20 research outputs found
Large-time behavior for viscous and nonviscous Hamilton-Jacobi equations forced by additive noise
We study the large-time behavior of the solutions to viscous and nonviscous Hamilton--Jacobi equations with additive noise and periodic spatial dependence. Under general structural conditions on the Hamiltonian, we show the existence of unique up to constants, global-in-time solutions, which attract any other solution
Some new results on Lipschitz regularization for parabolic equations
It is well known that the bounded solution u(t, x) of the heat equation posed in RNĂ(0,T) for any continuous initial condition becomes Lipschitz continuous as soon as t>0, even if the initial datum is not Lipschitz continuous. We investigate this Lipschitz regularization for both strictly and degenerate parabolic equations of HamiltonâJacobi type. We give proofs avoiding Bernsteinâs method which leads to new, less restrictive conditions on the Hamiltonian, i.e., the first-order term. We discuss also whether the Lipschitz constant depends on the oscillation for the initial datum or not. Finally, some important applications of this Lipschitz regularization are presented
Large Time Behavior of Periodic Viscosity Solutions for Uniformly Parabolic Integro-Differential Equations
International audienceIn this paper, we study the large time behavior of solutions of a class of parabolic fully nonlinear integro-differential equations in a periodic setting. In order to do so, we first solve the ergodic problem}(or cell problem), i.e. we construct solutions of the form . We then prove that solutions of the Cauchy problem look like those specific solutions as time goes to infinity. We face two key difficulties to carry out this classical program: (i) the fact that we handle the case of ''mixed operators'' for which the required ellipticity comes from a combination of the properties of the local and nonlocal terms and (ii) the treatment of the superlinear case (in the gradient variable). Lipschitz estimates previously proved by the authors (2012) and Strong Maximum principles proved by the third author (2012) play a crucial role in the analysis
Jointly invariant measures for the Kardar-Parisi-Zhang Equation
We give an explicit description of the jointly invariant measures for the KPZ
equation. These are couplings of Brownian motions with drift, and can be
extended to a process defined for all drift parameters simultaneously. We term
this process the KPZ horizon (KPZH). As a corollary of this description, we
resolve a recent conjecture of Janjigian, and the second and third authors by
showing the existence of a random, countably infinite dense set of directions
at which the Busemann process of the KPZ equation is discontinuous. This
signals instability and shows the failure of the one force--one solution
principle and the existence of at least two extremal semi-infinite polymer
measures in the exceptional directions. As the inverse temperature parameter
for the KPZ equation goes to , the KPZH converges to the
stationary horizon (SH) first introduced by Busani, and studied further by
Busani and the third and fourth authors. As , the KPZH
converges to a coupling of Brownian motions that differ by linear shifts, which
is a jointly invariant measure for the Edwards-Wilkinson fixed point.Comment: v2: minor changes, 42 pages, 3 figure
Aeronautical Engineering: A continuing bibliography with indexes, supplement 174
This bibliography lists 466 reports, articles and other documents introduced into the NASA scientific and technical information system in April 1984