Class Of Parabolic Problems Research Articles

A parabolic multiscale inverse problem approached via homogenization: A numerical method

A method based on homogenization is studied for the solution of a multiscale inverse problem. We consider a class of parabolic problems with highly oscillatory tensors that vary on a microscopic scale. We assume that the microscopic structure is known and seek to recover a macroscopic scalar parameterization of the multiscale tensor. Classical approaches, such as finite elements methods, would require mesh resolution for the direct problem down to the finest scale, that could lead to computational difficulties when implemented. So, starting from the full fine scale model, we solve the inverse problem for a coarse model obtained by homogenization, both theoretically and numerically. The input data, which consist on measurements from the fluxes and the solutions of the direct problem in a given time, are solely based on the original fine scale model. Uniqueness and stability of the inverse problem obtained via homogenization are established under some natural conditions for the fine scale model, and a link with this latter model is established by means of G-convergence. Error estimates are proven for the method.

Parabolic inequalities in inhomogeneous Orlicz-Sobolev spaces with gradients constraints and L 1-data

Abstract This work is devoted to the study of a new class of parabolic problems in inhomogeneous Orlicz spaces with gradient constraints and L 1-data. One proves the existence of the solution by studying the asymptotic behaviour as p goes to ∞, of a sequence of entropy solutions (up ) of some nonlinear parabolic equation in inhomogeneous Orlicz-Sobolev spaces with L 1-data involving the parameter p.

Решение анизотропных уравнений теплопроводности экспоненциальными схемами на крыловских полиномиальных подпространствах и подпространствах «сдвиг-обращение»

We assess performance of the exponential Krylov subspace methods for solving a class of parabolic problems with a strong anisotropy in coefficients. Different boundary conditions are considered, which have a direct impact on the smallest eigenvalue of the discretized operator and, hence, on the convergence behavior of the exponential Krylov subspace solvers. Restarted polynomial Krylov subspace methods and shift-and-invert Krylov subspace methods combined with algebraic multigrid are considered.

On the approximate controllability of parabolic problems with non-smooth coefficients

In this paper we prove the existence of approximate controls for certain classes of parabolic problems with non-smooth coefficients and discuss as examples the problem of approximate controllability for the heat flow in heterogeneous media such as, periodic composites, perforated domains or periodic microstructures separated by rough interfaces.

Estimates for parabolic equations with measure data in generalized Morrey spaces

In this paper, we prove the optimal generalized Morrey estimates for the spatial gradient of the solutions obtained by limits of approximations (SOLA) for a class of parabolic problems with right-hand side measure in a very general irregular domain. The nonlinearity is assumed to be merely measurable only in the time variable [Formula: see text] and belongs to the small bounded mean oscillation (BMO) class as functions of the spatial variable [Formula: see text].

Stability and convergence of difference schemes for multi-dimensional parabolic equations with variable coefficients and mixed derivatives

ABSTRACTWe present second-order difference schemes for a class of parabolic problems with variable coefficients and mixed derivatives. The solvability, stability and convergence of the schemes are rigorously analysed by the discrete energy method. Using the Richardson extrapolation technique, the fourth-order accurate numerical approximations both in time and space are obtained. It is noted that the Richardson extrapolation algorithms can preserve stability of the original difference scheme. Finally, numerical examples are carried out to verify the theoretical results.

On the Calderon-Zygmund Theory for Nonlinear Parabolic Problems with Nonstandard Growth Condition

We prove Calder´ on-Zygmund estimates for a class of parabolic problems whose model is the non-homogeneous parabolic p(x;t)-Laplacian equation @tu div ( jDuj p(x;t) 2 Du ) = f div ( jFj p(x;t) 2 F ) : More precisely, we will show that the spatial gradient Du is as integrable as the inhomogeneities f and F, i.e. jFj p(x;t) ;jfj 1 11 2 L q loc ) jFj p(x;t) 2 L q loc for any q > 1; where 1 is the lower bound for p(x;t). Moreover, it is possible to use this approach to establish the Calder´ on

A remark on uniqueness of weak solutions for some classes of parabolic problems

We prove some uniqueness results for weak solutions to some classes of nonlinear parabolic equations with homogeneous Cauchy-Dirichlet boundary condition. Precisely we consider operators with a first order term or operators which have just principal part depending on \(u\).

ESTIMATES FROM BELOW OF BLOW-UP TIME IN A PARABOLIC SYSTEM WITH GRADIENT TERM

Qualitative properties as blow up, decay bounds and extinction in finite time for solutions to linear and nonlinear parabolic problems have received much attention in the recent literature. We refer to the papers [1], [4], [5] and the references therein. We also mention the results of Galaktionov-Mitidieri-Pohoazev ([3], [13]) on blow-up and global existence for different classes of parabolic problems, including higher order parabolic equations. The blow-up phenomena of solutions to various nonlinear problems, particularly for parabolic systems, have been investigated by different authors. For results in this area, the reader can reference the book [19] and the survey paper

Convergence to SPDEs in Stratonovich Form

We consider the perturbation of parabolic operators of the form ∂ t + P(x, D) by large-amplitude highly oscillatory spatially dependent potentials modeled as Gaussian random fields. The amplitude of the potential is chosen so that the solution to the random equation is affected by the randomness at the leading order. We show that, when the dimension is smaller than the order of the elliptic pseudo-differential operator P(x, D), the perturbed parabolic equation admits a solution given by a Duhamel expansion. Moreover, as the correlation length of the potential vanishes, we show that the latter solution converges in distribution to the solution of a stochastic parabolic equation with multiplicative noise that should be interpreted in the Stratonovich sense. The theory of mild solutions for such stochastic partial differential equations is developed. The behavior described above should be contrasted to the case of dimensions larger than or equal to the order of the elliptic pseudo-differential operator P(x, D). In the latter case, the solution to the random equation converges strongly to the solution of a homogenized (deterministic) parabolic equation as is shown in [2]. A stochastic limit is obtained only for sufficiently small space dimensions in this class of parabolic problems.

Time behaviour of solutions of parabolic problems

For a large class of parabolic problems, we give a priori L -estimates of the solution u(t). We show that these Lp-norms may decrease exponentially when time goes to infinity

Non-linear mixed boundary value problems for parabolic partial differential equations

SynopsisA sufficient condition on the angles of a bounded open subset Ω of ℝn is given for the best possible regularity of a solution to a class of parabolic problems with non-linear mixed boundary conditions.