15,149 research outputs found
Oscillating waves and optimal smoothing effect for one-dimensional nonlinear scalar conservation laws
Lions, Perthame, Tadmor conjectured in 1994 an optimal smoothing effect for
entropy solutions of nonlinear scalar conservations laws . In this short paper
we will restrict our attention to the simpler one-dimensional case. First,
supercritical geometric optics lead to sequences of solutions
uniformly bounded in the Sobolev space conjectured. Second we give continuous
solutions which belong exactly to the suitable Sobolev space. In order to do so
we give two new definitions of nonlinear flux and we introduce fractional
spaces
Lasry-Lions regularization and a Lemma of Ilmanen
We provide a full self-contained proof of a famous Lemma of Ilmanen. This
proof is based on a regularisation procedure similar to Lasry-Lions
regularisation
Geodesics for a class of distances in the space of probability measures
In this paper, we study the characterization of geodesics for a class of
distances between probability measures introduced by Dolbeault, Nazaret and
Savar e. We first prove the existence of a potential function and then give
necessary and suffi cient optimality conditions that take the form of a coupled
system of PDEs somehow similar to the Mean-Field-Games system of Lasry and
Lions. We also consider an equivalent formulation posed in a set of probability
measures over curves
Long time average of first order mean field games and weak KAM theory
We show that the long time average of solutions of first order mean field
game systems in finite horizon is governed by an ergodic system of mean field
game type. The well-posedness of this later system and the uniqueness of the
ergodic constant rely on weak KAM theory
Existence and uniqueness for Mean Field Games with state constraints
In this paper, we study deterministic mean field games for agents who operate
in a bounded domain. In this case, the existence and uniqueness of Nash
equilibria cannot be deduced as for unrestricted state space because, for a
large set of initial conditions, the uniqueness of the solution to the
associated minimization problem is no longer guaranteed. We attack the problem
by interpreting equilibria as measures in a space of arcs. In such a relaxed
environment the existence of solutions follows by set-valued fixed point
arguments. Then, we give a uniqueness result for such equilibria under a
classical monotonicity assumption
Weak-strong uniqueness of solutions to entropy-dissipating reaction-diffusion equations
We establish a weak-strong uniqueness principle for solutions to
entropy-dissipating reaction-diffusion equations: As long as a strong solution
to the reaction-diffusion equation exists, any weak solution and even any
renormalized solution must coincide with this strong solution. Our assumptions
on the reaction rates are just the entropy condition and local Lipschitz
continuity; in particular, we do not impose any growth restrictions on the
reaction rates. Therefore, our result applies to any single reversible reaction
with mass-action kinetics as well as to systems of reversible reactions with
mass-action kinetics satisfying the detailed balance condition. Renormalized
solutions are known to exist globally in time for reaction-diffusion equations
with entropy-dissipating reaction rates; in contrast, the global-in-time
existence of weak solutions is in general still an open problem - even for
smooth data - , thereby motivating the study of renormalized solutions. The key
ingredient of our result is a careful adjustment of the usual relative entropy
functional, whose evolution cannot be controlled properly for weak solutions or
renormalized solutions.Comment: 32 page
Uniqueness Results for Nonlocal Hamilton-Jacobi Equations
We are interested in nonlocal Eikonal Equations describing the evolution of
interfaces moving with a nonlocal, non monotone velocity. For these equations,
only the existence of global-in-time weak solutions is available in some
particular cases. In this paper, we propose a new approach for proving
uniqueness of the solution when the front is expanding. This approach
simplifies and extends existing results for dislocation dynamics. It also
provides the first uniqueness result for a Fitzhugh-Nagumo system. The key
ingredients are some new perimeter estimates for the evolving fronts as well as
some uniform interior cone property for these fronts
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