The Structure of Internal Layers for Unstable Nonlinear Diffusion Equations

We study the structure of diffusive layers in solutions of unstable nonlinear diffusion equations. These equations are regularizations of the forward-backward heat equation and have diffusion coefficients that become negative. Such models include the Cahn-Hilliard equation and the pseudoparabolic viscous diffusion equation. Using singular perturbation methods we show that the balance between diffusion and higher-order regularization terms uniquely determines the interface structure in these equations. It is shown that the well-known “equal area” rule for the Cahn-Hilliard equation is a special case of a more general rule for shock construction in the viscous Cahn-Hilliard equation.     

Implementation of the “Hopscotch” technique in the numerical solution of diffusion problems

An algorithm is given for solving the time-dependent diffusion equation by the “hopscotch” method in cylindrical and Cartesian co-ordinates. Boundaries between media having different diffusion coefficients, solubilities and continuous “sinks” for the diffusing substances may be either along or between lines of grid points.     

Homogenization and concentration for a diffusion equation with large convection in a bounded domain

We consider the homogenization of a non-stationary convection–diffusion equation posed in a bounded domain with periodically oscillating coefficients and homogeneous Dirichlet boundary conditions. Assuming that the convection term is large, we give the asymptotic profile of the solution and determine its rate of decay. In particular, it allows us to characterize the “hot spot”, i.e., the precise asymptotic location of the solution maximum which lies close to the domain boundary and is also the point of concentration. Due to the competition between convection and diffusion, the position of the “hot spot” is not always intuitive as exemplified in some numerical tests.     

A mean field equation as limit of nonlinear diffusions with fractional Laplacian operators

In the limit of a nonlinear diffusion model involving the fractional Laplacian we get a “mean field” equation arising in superconductivity and superfluidity. For this equation, we obtain uniqueness, universal bounds and regularity results. We also show that solutions with finite second moment and radial solutions admit an asymptotic large time limiting profile which is a special self-similar solution: the “elementary vortex patch”.     

On the Existence of Smooth Solutions for Fully Nonlinear Parabolic Equations with Measurable “Coefficients” without Convexity Assumptions

We show that for any uniformly parabolic fully nonlinear second-order equation with bounded measurable “coefficients” and bounded “free” term in any cylindrical smooth domain with smooth boundary data one can find an approximating equation which has a unique continuous solution with the first derivatives bounded and the second spacial derivatives locally bounded. The approximating equation is constructed in such a way that it modifies the original one only for large values of the unknown function and its spacial derivatives.     

A necessary condition for a power series to be a formal solution of a singular linear differential equation of order k

We obtain a necessary condition on the coefficients of a formal power series, which is a formal solution of a nontrivial singular linear differential equation of order k, with analytic coefficients and prove a “uniqueness” theorem for the differential equation.     

Asymptotic analysis of P.D.E.s with wide–band noise disturbances,and expansion of the moments

We present an asymptotic analysis -in the “ white-noise limit”- of a linear parabolic partial differential equation, whose coefficients are perturbed by a wide-band noise. After having studied some ergodic properties of a class of diffusion processes, we prove the convergence in law towards the solution of an Ito stochastic P.D.E. We then establish an expansion in powers of Δ ( 1/Δ being a measure of the bandwith of the driving noise) of the first moment of the solution     

Homogenization of “Viscous” Hamilton–Jacobi Equations in Stationary Ergodic Media

ABSTRACT

We study the homogenization of “viscous” Hamilton–Jacobi equations in stationary ergodic media. The “viscosity” and the spatial oscillations are assumed to be of the same order. We identify the asymptotic (effective) equation, which is a first-order deterministic Hamilton–Jacobi equation. We also provide examples that show that the associated macroscopic problem does not admit suitable solutions (correctors). Finally, we present as applications results about large deviations of diffusion processes and front propagation (asymptotics of reaction-diffusion equations) in random environments.     

Application of Nilpotent Approximation and the Orbit Method to the Search of the Diagonal Asymptotics of Sub-Riemannian Heat Kernels

Siberian Mathematical Journal - We propose a general scheme for the search of a fundamental solution to the hypoelliptic diffusion equation in a “sufficiently good” sub-Riemannian...     

On the dynamic programming principle for uniformly nondegenerate stochastic differential games in domains and the Isaacs equations

We prove the dynamic programming principle for uniformly nondegenerate stochastic differential games in the framework of time-homogeneous diffusion processes considered up to the first exit time from a domain. In contrast with previous results established for constant stopping times we allow arbitrary stopping times and randomized ones as well. There is no assumption about solvability of the the Isaacs equation in any sense (classical or viscosity). The zeroth-order “coefficient” and the “free” term are only assumed to be measurable in the space variable. We also prove that value functions are uniquely determined by the functions defining the corresponding Isaacs equations and thus stochastic games with the same Isaacs equation have the same value functions.     

A Remark on the Global Existence of a Third Order Dispersive Flow into Locally Hermitian Symmetric Spaces

We prove time-global existence of solutions to the initial value problem for a third order dispersive flow into compact locally Hermitian symmetric spaces. The equation under consideration generalizes two-sphere-valued completely integrable systems modeling the motion of vortex filament. Unlike one-dimensional Schrödinger maps, our third order equation is not completely integrable under the curvature condition on the target manifold in general. The idea of our proof is to exploit two conservation laws and an “almost conserved quantity” which prevents the formation of a singularity in finite time.     

The C-finite ansatz

A C-finite sequence is a sequence satisfying a linear recurrence equation with constant coefficients. While it is trivial to multiply two C-finite sequences (just like integers), it is not quite so trivial to “factorize” them, or to decide whether they are “prime”. Here we address these problems.     

Second-order linear differential equations in a Banach space and splitting operators

We consider a second-order linear differential equation whose coefficients are bounded operators acting in a complex Banach space. For this equation with a bounded right-hand side, we study the question on the existence of solutions which are bounded on the whole real axis. An asymptotic behavior of solutions is also explored. The research is conducted under condition that the corresponding “algebraic” operator equation has separated roots or provided that an operator placed in front of the first derivative in the equation has a small norm. In the latter case we apply the method of similar operators, i.e., the operator splitting theorem. To obtain the main results we make use of theorems on the similarity transformation of a second order operator matrix to a block-diagonal matrix.     

Analytical examples of diffusive waves generated by a traveling wave

We construct analytical solutions for a system composed of a reaction–diffusion equation coupled with a purely diffusive equation. The question is to know if the traveling wave solutions of the reaction–diffusion equation can generate a traveling wave for the diffusion equation. Our motivation comes from the calcic wave, generated after fertilization within the egg cell endoplasmic reticulum, and propagating within the egg cell. We consider both the monostable (Fisher–KPP type) and bistable cases. We use a piecewise linear reaction term so as to build explicit solutions, which leads us to compute exponential tails whose exponents are roots of second-, third-, or fourth-order polynomials. These raise conditions on the coefficients for existence of a traveling wave of the diffusion equation. The question of positivity and monotonicity is only partially answered.     

Analytical solution of a spatially variable coefficient advection–diffusion equation in up to three dimensions

Analytical solutions are provided for the two- and three-dimensional advection–diffusion equation with spatially variable velocity and diffusion coefficients. We assume that the velocity component is proportional to the distance and that the diffusion coefficient is proportional to the square of the corresponding velocity component. There is a simple transformation which reduces the spatially variable equation to a constant coefficient problem for which there are available a large number of known analytical solutions for general initial and boundary conditions. These solutions are also solutions to the spatially variable advection–diffusion equation. The special form of the spatial coefficients has practical relevance and for divergent free flow represent corner or straining flow. Unlike many other analytical solutions, we use the transformation to obtain solutions of the spatially variable coefficient advection–diffusion equation in two and three dimensions. The analytical solutions, which are simple to evaluate, can be used to validate numerical models for solving the advection–diffusion equation with spatially variable coefficients. For numerical schemes which cannot handle flow stagnation points, we provide analytical solution to the spatially variable coefficient advection–diffusion equation for two-dimensional corner flow which contains an impermeable flow boundary. The impermeable flow boundary coincides with a streamline along which the fluid velocity is finite but the concentration vanishes. This example is useful for validating numerical schemes designed to predict transport around a curved boundary.     

Extended Lagrangian approach for the defocusing nonlinear Schrödinger equation

We study the defocusing nonlinear Schrödinger (NLS) equation written in hydrodynamic form through the Madelung transform. From the mathematical point of view, the hydrodynamic form can be seen as the Euler–Lagrange equations for a Lagrangian submitted to a differential constraint corresponding to the mass conservation law. The dispersive nature of the NLS equation poses some major numerical challenges. The idea is to introduce a two‐parameter family of extended Lagrangians, depending on a greater number of variables, whose Euler–Lagrange equations are hyperbolic and accurately approximate NLS equation in a certain limit. The corresponding hyperbolic equations are studied and solved numerically using Godunov‐type methods. Comparison of exact and asymptotic solutions to the one‐dimensional cubic NLS equation (“gray” solitons and dispersive shocks) and the corresponding numerical solutions to the extended system was performed. A very good accuracy of such a hyperbolic approximation was observed.     

Équations du filtrage non linéaire de la prédiction et du lissage

We establish equations of non linear filtering, prediction (extrapolation) and smoothing (interpolation) in the case where the signal is a non degenerate diffusion process, and the observation is a noisy functional of the signal. We consider both the case of observation noise correlated with the signal, and the opposite case where we establish “robust” form of the equations. We study finally the case of unbounded coefficients, and the case where there is a feedback from the observation to the signal.     

Polynomial asymptotics near zero points of solutions of general elliptic equations

We extend a classical result of Lipman Bers concerning the local behavior of solutions to a wide class of elliptic equations, systems and inequalities with singular coefficients. The main theorem states that near any zero point of finite vanishing order, the solution is asymptotic to a homogeneous polynomial solution of the “osculating” equation, under mild hypotheses on coefficients. The analysis invloves homothety blow-up arguments with the aid of some elements of Lyapunov exponents in the theory of dynamical systems.     

The Propagation of Finite-Amplitude Waves in a Model Boundary Layer

Finite-amplitude wave propagation is considered in flows of boundary-layer type when the wavelength is long compared to the boundary layer thickness. In this limit, the evolution of the amplitude is governed by the Benjamin-Ono equation and we have computed the coefficients of its nonlinear and dispersive terms for the specific case of Tietjens's model. The propagation of wave packets is also considered, and it is found that for packets centered about an O(1) wavenumber questions again arise relative to long waves, except that now the packet-induced mean flow is the “long wave.” By introducing an appropriate scaling for the far field and employing multiple scales in the direction transverse to the flow, it is shown how the mean-flow distortion can be made to vanish at infinity.     

Strichartz Estimates on Asymptotically de Sitter Spaces

In this article we prove a family of local (in time) weighted Strichartz estimates with derivative losses for the Klein–Gordon equation on asymptotically de Sitter spaces and provide a heuristic argument for the non-existence of a global dispersive estimate on these spaces. The weights in the estimates depend on the mass parameter and disappear in the “large mass” regime. We also provide an application of these estimates to establish small-data global existence for a class of semilinear equations on these spaces.