Carlota M. Cuesta

Traveling Waves of a Kinetic Transport Model for the KPP-Fisher Equation

A reactive kinetic transport equation whose macroscopic limit is the KPP-Fisher equation is considered. In a scale where collisions occur at a faster rate than reactions, existence of traveling waves close to those of the KPP-Fisher equation is shown. The method adapts a micro-macro decomposition in the spirit of the work of Caflisch and Nicolaenko for the Boltzmann equation. Stability of these waves is shown for perturbations in a weighted

Small- and Waiting-Time Behavior of a Darcy Flow Model with a Dynamic Pressure Saturation Relation

L^2

Weak Shocks for a One-Dimensional BGK Kinetic Model for Conservation Laws

-space, where the weight function is exponential and such that the (macroscopic) linearized operator in the weighted space is self-adjoint and negative definite. Similar approaches to stability of traveling waves are well known for the KPP-Fisher equation.

Analysis of Oscillations in a Drainage Equation

We address the small-time evolution of interfaces (fronts) for the pseudoparabolic generalization \[ {\partial u\over \partial t} = {\partial\over \partial x} \left( u^\alpha {\partial u\over \partial x} + u^\beta {\partial^2 u \over \partial x \partial t} \right) \] of the porous-medium equation, identifying regimes in which the local behavior remains fixed for some finite time and others in which it changes instantaneously. A number of phenomena beyond those exhibited by the porous-medium equation are elucidated, including retreating fronts and novel types of local behavior. Related results for the important limit case \[ {\partial u\over \partial t} ={\partial\over \partial x}\left(u^\beta {\partial^2 u \over \partial x \partial t} \right) \] are also described.

Fluid accumulation in thin-film flows driven by surface tension and gravity

For one-dimensional kinetic BGK models, regarded as relaxation models for scalar conservation laws with genuinely nonlinear fluxes, existence of small amplitude traveling waves is proven. Dynamic stability of these kinetic shock profiles is shown by extending a classical energy method for viscous regularizations of conservation laws.

A pseudo-spectral method for a non-local KdV–Burgers equation posed on R

We analyze rigorously the drainage equation

STABILITY OF SOLITARY WAVES IN A SEMICONDUCTOR DRIFT-DIFFUSION MODEL ∗

(\frac{d^3\Phi}{d\tau^3} +1)\Phi^3 = 1

Self-Similar Lifting and Persistent Touch-Down Points in the Thin-Film Equation

. It is known that all solutions that do not satisfy

Interfaces determined by capillarity and gravity in a two-dimensional porous medium

\Phi\to 1

Existence of Solutions Describing Accumulation in a Thin-Film Flow ∗

as