1,195 research outputs found
Asymptotic stability of traveling wave solutions for nonlocal viscous conservation laws with explicit decay rates
We consider scalar conservation laws with nonlocal diffusion of Riesz-Feller
type such as the fractal Burgers equation. The existence of traveling wave
solutions with monotone decreasing profile has been established recently (in
special cases). We show the local asymptotic stability of these traveling wave
solutions in a Sobolev space setting by constructing a Lyapunov functional.
Most importantly, we derive the algebraic-in-time decay of the norm of such
perturbations with explicit algebraic-in-time decay rates
Systems of Points with Coulomb Interactions
Large ensembles of points with Coulomb interactions arise in various settings
of condensed matter physics, classical and quantum mechanics, statistical
mechanics, random matrices and even approximation theory, and give rise to a
variety of questions pertaining to calculus of variations, Partial Differential
Equations and probability. We will review these as well as "the mean-field
limit" results that allow to derive effective models and equations describing
the system at the macroscopic scale. We then explain how to analyze the next
order beyond the mean-field limit, giving information on the system at the
microscopic level. In the setting of statistical mechanics, this allows for
instance to observe the effect of the temperature and to connect with
crystallization questions.Comment: 30 pages, to appear as Proceedings of the ICM201
A Generalization of the Hopf-Cole Transformation
A generalization of the Hopf-Cole transformation and its relation to the
Burgers equation of integer order and the diffusion equation with quadratic
nonlinearity are discussed. The explicit form of a particular analytical
solution is presented. The existence of the travelling wave solution and the
interaction of nonlocal perturbation are considered. The nonlocal
generalizations of the one-dimensional diffusion equation with quadratic
nonlinearity and of the Burgers equation are analyzed
- …