Results 1  10
of
79
A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations
 475 (1994) MR 94j:65136
"... Abstract. Using the abstract framework of [9] we analyze a residual a posteriori error estimator for spacetime finite element discretizations of quasilinear parabolic pdes. The estimator gives global upper and local lower bounds on the error of the numerical solution. The finite element discretizat ..."
Abstract

Cited by 48 (2 self)
 Add to MetaCart
Abstract. Using the abstract framework of [9] we analyze a residual a posteriori error estimator for spacetime finite element discretizations of quasilinear parabolic pdes. The estimator gives global upper and local lower bounds on the error of the numerical solution. The finite element discretizations in particular cover the socalled θscheme, which includes the implicit and explicit Euler methods and the CrankNicholson scheme. 1.
A Review of A Posteriori Error Estimation
 and Adaptive MeshRefinement Techniques, Wiley & Teubner
, 1996
"... linear parabolic equations ..."
The Willmore flow for near spheres
 Differential and Integral Equations
, 2001
"... Abstract. The Willmore flow leads to a quasilinear evolution equation of fourth order. We study existence, uniqueness and regularity of solutions. Moreover, we prove that solutions exist globally and converge exponentially fast to a sphere, provided that they are initially close to a sphere. 1. ..."
Abstract

Cited by 22 (2 self)
 Add to MetaCart
Abstract. The Willmore flow leads to a quasilinear evolution equation of fourth order. We study existence, uniqueness and regularity of solutions. Moreover, we prove that solutions exist globally and converge exponentially fast to a sphere, provided that they are initially close to a sphere. 1.
Spinodal Decomposition for the CahnHilliard Equation in Higher Dimensions. Part II: Nonlinear Dynamics
 Part I: Probability and wavelength estimate. Communications in Mathematical Physics
, 1997
"... This paper is the second in a series of two papers addressing the phenomenon of spinodal decomposition for the CahnHilliard equation u t = \Gamma\Delta(" 2 \Deltau + f(u)) in\Omega ; @u @ = @ \Deltau @ = 0 on @\Omega ; where\Omega ae R n , n 2 f1; 2; 3g, is a bounded domain with suffici ..."
Abstract

Cited by 21 (9 self)
 Add to MetaCart
This paper is the second in a series of two papers addressing the phenomenon of spinodal decomposition for the CahnHilliard equation u t = \Gamma\Delta(" 2 \Deltau + f(u)) in\Omega ; @u @ = @ \Deltau @ = 0 on @\Omega ; where\Omega ae R n , n 2 f1; 2; 3g, is a bounded domain with sufficiently smooth boundary, and f is cubiclike, for example f(u) = u \Gamma u 3 . Using the results of [22] the nonlinear CahnHilliard equation will be discussed. This equation generates a nonlinear semiflow in certain affine subspaces of H 2(\Omega\Gamma6 In a neighborhood U " with size proportional to " n around the constant solution ¯ u 0 j ¯, where ¯ lies in the spinodal region, we observe the following behavior. Within a local inertial manifold N " containing ¯ u 0 j ¯ there exists a finitedimensional invariant manifold M " which dominates the behavior of all solutions starting with initial conditions from a small ball around ¯ u 0 j ¯ with probability almost 1. The dimension of M...
Parabolic Equations: Asymptotic Behavior and Dynamics on Invariant Manifolds
 HANDBOOK ON DYNAMICAL SYSTEMS
, 2002
"... results are given without proofs, sometimes short sketches are included to help the reader's intuition. Applications to (1.1), (1.2) are given with more details. We rst introduce basic notation and denitions. In the whole section Y is an ordered Banach space with norm k k and order cone Y+ . Reca ..."
Abstract

Cited by 18 (0 self)
 Add to MetaCart
results are given without proofs, sometimes short sketches are included to help the reader's intuition. Applications to (1.1), (1.2) are given with more details. We rst introduce basic notation and denitions. In the whole section Y is an ordered Banach space with norm k k and order cone Y+ . Recall that an order cone is a closed convex cone such that Y+ \ ( Y+ ) = f0g. We assume that Y is strongly ordered which means that int Y+ , the interior of Y+ , is nonempty. For x; y 2 Y we write x y x < y x y if y x 2 Y+ ; if x y and x 6= y ; if y x 2 int Y+ : The reversed signs are used in the usual way. Two points are said to be related (or ordered) if they are related by or . The notation A B (similarly for < and ) between two sets means that x y whenever x 2 A and y 2 B. A mapping F : D(F ) Y ! Y is said to be monotone if x; y 2 D(F ) and x y imply F (x) F (y). It is called strongly monotone if x; y 2 D(F ) and x < y imply F (x) F (y). A linear strongly mon...
Sufficient Conditions For Exponential Stability And Dichotomy Of Evolution Equations
 Forum Math
, 1998
"... . We present several sufficient conditions for exponential stability and dichotomy of solutions of the evolution equation u 0 (t) = A(t)u(t) () on a Banach space X . Our main theorem says that if the operators A(t) generate analytic semigroups on X having exponential dichotomy with uniform const ..."
Abstract

Cited by 13 (7 self)
 Add to MetaCart
. We present several sufficient conditions for exponential stability and dichotomy of solutions of the evolution equation u 0 (t) = A(t)u(t) () on a Banach space X . Our main theorem says that if the operators A(t) generate analytic semigroups on X having exponential dichotomy with uniform constants and A(\Delta) has a sufficiently small Holder constant, then () has exponential dichotomy. We further study robustness of exponential dichotomy under time dependent unbounded Miyaderatype perturbations. Our main tool is a characterization of exponential dichotomy of evolution families by means of the spectra of the socalled evolution semigroup on C 0 (R; X) or L 1 (R; X). 1. Introduction and preliminaries Exponential dichotomy is one of the fundamental asymptotic properties of solutions of the linear Cauchy problem (CP ) ae d dt u(t) = A(t)u(t); t ? s; u(s) = x in a Banach space X. It also plays an important role in the investigation of qualitative properties of nonlinear evolut...
L p spectral theory of higherorder elliptic differential operators
 Bull. London Math. Soc
, 1997
"... 2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518 ..."
Abstract

Cited by 13 (1 self)
 Add to MetaCart
2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518
Maximal regularity for a free boundary problem
 Nonl. Diff. Equs. Appl
, 1995
"... This paper is concerned with the motion of an incompressible fluid in a rigid porous medium of infinite extent. The fluid is bounded below by a fixed, impermeable layer and above by a free surface moving under the influence of gravity. The laminar flow is governed by Darey's law. We prove existence ..."
Abstract

Cited by 8 (6 self)
 Add to MetaCart
This paper is concerned with the motion of an incompressible fluid in a rigid porous medium of infinite extent. The fluid is bounded below by a fixed, impermeable layer and above by a free surface moving under the influence of gravity. The laminar flow is governed by Darey's law. We prove existence of a unique maximal classical solutipn, using methods from the theory of maximal regularity, analytic semigroups, and Fourier multipliers. Moreover, we describe a state space which can be considered as domain of parabolicity for the problem under consideration. 1 Introduction and main result In this paper we investigate a class of free boundary problems, which can be described as follows. Let Given f C ~, define: = {f E BC2(R) " inf f(x)> 0}.
SelfIntersections For The Surface Diffusion And The VolumePreserving Mean Curvature Flow
 Mean Curvature Flow. Differential and Integral Equations 13 (2000
, 2000
"... We prove that the surfacediffusion flow and the volumepreserving mean curvature flow can drive embedded hypersurfaces to selfintersections. ..."
Abstract

Cited by 8 (6 self)
 Add to MetaCart
We prove that the surfacediffusion flow and the volumepreserving mean curvature flow can drive embedded hypersurfaces to selfintersections.
A New Approach to the Regularity of Solutions for Parabolic Equations
 Lecture Notes in Pure and
"... In this note we describe a new approach to establish regularity properties for solutions of parabolic equations. It is based on maximal regularity and the implicit function theorem. 1. ..."
Abstract

Cited by 8 (8 self)
 Add to MetaCart
In this note we describe a new approach to establish regularity properties for solutions of parabolic equations. It is based on maximal regularity and the implicit function theorem. 1.