57 research outputs found
On systems of continuity equations with nonlinear diffusion and nonlocal drifts
This paper is devoted to existence and uniqueness results for classes of
nonlinear diffusion equations (or systems) which may be viewed as regular
perturbations of Wasserstein gradient flows. First, in the case. where the
drift is a gradient (in the physical space), we obtain existence by a
semi-implicit Jordan-Kinderlehrer-Otto scheme. Then, in the nonpotential case,
we derive existence from a regularization procedure and parabolic energy
estimates. We also address the uniqueness issue by a displacement convexity
argument
Nonlinear diffusion equations with degenerate fast-decay mobility by coordinate transformation
We prove an existence and uniqueness result for solutions to nonlinear
diffusion equations with degenerate mobility posed on a bounded interval for a
certain density . In case of \emph{fast-decay} mobilities, namely mobilities
functions under a Osgood integrability condition, a suitable coordinate
transformation is introduced and a new nonlinear diffusion equation with linear
mobility is obtained. We observe that the coordinate transformation induces a
mass-preserving scaling on the density and the nonlinearity, described by the
original nonlinear mobility, is included in the diffusive process. We show that
the rescaled density is the unique weak solution to the nonlinear
diffusion equation with linear mobility. Moreover, the results obtained for the
density allow us to motivate the aforementioned change of variable and
to state the results in terms of the original density without prescribing
any boundary conditions
Long-time behavior of a finite volume discretization for a fourth order diffusion equation
We consider a non-standard finite-volume discretization of a strongly
non-linear fourth order diffusion equation on the -dimensional cube, for
arbitrary . The scheme preserves two important structural properties
of the equation: the first is the interpretation as a gradient flow in a mass
transportation metric, and the second is an intimate relation to a linear
Fokker-Planck equation. Thanks to these structural properties, the scheme
possesses two discrete Lyapunov functionals. These functionals approximate the
entropy and the Fisher information, respectively, and their dissipation rates
converge to the optimal ones in the discrete-to-continuous limit. Using the
dissipation, we derive estimates on the long-time asymptotics of the discrete
solutions. Finally, we present results from numerical experiments which
indicate that our discretization is able to capture significant features of the
complex original dynamics, even with a rather coarse spatial resolution.Comment: 27 pages, minor change
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