963 research outputs found
Renormalized Solutions to a Nonlinear Parabolic-Elliptic System
The aim of this paper is to show the existence of renormalized solutions to a parabolicelliptic
system with unbounded diffusion coefficients. This system may be regarded as a modified
version of the well-known thermistor problem; in this case, the unknowns are the temperature in a
conductor and the electrical potential.Ministerio de Ciencia y TecnologÃa BFM2003-01187Junta de AndalucÃa FQM-31
Existence and uniqueness results for a nonlinear stationary system
We prove a few existence results of a solution for a static system with a
coupling of thermoviscoelastic type. As this system involves coupling
terms we use the techniques of renormalized solutions for elliptic equations
with data. We also prove partial uniqueness results
Cross-diffusion systems with entropy structure
Some results on cross-diffusion systems with entropy structure are reviewed.
The focus is on local-in-time existence results for general systems with
normally elliptic diffusion operators, due to Amann, and global-in-time
existence theorems by Lepoutre, Moussa, and co-workers for cross-diffusion
systems with an additional Laplace structure. The boundedness-by-entropy method
allows for global bounded weak solutions to certain diffusion systems.
Furthermore, a partial result on the uniqueness of weak solutions is recalled,
and some open problems are presented
Potential estimates and quasilinear parabolic equations with measure data
In this paper, we study the existence and regularity of the quasilinear
parabolic equations: in
, and a bounded domain
. Here , the nonlinearity
fulfills standard growth conditions and term is a continuous function
and is a radon measure. Our first task is to establish the existence
results with , for . We next obtain global
weighted-Lorentz, Lorentz-Morrey and Capacitary estimates on gradient of
solutions with , under minimal conditions on the boundary of domain
and on nonlinearity . Finally, due to these estimates, we solve the
existence problems with for .Comment: 120
Regularity estimates for singular parabolic measure data problems with sharp growth
We prove global gradient estimates for parabolic -Laplace type equations
with measure data, whose model is where is a signed Radon measure with finite total
mass. We consider the singular case
and give possibly minimal conditions on the nonlinearity and the boundary of
, which guarantee the regularity results for such measure data
problems.Comment: 28 page
Compressive Space-Time Galerkin Discretizations of Parabolic Partial Differential Equations
We study linear parabolic initial-value problems in a space-time variational
formulation based on fractional calculus. This formulation uses "time
derivatives of order one half" on the bi-infinite time axis. We show that for
linear, parabolic initial-boundary value problems on , the
corresponding bilinear form admits an inf-sup condition with sparse tensor
product trial and test function spaces. We deduce optimality of compressive,
space-time Galerkin discretizations, where stability of Galerkin approximations
is implied by the well-posedness of the parabolic operator equation. The
variational setting adopted here admits more general Riesz bases than previous
work; in particular, no stability in negative order Sobolev spaces on the
spatial or temporal domains is required of the Riesz bases accommodated by the
present formulation. The trial and test spaces are based on Sobolev spaces of
equal order with respect to the temporal variable. Sparse tensor products
of multi-level decompositions of the spatial and temporal spaces in Galerkin
discretizations lead to large, non-symmetric linear systems of equations. We
prove that their condition numbers are uniformly bounded with respect to the
discretization level. In terms of the total number of degrees of freedom, the
convergence orders equal, up to logarithmic terms, those of best -term
approximations of solutions of the corresponding elliptic problems.Comment: 26 page
- …