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Register a new userRate of convergence for periodic homogenization of convex Hamilton-Jacobi equations in one dimension
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Son N.T. Tu
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Let $u^varepsilon$ and $u$ be viscosity solutions of the oscillatory Hamilton-Jacobi equation and its corresponding effective equation. Given bounded, Lipschitz initial data, we present a simple proof to obtain the optimal rate of convergence $mathcal{O}(varepsilon)$ of $u^varepsilon rightarrow u$ as $varepsilon rightarrow 0^+$ for a large class of convex Hamiltonians $H(x,y,p)$ in one dimension. This class includes the Hamiltonians from classical mechanics with separable potential. The proof makes use of optimal control theory and a quantitative version of the ergodic theorem for periodic functions in dimension $n = 1$.
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Let $(Omega, mu)$ be a probability space endowed with an ergodic action, $tau$ of $( {mathbb R} ^n, +)$. Let $H(x,p; omega)=H_omega(x,p)$ be a smooth Hamiltonian on $T^* {mathbb R} ^n$ parametrized by $omegain Omega$ and such that $ H(a+x,p;tau_aomega)=H(x,p;omega)$. We consider for an initial condition $fin C^0 ( {mathbb R}^n)$, the family of variational solutions of the stochastic Hamilton-Jacobi equations $$left{ begin{aligned} frac{partial u^{ varepsilon }}{partial t}(t,x;omega)+Hleft (frac{x}{ varepsilon } , frac{partial u^varepsilon }{partial x}(t,x;omega);omega right )=0 & u^varepsilon (0,x;omega)=f(x)& end{aligned} right .$$ Under some coercivity assumptions on $p$ -- but without any convexity assumption -- we prove that for a.e. $omega in Omega$ we have $C^0-lim u^{varepsilon}(t,x;omega)=v(t,x)$ where $v$ is the variational solution of the homogenized equation $$left{ begin{aligned} frac{partial v}{partial t}(x)+{overline H}left (frac{partial v }{partial x}(x) right )=0 & v (0,x)=f(x)& end{aligned} right.$$
We consider the specified stochastic homogenization of first order evolutive Hamilton-Jacobi equations on a very simple junction, i.e the real line with a junction at the origin. Far from the origin, we assume that the considered hamiltonian is closed to given stationary ergodic hamiltonians (which are different on the left and on the right). Near the origin, there is a perturbation zone which allows to pass from one hamiltonian to the other. The main result of this paper is a stochastic homogenization as the length of the transition zone goes to zero. More precisely, at the limit we get two deterministic right and left hamiltonians with a deterministic junction condition at the origin. The main difficulty and novelty of the paper come from the fact that the hamiltonian is not stationary ergodic. Up to our knowledge, this is the first specified stochastic homogenization result. This work is motivated by traffic flow applications.
We study the stochastic homogenization for a Cauchy problem for a first-order Hamilton-Jacobi equation whose operator is not coercive w.r.t. the gradient variable. We look at Hamiltonians like $H(x,sigma(x)p,omega)$ where $sigma(x)$ is a matrix associated to a Carnot group. The rescaling considered is consistent with the underlying Carnot group structure, thus anisotropic. We will prove that under suitable assumptions for the Hamiltonian, the solutions of the $varepsilon$-problem converge to a deterministic function which can be characterized as the unique (viscosity) solution of a suitable deterministic Hamilton-Jacobi problem.
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.
For any compact connected manifold $M$, we consider the generalized contact Hamiltonian $H(x,p,u)$ defined on $T^*Mtimesmathbb R$ which is conex in $p$ and monotonically increasing in $u$. Let $u_epsilon^-:Mrightarrowmathbb R$ be the viscosity solution of the parametrized contact Hamilton-Jacobi equation [ H(x,partial_x u_epsilon^-(x),epsilon u_epsilon^-(x))=c(H) ] with $c(H)$ being the Ma~ne Critical Value. We prove that $u_epsilon^-$ converges uniformly, as $epsilonrightarrow 0_+$, to a specfic viscosity solution $u_0^-$ of the critical equation [ H(x,partial_x u_0^-(x),0)=c(H) ] which can be characterized as a minimal combination of associated Peierls barrier functions.
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