arXiv subjects

Panagiotis E. Souganidis

Publications and source records attributed to Panagiotis E. Souganidis.

49 records · Page 3Linked to original sources

arXiv2012

Stochastic averaging lemmas for kinetic equations

We develop a class of averaging lemmas for stochastic kinetic equations. The velocity is multiplied by a white noise which produces a remarkable change in time scale. Compared to the deterministic case and as far as we work in $L^2$, the nature of regularity on averages is not changed in this stochastic kinetic equation and stays in the range of fractional Sobolev spaces at the price of an additional expectation. However all the exponents are changed; either time decay rates are slower (when the right hand side belongs to $L^2$), or regularity is better when the right hand side contains derivatives. These changes originate from a different space/time scaling in the deterministic and stochastic cases. Our motivation comes from scalar conservation laws with stochastic fluxes where the structure under consideration arises naturally through the kinetic formulation of scalar conservation laws.

math.AP↗

arXiv2012

Stochastic homogenization of level-set convex Hamilton-Jacobi equations

We present a simple new proof for the stochastic homogenization of quasiconvex (level-set convex) Hamilton-Jacobi equations set in stationary ergodic environments. Our approach, which is new even in the convex case, yields more information about the qualitative behavior of the effective nonlinearity.

math.AP↗

arXiv2012

Concentration phenomena for neutronic multigroup diffusion in random environments

We study the asymptotic behavior of the principal eigenvalue of a weakly coupled, cooperative linear elliptic system in a stationary ergodic heterogeneous medium. The system arises as the so-called multigroup diffusion model for neutron flux in nuclear reactor cores, the principal eigenvalue determining the criticality of the reactor in a stationary state. Such systems have been well-studied in recent years in the periodic setting, and the purpose of this work is to obtain results in random media. Our approach connects the linear eigenvalue problem to a system of quasilinear viscous Hamilton-Jacobi equations. By homogenizing the latter, we characterize the asymptotic behavior of the eigenvalue of the linear problem and exhibit some concentration behavior of the eigenfunctions.

math.AP↗

arXiv2012

Stochastic homogenization of $L^\infty$ variational problems

We present a homogenization result for $L^\infty$ variational problems in general stationary ergodic random environments. By introducing a generalized notion of distance function (a special solution of an associated eikonal equation) and demonstrating a connection to absolute minimizers of the variational problem, we obtain the homogenization result as a consequence of the fact that the latter homogenizes in random environments.

math.AP↗

arXiv2012

A homogenization approach for the motion of motor proteins

We consider the asymptotic behavior of an evolving weakly coupled Fokker-Planck system of two equations set in a periodic environment. The magnitudes of the diffusion and the coupling are respectively proportional and inversely proportional to the size of the period. We prove that, as the period tends to zero, the solutions of the system either propagate (concentrate) with a fixed constant velocity (determined by the data) or do not move at all. The system arises in the modeling of motor proteins which can take two different states. Our result implies that, in the limit, the molecules either move along a filament with a fixed direction and constant speed or remain immobile.

math.AP↗

arXiv2011

Homogenization and enhancement of the $G-$equation in random environments

We study the homogenization of a $G$-equation which is advected by a divergence free stationary vector field in a general ergodic random environment. We prove that the averaged equation is an anisotropic deterministic G-equation and we give necessary and sufficient conditions in order to have enhancement. Since the problem is not assumed to be coercive it is not possible to have uniform bounds for the solutions. In addition, as we show, the associated minimal (first passage) time function does not satisfy, in general, the uniform integrability condition which is necessary to apply the sub-additive ergodic theorem. We overcome these obstacles by (i) establishing a new reachability (controllability) estimate for the minimal function and (ii) constructing, for each direction and almost surely, a random sequence which has both a long time averaged limit (due to the sub-additive ergodic theorem) and stays (in the same sense) asymptotically close to the minimal time.

math.OC↗

arXiv2011

Stochastic homogenization of Hamilton-Jacobi and degenerate Bellman equations in unbounded environments

We consider the homogenization of Hamilton-Jacobi equations and degenerate Bellman equations in stationary, ergodic, unbounded environments. We prove that, as the microscopic scale tends to zero, the equation averages to a deterministic Hamilton-Jacobi equation and study some properties of the effective Hamiltonian. We discover a connection between the effective Hamiltonian and an eikonal-type equation in exterior domains. In particular, we obtain a new formula for the effective Hamiltonian. To prove the results we introduce a new strategy to obtain almost sure homogenization, completing a program proposed by Lions and Souganidis that previously yielded homogenization in probability. The class of problems we study is strongly motivated by Sznitman's study of the quenched large deviations of Brownian motion interacting with a Poissonian potential, but applies to a general class of problems which are not amenable to probabilistic tools.

math.AP↗

arXiv2011

Singular Hamilton-Jacobi equation for the tail problem

In this paper we study the long time-long range behavior of reaction diffusion equations with negative square root -type reaction terms. In particular we investigate the exponential behavior of the solutions after a standard hyperbolic scaling. This leads to a Hamilton-Jacobi variational inequality with an obstacle that depends on the solution itself and defines the open set where the limiting solution does not vanish. Counter-examples show a nontrivial lack of uniqueness for the variational inequality depending on the conditions imposed on the boundary of this open set. Both Dirichlet and state constraints boundary conditions play a role. When the competition term does not change sign, we can identify the limit, while, in general, we find lower and upper bounds for the limit. Although models of this type are rather old and extinction phenomena are as important as blow-up, our motivation comes from the so-called "tail problem" in population biology. One way to avoid meaningless exponential tails, is to impose extra-mortality below a given survival threshold. Our study shows that the precise form of this extra-mortality term is asymptotically irrelevant and that, in the survival zone, the population profile is impacted by the survival threshold (except in the very particular case when the competition term is non-positive).

math.AP↗

arXiv2010

Homogenization and enhancement for the G-equation

We consider the so-called G-equation, a level set Hamilton-Jacobi equation, used as a sharp interface model for flame propagation, perturbed by an oscillatory advection in a spatio-temporal periodic environment. Assuming that the advection has suitably small spatial divergence, we prove that, as the size of the oscillations diminishes, the solutions homogenize (average out) and converge to the solution of an effective anisotropic first-order (spatio-temporal homogeneous) level set equation. Moreover we obtain a rate of convergence and show that, under certain conditions, the averaging enhances the velocity of the underlying front. We also prove that, at scale one, the level sets of the solutions of the oscillatory problem converge, at long times, to the Wulff shape associated with the effective Hamiltonian. Finally we also consider advection depending on position at the integral scale.

math.AP↗

arXiv2009

Ergodic problems and periodic homogenization for fully nonlinear equations in half-space type domains with Neumann boundary conditions

We study periodic homogenization problems for second-order pde in half-space type domains with Neumann boundary conditions. In particular, we are interested in "singular problems" for which it is necessary to determine both the homogenized equation and boundary conditions. We provide new results for fully nonlinear equations and boundary conditions. Our results extend previous work of Tanaka in the linear, periodic setting in half-spaces parallel to the axes of the periodicity, and of Arisawa in a rather restrictive nonlinear periodic framework. The key step in our analysis is the study of associated ergodic problems in domains with similar structure.

math.AP↗

arXiv2009

Phasefield theory for fractional diffusion-reaction equations and applications

This paper is concerned with diffusion-reaction equations where the classical diffusion term, such as the Laplacian operator, is replaced with a singular integral term, such as the fractional Laplacian operator. As far as the reaction term is concerned, we consider bistable non-linearities. After properly rescaling (in time and space) these integro-differential evolution equations, we show that the limits of their solutions as the scaling parameter goes to zero exhibit interfaces moving by anisotropic mean curvature. The singularity and the unbounded support of the potential at stake are both the novelty and the challenging difficulty of this work.

math.AP↗

arXiv2008

Convergence of nonlocal threshold dynamics approximations to front propagation

In this note we prove that appropriately scaled threshold dynamics-type algorithms corresponding to the fractional Laplacian of order $α\in (0,2)$ converge to moving fronts. When $α\geqq 1$ the resulting interface moves by weighted mean curvature, while for $α<1$ the normal velocity is nonlocal of ``fractional-type.'' The results easily extend to general nonlocal anisotropic threshold dynamics schemes.

math.AP↗

arXiv2007

Asymmetric potentials and motor effect: a large deviation approach

We provide a mathematical analysis of appearance of the concentrations (as Dirac masses) of the solution to a Fokker-Planck system with asymmetric potentials. This problem has been proposed as a model to describe motor proteins moving along molecular filaments. The components of the system describe the densities of the different conformations of the proteins. Our results are based on the study of a Hamilton-Jacobi equation arising, at the zero diffusion limit, after an exponential transformation change of the phase function that rises a Hamilton-Jacobi equation. We consider different classes of conformation transitions coefficients (bounded, unbounded and locally vanishing).

math.AP↗