Front propagation, biological problems and related topics: viscosity solution methods for asymptotic analysis

Contents

Abstract

P. Cardaliaguet (Université Paris Dauphine, France)
Title：Homogenization of the G equation in random environments
Abstract：
In this joint work with P. Souganidis, 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 random environment. Assuming that the advection is divergence free, 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 show that, under certain conditions, the averaging enhances the velocity of the underlying front.
C.-C. Chen (National Taiwan University)
Title：Travelling wave solutions of the 3-species Lotka-Volterra competition-di ffusion system
Abstract：
This is a joint work with Chueh-Hsin Chang. The existence of a travelling wave solutions of the 3-species Lotka-Volterra competition-diff usion system is established. A travelling wave solution can be considered as a heteroclinic orbit of a vector field in the six dimensional Euclidean space. Under suitable assumptions on the parameters of the equations, we apply a bifurcation theory of heteroclinic orbits to show that a 3-species travelling wave can bifurcate from two 2-species waves which connect to a common equilibrium. The three components of the 3-species wave obtained are positive and have the profiles that one component connects a positive state to zero, one component connects zero to a positive state, and the third component is a pulse between the previous two with a long middle part close to a positive constant. As concrete examples of application of our result, we find several explicit regions of the parameters of the equations where the bifurcations of 3-species travelling waves occur.
N. Dirr (Cardiff University, UK)
Title：Front propoagation in a random environment
Abstract：
We consider the evolution of an interface driven by the competition between mean curvature flow and an external random field which changes sign and oscillates on a small scale. We are interested in deriving the effective equation on a large scale. The scaling is such that the expected limit evolution does not contain a curvature term, i.e. a singular homogenization problem. We motivate the model, review existig results both in the periodic and random case, explain the specific features of the random case and speak about connections with statistical mechanics and with homogenization for viscosity solutions.
Y. Giga (The University of Tokyo)
Title：Eikonal equations in metric spaces
Abstract：
Eikonal equations are very fundamental when we discuss a propagation of an interface or a wave.In a simplest case one considers the Eikonal equations in a domain in an Euclidean space ora manifold or even more generally in a Banach space. However, it is also important to consider these equations in a singular manifold or more generally in Wasserstein metric spaceor in a topological network where there is no strong tangent vector space structure. Thus it is useful to considerthe Eikonal equation |Du|=f in a domain of a general complete metric space. In this talk we introduce a notion of a viscosity solution called a metric viscosity solutions and establish a comparison theorem.The existence ofa solution for a given boundary data is established byconstructing a value function of the corresponding control theory. For zero boundary data the solution is the geodesic distance function from the boundary which is as expected. The consistency of usual solutions in an Euclidean space is also studied. Our formulation applies to a general complete metric space and is expected to have a wide application like social network.This is a joint work with Nao Hamamuki (U. Tokyo) and Atsushi Nakayasu(U. Tokyo).
N. Hamamuki （The University of Tokyo)
Title：Hamilton-Jacobi equations with discontinuous source terms
Abstract：
We discuss the initial value problem for a Hamilton-Jacobi equation whose Hamiltonian is discontinuous with respect to the state variable.

The problem stems from crystal growth phenomena with an external supply of crystal molecules. Such a source of supply is called a step source. A typical equation has a semicontinuous external force term, but we cannot expect the uniqueness of solutions in the standard viscosity sense. In this talk I introduce a suitable notion of solutions and establish comparison principles and existence theorems.

We also study a representation formula of the solution as a value function of an optimal control problem with a semicontinuous running cost function.

This is a joint work with Professor Yoshikazu Giga.
N. Ikoma (Waseda University)
Title：Eigenvalue problem for radially symmetric fully nonlinear operator
Abstract：
In this talk, we are concerned with the eigenvalue problem for fully nonlinear second-order elliptic operator on finite intervals or on the ball. We study the existence of not only the principal eigenpair (pair of an eigenvalue and its corresponding eigenfunction) but also higher eigenpairs. Moreover, the simplicity and completeness of eigenpairs (radial eigenpairs in the higher dimension case) are also considered. This is a joint work with Hitoshi Ishii (Waseda University).
K. Ishii (Kobe University)
Title：An area minimizing scheme for anisotropic mean curvature flow
Abstract：
We consider an area minimizing scheme for anisotropic mean curvature flow originally due to Chambolle [1]. We show the convergence of the scheme to anisotropic mean curvature flow in the sense of Hausdorff distance by the level set method provided that no fattening occurs.
This talk is based on my joint work with Tokuhito Eto (Yahoo JAPAN) and Yoshikazu Giga (The University of Tokyo, JAPAN).
Reference
[1] A. Chambolle, An algorithm for mean curvature motion, Interfaces Free Bound., 6 (2004), 195–218. 1
S. Mirrahimi (Université Pierre et Marie Curie)
Title：Asymptotic dynamics of a population density under selection-mutation: a Hamilton-Jacobi approach
Abstract：
This work is devoted to the mathematical study of concentration phenomena, which appear in problems related to population dynamics. We study the adaptive dynamics of a quantitative trait depending on ecological parameters of the environment, such as the nutrients. The mathematical modeling of these problems gives rise to parabolic equations with small diffusion. The presence of a small term leads to multi-scale models. The asymptotic solutions of these equations concentrate on one or several points in the trait space that are evolving in time. We give a description of the dynamics of Dirac masses, using a Hamilton-Jacobi formulation. This is based on a joint work with Guy Barles and Benoit Perthame.
H. Mitake （Hiroshima University)
Title：On the large time behavior of solutions of weakly coupled systems of Hamilton--Jacobi equations
Abstract：
We discuss the large-time behavior of viscosity solutions of quasi-monotone weakly coupled systems of Hamilton--Jacobi equations on the n-dimensional unit torus. This kind of asymptotic problems for single Hamilton-Jacobi equations have been widely studied since the last decade and it is well-known that the solution converges to a stationary state which is characterized by the so-called ergodic problem under rather general conditions. When we consider the system of equations, it seems that known methods are not able to be applied at least directly.

In the talk, we will first derive the large time asymptotics heuristically and later discuss how to justify it rigorously.
H. Ninomiya (Meiji University)
Title：The traveling spots and rotating waves of the wave front interaction model
Abstract：
Recent experimental studies of photosensitive Belousov-Zhabotinskii reaction has revealed the existence of localized propagating wave. The propagating wave are unstable, but can be stabilized by using a feedback control to continually adjust the excitability of the medium. To study the propagating wave, we deal with the wave front interaction model proposed by Zykov and Showalter. The wave front interaction model consists of two different reduced first order systems of two ordinary differential equations which represent the front and the back of the propagating wave respectively. Using the wave front interaction model, we show the existence of the traveling spots in the plane and the rotating waves in the disk.
P. E. Souganidis (University of Chicago, USA)
Title：Stochastic homogenization revisited and applications to unbounded environments and L∞ variational problems
Abstract：
Some time ago Lions and Souganidis proposed a different approach to obtain the homogenization for Hamilton-Jacobi and viscous Hamilton-Jacobi equations in random environments. This methods yields convergence in probability. In recent joint work with Scott Armstrong we were able to upgrade the convergence to a.s. and to use this to study homogenization in bounded random environments as well as for L∞ variational problems including the ∞ Laplacian.
B. Perthame (Université Pierre et Marie Curie)
Title：Concentration phenomena in nonlocal PDEs and adaptive evolution in population biology (Part 1 - Part 3)
Abstract：
TBA
N. Pozar (The University of Tokyo)
Title：Homogenization of the Hele-Shaw problem in random and periodic media
Abstract：
We study a (one-phase) Hele-Shaw problem, which models a variety of phenomena, including the motion of a fluid injected in between two parallel plates or the flow of a fluid trough a porous medium. Our goal is to present new results on the homogenization of this problem in various settings.

First, we investigate the long-time behavior of an exterior Hele-Shaw problem in random media with a free boundary velocity that depends on the position. A natural rescaling of solutions that is compatible with the evolution of the free boundary leads to the homogenization of the free boundary velocity. By studying a limit obstacle problem for a Hele-Shaw problem with a point source, we are able to show the uniform convergence of the rescaled viscosity solution to a self-similar limit profile and we deduce that the rescaled free boundary uniformly approaches a sphere.

The next part is devoted to a discussion of possible generalizations of the previous homogenization results. One interesting direction is the exterior Hele-Shaw problem in random and periodic media in a setting where the free boundary velocity is merely nonnegative. The possibility that the free boundary will stop results in a loss of uniqueness of the viscosity solutions, and the flow can leave bubbles behind that will persist for all time.

The problem gets more technically challenging if we allow time-dependent free boundary velocity. In this setting, the problem does not admit an obstacle problem formulation and hence one needs to rely exclusively on the viscosity methods.