On the domain dependence of solutions to the Navier–Stokes equations of a two‐dimensional compressible flow

We consider the Navier–Stokes equations for compressible, barotropic flow in two space dimensions, with pressure satisfying p(?)=a?log^d(?) for large ?, here d>1 and a>0. After introducing useful tools from the theory of Orlicz spaces, we prove a compactness result for the solution set of the equations with respect to the variation of the underlying bounded spatial domain. Especially, we get a general existence theorem for the system in question with no restrictions on smoothness of the bounded spatial domain. Copyright © 2009 John Wiley & Sons, Ltd.     

Stationary solutions of the non-linear Boltzmann equation in a bounded spatial domain

A contraction mapping (or, alternatively, an implicit function theory) argument is applied in combination with the Fredholm alternative to prove the existence of a unique stationary solution of the non-linear Boltzmann equation on a bounded spatial domain under a rather general reflection law at the piecewise C¹ boundary. The boundary data are to be small in a weighted L_∞-norm.     

A theoretical analysis of a new finite volume scheme for second order hyperbolic equations on general nonconforming multidimensional spatial meshes

We consider the wave equation, on a multidimensional spatial domain. The discretization of the spatial domain is performed using a general class of nonconforming meshes which has been recently studied for stationary anisotropic heterogeneous diffusion problems, see Eymard et al. (IMAJ Numer Anal 30 (2010), 1009–1043). The discretization in time is performed using a uniform mesh. We derive a new implicit finite volume scheme approximating the wave equation and we prove error estimates of the finite volume approximate solution in several norms which allow us to derive error estimates for the approximations of the exact solution and its first derivatives. We prove in particular, when the discrete flux is calculated using a stabilized discrete gradient, the convergence order is \begin{align*} h_\mathcal{D}\end{align*} (resp. k) is the mesh size of the spatial (resp. time) discretization. This estimate is valid under the regularity assumption \begin{align*}u\in C^3(\lbrack 0,T\rbrack;C^2(\overline{\Omega}))\end{align*} for the exact solution u. The proof of these error estimates is based essentially on a comparison between the finite volume approximate solution and an auxiliary finite volume approximation. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Solution of the spatial Tricomi problem for a singular mixed-type equation by the method of integral equations

We consider a mixed-type singular differential equation in a bounded spatial domain in a specific form and prove the unique solvability of the Tricomi problem for the mentioned equation.     

Large deviations for functionals of spatial point processes with applications to random packing and spatial graphs

Functionals of spatial point process often satisfy a weak spatial dependence condition known as stabilization. We prove general Donsker–Varadhan large deviation principles (LDP) for such functionals and show that the general result can be applied to prove LDPs for various particular functionals, including those concerned with random packing, nearest neighbor graphs, and lattice versions of the Voronoi and sphere of influence graphs.     

An analysis of the Wigner–Poisson problem with inflow boundary conditions

We present a study of the Wigner–Poisson problem in a bounded spatial domain with non-homogeneous and time-dependent “inflow” boundary conditions. This system of nonlinearly coupled equations is a mathematical model for quantum transport of charges in a semiconductor with external contacts. We prove well-posedness of the linearized n-dimensional problem as well as existence and uniqueness of a global-in-time, regular solution of the one-dimensional nonlinear problem.     

Small random perturbations of a dynamical system with blow-up

We study small random perturbations by additive white-noise of a spatial discretization of a reaction–diffusion equation with a stable equilibrium and solutions that blow up in finite time. We prove that the perturbed system blows up with total probability and establish its order of magnitude and asymptotic distribution. For initial data in the domain of explosion we prove that the explosion time converges to the deterministic one while for initial data in the domain of attraction of the stable equilibrium we show that the system exhibits metastable behavior.     

The spatial critical points not moving along the heat flow

We consider solutions of the heat equation, in domains inR ^N, and their spatial critical points. In particular, we show that a solutionu has a spatial critical point not moving along the heat flow if and only ifu satisfies some balance law. Furthermore, in the case of Dirichlet, Neumann, and Robin homogeneous initial-boundary value problems on bounded domains, we prove that if the origin is a spatial critical point never moving for sufficiently many compactly supported initial data satisfying the balance law with respect to the origin, then the domain must be a ball centered at the origin.     

Viscosity solutions of the <Emphasis Type="Italic">p</Emphasis>-Laplace equation with nonlinear source

In the present paper we consider the Dirichlet problem in a convex domain for the multidimensional p-Laplace equation with nonlinear source. We prove the existence of the unique continuous viscosity solution under quite general assumptions on the structure of the source. Received: 18 January 2006     

Interface boundary value problem for the Navier-Stokes equations in thin two-layer domains

We study a system of 3D Navier-Stokes equations in a two-layer parallelepiped-like domain with an interface coupling of the velocities and mixed (free/periodic) boundary condition on the external boundary. The system under consideration can be viewed as a simplified model describing some features of the mesoscale interaction of the ocean and atmosphere. In case when our domain is thin (of order ε), we prove the global existence of the strong solutions corresponding to a large set of initial data and forcing terms (roughly, of order ε^−2/3). We also give some results concerning the large time dynamics of the solutions. In particular, we prove a spatial regularity of the global weak attractor.     

Global existence of weak solution for the 2-D Ericksen–Leslie system

We prove the global existence of weak solution for two dimensional Ericksen–Leslie system with the Leslie stress and general Ericksen stress under the physical constrains on the Leslie coefficients. We also prove the local well-posedness of the Ericksen–Leslie system in two and three spatial dimensions.     

Strong solutions to the incompressible magnetohydrodynamic equations

In this paper, we are concerned with strong solutions to the Cauchy problem for the incompressible Magnetohydrodynamic equations. By the Galerkin method, energy method and the domain expansion technique, we prove the local existence of unique strong solutions for general initial data, develop a blow‐up criterion for local strong solutions and prove the global existence of strong solutions under the smallness assumption of initial data. The initial data are assumed to satisfy a natural compatibility condition and allow vacuum to exist. Copyright © 2010 John Wiley & Sons, Ltd.     

Optimization of Schwarz waveform relaxation over short time windows

Schwarz waveform relaxation algorithms (SWR) are naturally parallel solvers for evolution partial differential equations. They are based on a decomposition of the spatial domain into subdomains, and a partition of the time interval of interest into time windows. On each time window, an iteration, during which subproblems are solved in space-time subdomains, is then used to obtain better and better approximations of the overall solution. The information exchange between subdomains in space-time is performed through classical or optimized transmission conditions (TCs). We analyze in this paper the optimization problem when the time windows are short. We use as our model problem the optimized SWR algorithm with Robin TCs applied to the heat equation. After a general convergence analysis using energy estimates, we prove that in one spatial dimension, the optimized Robin parameter scales like the inverse of the length of the time window, which is fundamentally different from the known scaling on general bounded time windows, which is like the inverse of the square root of the time window length. We illustrate our analysis with a numerical experiment.     

Spatial critical points not moving along the heat flow II: The centrosymmetric case

We consider solutions of initial-boundary value problems for the heat equation on bounded domains in and their spatial critical points as in the previous paper [MS]. In Dirichlet, Neumann, and Robin homogeneous initial-boundary value problems on bounded domains, it is proved that if the origin is a spatial critical point never moving for sufficiently many compactly supported initial data being centrosymmetric with respect to the origin, then the domain must be centrosymmetric with respect to the origin. Furthermore, we consider spatial zero points instead of spatial critical points, and prove some similar symmetry theorems. Also, it is proved that these symmetry theorems hold for initial-boundary value problems for the wave equation. Received October 31, 1997; in final form February 3, 1998     

On the sufficient condition for regularity of a boundary point for singular parabolic equations with non‐standard growth

We investigate the continuity of solutions for general nonlinear parabolic equations with non‐standard growth near a nonsmooth boundary of a cylindrical domain. We prove a sufficient condition for regularity of a boundary point.     

Spectral stability of general non-negative self-adjoint operators with applications to Neumann-type operators

We prove a stability theorem for the eigenvalues of general non-negative self-adjoint linear operators with compact resolvents and by applying it we prove a sharp stability result for the dependence of the eigenvalues of second order uniformly elliptic linear operators with homogeneous Neumann boundary conditions upon domain perturbation.     

Fully nonlinear elliptic equations and the dini condition

Abstract

We investigate transmission problems with strongly Lipschitz interfaces for the Dirac equation by establishing spectral estimates on an associated boundary singular integral operator, the rotation operator. Using Rellich estimates we obtain angular spectral estimates on both the essential and full spectrum for general bi-oblique transmission problems. Specializing to the normal transmission problem, we investigate transmission problems for Maxwell's equations using a nilpotent exterior/interior derivativeoperator. The fundamental commutation properties for this operator with the two basic reflection operators are proved. We show how the L ₂spectral estimates are inherited for the domain of the exterior/interior derivative operator and prove some complementary eigenvalue estimates. Finally we use a general algebraic theorem to prove a regularity property needed for Maxwell's equations.     

On the optimal control of a class of non-Newtonian fluids

We consider optimal control problems of systems governed by stationary, incompressible generalized Navier–Stokes equations with shear dependent viscosity in a two-dimensional or three-dimensional domain. We study a general class of viscosity functions with shear-thinning and shear-thickening behavior. We prove an existence result for such class of optimal control problems.     

ON THE EXISTENCE OF GLOBAL GENERAL SOLUTIONS OF POLYNOMIAL SYSTEMS

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Solutions to a gradient-dependent integro-differential parabolic problem arising in the pricing of financial options in a Lévy market

We study an integro-differential parabolic problem modeling a process with jumps arising in financial mathematics. Under suitable conditions, we prove the existence of solutions in a general domain by proving a uniform bound on an iterative sequence of solutions and then applying the Arzelà–Ascoli theorem.