4,161 research outputs found
Asymptotic behaviour of solutions to fractional diffusion-convection equations
We consider a convection-diffusion model with linear fractional diffusion in
the sub-critical range. We prove that the large time asymptotic behavior of the
solution is given by the unique entropy solution of the convective part of the
equation. The proof is based on suitable a-priori estimates, among which
proving an Oleinik type inequality plays a key role.Comment: 24
Incompressible flow in porous media with fractional diffusion
In this paper we study the heat transfer with a general fractional diffusion
term of an incompressible fluid in a porous medium governed by Darcy's law. We
show formation of singularities with infinite energy and for finite energy we
obtain existence and uniqueness results of strong solutions for the
sub-critical and critical cases. We prove global existence of weak solutions
for different cases. Moreover, we obtain the decay of the solution in ,
for any , and the asymptotic behavior is shown. Finally, we prove the
existence of an attractor in a weak sense and, for the sub-critical dissipative
case with , we obtain the existence of the global attractor
for the solutions in the space for any
Multigrid waveform relaxation for the time-fractional heat equation
In this work, we propose an efficient and robust multigrid method for solving
the time-fractional heat equation. Due to the nonlocal property of fractional
differential operators, numerical methods usually generate systems of equations
for which the coefficient matrix is dense. Therefore, the design of efficient
solvers for the numerical simulation of these problems is a difficult task. We
develop a parallel-in-time multigrid algorithm based on the waveform relaxation
approach, whose application to time-fractional problems seems very natural due
to the fact that the fractional derivative at each spatial point depends on the
values of the function at this point at all earlier times. Exploiting the
Toeplitz-like structure of the coefficient matrix, the proposed multigrid
waveform relaxation method has a computational cost of
operations, where is the number of time steps and is the number of
spatial grid points. A semi-algebraic mode analysis is also developed to
theoretically confirm the good results obtained. Several numerical experiments,
including examples with non-smooth solutions and a nonlinear problem with
applications in porous media, are presented
Porous medium equation with nonlocal pressure
We provide a rather complete description of the results obtained so far on
the nonlinear diffusion equation , which describes a flow through a porous medium driven by a
nonlocal pressure. We consider constant parameters and , we assume
that the solutions are non-negative, and the problem is posed in the whole
space. We present a theory of existence of solutions, results on uniqueness,
and relation to other models. As new results of this paper, we prove the
existence of self-similar solutions in the range when and , and the
asymptotic behavior of solutions when . The cases and were
rather well known.Comment: 24 pages, 2 figure
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