Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSharp Decay Estimates and Vanishing Viscosity for Diffusive Hamilton-Jacobi Equations
147
0
0.0
(
0
)
Authors
Said Benachour
Ask ChatGPT about the research
No Arabic abstract
Sharp temporal decay estimates are established for the gradient and time derivative of solutions to a viscous Hamilton-Jacobi equation as well the associated Hamilton-Jacobi equation. Special care is given to the dependence of the estimates on the viscosity. The initial condition being only continuous and either bounded or non-negative. The main requirement on the Hamiltonians is that it grows superlinearly or sublinearly at infinity, including in particular H(r) = r^p for r non-negatif and p positif and different from 1.
rate research
Read More
We investigate the convergence rate in the vanishing viscosity process of the solutions to the subquadratic state-constraint Hamilton-Jacobi equations. We give two different proofs of the fact that, for nonnegative Lipschitz data that vanish on the boundary, the rate of convergence is $mathcal{O}(sqrt{varepsilon})$ in the interior. Moreover, the one-sided rate can be improved to $mathcal{O}(varepsilon)$ for nonnegative compactly supported data and $mathcal{O}(varepsilon^{1/p})$ (where $1<p<2$ is the exponent of the gradient term) for nonnegative data $fin mathrm{C}^2(overline{Omega})$ such that $f = 0$ and $Df = 0$ on the boundary. Our approach relies on deep understanding of the blow-up behavior near the boundary and semiconcavity of the solutions.
We prove sharp pointwise decay estimates for critical Dirac equations on $mathbb{R}^n$ with $ngeq 2$. They appear for instance in the study of critical Dirac equations on compact spin manifolds, describing blow-up profiles, and as effective equations in honeycomb structures. For the latter case, we find excited states with an explicit asymptotic behavior. Moreover, we provide some classification results both for ground states and for excited states.
We study in this series of articles the Kardar-Parisi-Zhang (KPZ) equation $$ partial_t h(t,x)= uDelta h(t,x)+lambda V(| abla h(t,x)|) +sqrt{D}, eta(t,x), qquad xin{mathbb{R}}^d $$ in $dge 1$ dimensions. The forcing term $eta$ in the right-hand side is a regularized white noise. The deposition rate $V$ is assumed to be isotropic and convex. Assuming $V(0)ge 0$, one finds $V(| abla h|)ltimes | abla h|^2$ for small gradients, yielding the equation which is most commonly used in the literature. The present article, a continuation of [24], is dedicated to a generalization of the PDE estimates obtained in the previous article to the case of a deposition rate $V$ with polynomial growth of arbitrary order at infinity, for which in general the Cole-Hopf transformation does not allow any more a comparison to the heat equation. The main tool here instead is the representation of $h$ as the solution of some minimization problem through the Hamilton-Jacobi-Bellman formalism. This sole representation turns out to be powerful enough to produce local or pointwise estimates in ${cal W}$-spaces of functions with locally bounded averages, as in [24], implying in particular global existence and uniqueness of solutions.
Global classical solutions to the viscous Hamilton-Jacobi equation with homogenious Dirichlet boundary conditions are shown to converge to zero at the same speed as the linear heat semigroup when p > 1. For p = 1, an exponential decay to zero is also obtained in one space dimension but the rate depends on a and differs from that of the linear heat equation. Finally, if 0 < p < 1 and a < 0, finite time extinction occurs for non-negative solutions.
We study the singular locus of solutions to Hamilton-Jacobi equations with a Hamiltonian independent of $u$. In a previous paper, we proved that the singular locus is what we call a balanced split locus. In this paper, we find and classify all balanced split sets, identifying the cases where the only balanced split locus is the singular locus, and the cases where this doesnt hold. This clarifies the relationship between viscosity solutions and the more classical approach of characteristics and shocks.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
