1,405 research outputs found
Traveling Wave Solutions of Advection-Diffusion Equations with Nonlinear Diffusion
We study the existence of particular traveling wave solutions of a nonlinear
parabolic degenerate diffusion equation with a shear flow. Under some
assumptions we prove that such solutions exist at least for propagation speeds
c {\in}]c*, +{\infty}, where c* > 0 is explicitly computed but may not be
optimal. We also prove that a free boundary hy- persurface separates a region
where u = 0 and a region where u > 0, and that this free boundary can be
globally parametrized as a Lipschitz continuous graph under some additional
non-degeneracy hypothesis; we investigate solutions which are, in the region u
> 0, planar and linear at infinity in the propagation direction, with slope
equal to the propagation speed.Comment: 40 pages, 1 figur
Regularity estimates for the solution and the free boundary to the obstacle problem for the fractional Laplacian
We use a characterization of the fractional Laplacian as a Dirichlet to
Neumann operator for an appropriate differential equation to study its obstacle
problem. We write an equivalent characterization as a thin obstacle problem. In
this way we are able to apply local type arguments to obtain sharp regularity
estimates for the solution and study the regularity of the free boundary
Random homogenization of an obstacle problem
We study the homogenization of an obstacle problem in a perforated domain.
The holes are periodically distributed but have random size and shape. The
capacity of the holes is assumed to be stationary ergodic. As in the periodic
case, we show that the asymptotic behavior of the solutions is described by an
elliptic equation involving an additional term that takes into account the
effects of the obstacle.Comment: 28 page
Fractional elliptic equations, Caccioppoli estimates and regularity
Let be a uniformly elliptic operator
in divergence form in a bounded domain . We consider the fractional
nonlocal equations Here , , is the fractional power of and
is the conormal derivative of with respect to the
coefficients . We reproduce Caccioppoli type estimates that allow us to
develop the regularity theory. Indeed, we prove interior and boundary Schauder
regularity estimates depending on the smoothness of the coefficients ,
the right hand side and the boundary of the domain. Moreover, we establish
estimates for fundamental solutions in the spirit of the classical result by
Littman--Stampacchia--Weinberger and we obtain nonlocal integro-differential
formulas for . Essential tools in the analysis are the semigroup
language approach and the extension problem.Comment: 37 page
Nonlinear porous medium flow with fractional potential pressure
We study a porous medium equation, with nonlocal diffusion effects given by
an inverse fractional Laplacian operator. We pose the problem in n-dimensional
space for all t>0 with bounded and compactly supported initial data, and prove
existence of a weak and bounded solution that propagates with finite speed, a
property that is nor shared by other fractional diffusion models.Comment: 32 pages, Late
On a price formation free boundary model by Lasry & Lions: The Neumann problem
We discuss local and global existence and uniqueness for the price formation
free boundary model with homogeneous Neumann boundary conditions introduced by
Lasry & Lions in 2007. The results are based on a transformation of the problem
to the heat equation with nonstandard boundary conditions. The free boundary
becomes the zero level set of the solution of the heat equation. The
transformation allows us to construct an explicit solution and discuss the
behavior of the free boundary. Global existence can be verified under certain
conditions on the free boundary and examples of non-existence are given
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