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Register a new userSharp Decay Estimates and Vanishing Viscosity for Diffusive Hamilton-Jacobi Equations
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Said Benachour
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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.
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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.
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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.
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