128 research outputs found
A Partial Integrodifferential Equation in Granular Matter and Its Connection with a Stochastic Model
Refined asymptotics for the infinite heat equation with homogeneous Dirichlet boundary conditions
The nonnegative viscosity solutions to the infinite heat equation with
homogeneous Dirichlet boundary conditions are shown to converge as time
increases to infinity to a uniquely determined limit after a suitable time
rescaling. The proof relies on the half-relaxed limits technique as well as
interior positivity estimates and boundary estimates. The expansion of the
support is also studied
Moment bounds for the Smoluchowski equation and their consequences
We prove uniform bounds on moments X_a = \sum_{m}{m^a f_m(x,t)} of the
Smoluchowski coagulation equations with diffusion, valid in any dimension. If
the collision propensities \alpha(n,m) of mass n and mass m particles grow more
slowly than (n+m)(d(n) + d(m)), and the diffusion rate d(\cdot) is
non-increasing and satisfies m^{-b_1} \leq d(m) \leq m^{-b_2} for some b_1 and
b_2 satisfying 0 \leq b_2 < b_1 < \infty, then any weak solution satisfies X_a
\in L^{\infty}(\mathbb{R}^d \times [0,T]) \cap L^1(\mathbb{R}^d \times [0,T])
for every a \in \mathbb{N} and T \in (0,\infty), (provided that certain moments
of the initial data are finite). As a consequence, we infer that these
conditions are sufficient to ensure uniqueness of a weak solution and its
conservation of mass.Comment: 30 page
On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density
We are concerned with the long time behaviour of solutions to the fractional
porous medium equation with a variable spatial density. We prove that if the
density decays slowly at infinity, then the solution approaches the
Barenblatt-type solution of a proper singular fractional problem. If, on the
contrary, the density decays rapidly at infinity, we show that the minimal
solution multiplied by a suitable power of the time variable converges to the
minimal solution of a certain fractional sublinear elliptic equation.Comment: To appear in DCDS-
A theory of -dissipative solvers for scalar conservation laws with discontinuous flux
We propose a general framework for the study of contractive semigroups
of solutions to conservation laws with discontinuous flux. Developing the ideas
of a number of preceding works we claim that the whole admissibility issue is
reduced to the selection of a family of "elementary solutions", which are
certain piecewise constant stationary weak solutions. We refer to such a family
as a "germ". It is well known that (CL) admits many different contractive
semigroups, some of which reflects different physical applications. We revisit
a number of the existing admissibility (or entropy) conditions and identify the
germs that underly these conditions. We devote specific attention to the
anishing viscosity" germ, which is a way to express the "-condition" of
Diehl. For any given germ, we formulate "germ-based" admissibility conditions
in the form of a trace condition on the flux discontinuity line (in the
spirit of Vol'pert) and in the form of a family of global entropy inequalities
(following Kruzhkov and Carrillo). We characterize those germs that lead to the
-contraction property for the associated admissible solutions. Our
approach offers a streamlined and unifying perspective on many of the known
entropy conditions, making it possible to recover earlier uniqueness results
under weaker conditions than before, and to provide new results for other less
studied problems. Several strategies for proving the existence of admissible
solutions are discussed, and existence results are given for fluxes satisfying
some additional conditions. These are based on convergence results either for
the vanishing viscosity method (with standard viscosity or with specific
viscosities "adapted" to the choice of a germ), or for specific germ-adapted
finite volume schemes
Second-order -regularity in nonlinear elliptic problems
A second-order regularity theory is developed for solutions to a class of
quasilinear elliptic equations in divergence form, including the -Laplace
equation, with merely square-integrable right-hand side. Our results amount to
the existence and square integrability of the weak derivatives of the nonlinear
expression of the gradient under the divergence operator. This provides a
nonlinear counterpart of the classical -coercivity theory for linear
problems, which is missing in the existing literature. Both local and global
estimates are established. The latter apply to solutions to either Dirichlet or
Neumann boundary value problems. Minimal regularity on the boundary of the
domain is required. If the domain is convex, no regularity of its boundary is
needed at all
Local and global behaviour of nonlinear equations with natural growth terms
This paper concerns a study of the pointwise behaviour of positive solutions
to certain quasi-linear elliptic equations with natural growth terms, under
minimal regularity assumptions on the underlying coefficients. Our primary
results consist of optimal pointwise estimates for positive solutions of such
equations in terms of two local Wolff's potentials.Comment: In memory of Professor Nigel Kalto
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