressible magnetohydrodynamic equations with Coulomb force

In this note, for the case of , we prove the existence of global-in-time finite energy weak solution of the equations of a two-dimensional magnetohydrodynamics with Coulomb force, where γ denotes the adiabatic exponent. The value is the optimal lower bound of γ to establish global-in-time finite energy weak solution under current frame.     

Discontinuous viscosity solutions to dirichlet problems for hamilton-jacob1 equations with

We introduce a new formulation of Dirichlet problem for a class of first order, nonlinear equations containing the minimum time problem, whose solution is expected to be discontinuous. We prove existence, uniqueness and representation formulas for the solution in the sense of viscosity solutions. Our method relies on a new way of prescribing the boundary condition, the use of recent ideas of Barron-Jensen [8] and Barles [5] , and the derivation of a "backwards" dynamic programming principle. We use the same ideas to prove uniqueness for the usual Dicchlet type formulation, following Ishii [13] and Bales-Perthame [6], under additional regularity conditions on the domain.