The convergence to non-diffusive self-similar solutions is investigated for non-negative solutions to the Cauchy problem $\partial_t u = Δ_p u + |\nabla u|^q$ when the initial data converge to zero at infinity. Sufficient conditions on the exponents $p>2$ and $q>1$ are given that guarantee that the diffusion becomes negligible for large times and the $L^\infty$-norm of $u(t)$ converges to a positive value as $t\to\infty$.
Convergence to a single steady state is shown for non-negative and radially symmetric solutions to a diffusive Hamilton-Jacobi equation with homogeneous Dirichlet boundary conditions, the diffusion being the $p$-Laplacian operator, $p\ge 2$, and the source term a power of the norm of the gradient of $u$. As a first step, the radially symmetric and non-increasing stationary solutions are characterized.
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.
In historical mathematics and physics, the Kardar-Parisi-Zhang equation or a quasilinear stationary version of a time-dependent viscous Hamilton-Jacobi equation in growing interface and universality classes, is also known by the different name as the quasilinear Riccati type equation. The existence of solutions to this type of equation under some assumptions and requirements, still remains an interesting open problem at the moment. In our previous studies \cite{MP2018, MPT2019}, we obtained the global bounds and gradient estimates for quasilinear elliptic equations with measure data. There hav…
We consider the diffusive Hamilton-Jacobi equation $u_t - Δu = |\nabla u|^p$ in a bounded planar domain with zero Dirichlet boundary condition. It is known that, for $p>2$, the solutions to this problem can exhibit gradient blow-up (GBU) at the boundary. In this paper we study the possibility of the GBU set being reduced to a single point. In a previous work [Y.-X. Li, Ph. Souplet, 2009], it was shown that single point GBU solutions can be constructed in very particular domains, i.e.~locally flat domains and disks. Here, we prove the existence of single point GBU solutions in a large class of …
A classification of the behavior of the solutions $f(\cdot,a)$ to the ordinary differential equation $(|f'|^{p-2} f')' + f - |f'|^{p-1} = 0$ in $(0,\infty)$ with initial condition $f(0,a)=a$ and $f'(0,a)=0$ is provided, according to the value of the parameter $a>0$, the exponent $p$ ranging in $(1,2)$. There is a threshold value $a_*$ which separates different behaviors of $f(\cdot,a)$: if $a>a_*$ then $f(\cdot,a)$ vanishes at least once in $(0,\infty)$ and takes negative values while $f(\cdot,a)$ is positive in $(0,\infty)$ and decays algebraically to zero as $r\to\infty$ if $a\in (0,a_*)$. A…
We study whether the solutions of a parabolic equation with diffusion given by the fractional Laplacian and a dominating gradient term satisfy Dirichlet boundary data in the classical sense or in the generalized sense of viscosity solutions. The Dirichlet problem is well posed globally in time when boundary data is assumed to be satisfied in the latter sense. Thus, our main results are \emph{a)} the existence of solutions which satisfy the boundary data in the classical sense for a small time, for all Hölder-continuous initial data, with Hölder exponent above a critical a value, and \emph{b)} …
In this article, we are interested in the Dirichlet problem for parabolic viscous Hamilton-Jacobi Equations. It is well-known that the gradient of the solution may blow up in finite time on the boundary of the domain, preventing a classical extension of the solution past this singularity. This behavior comes from the fact that one cannot prescribe the Dirichlet boundary condition for all time and, in order to define a solution globally in time, one has to use "generalized boundary conditions" in the sense of viscosity solution. In this work, we treat the case when the diffusion operator is the…
We consider the diffusive Hamilton-Jacobi equation $$u_t-Δu=|\nabla u|^p,$$ with Dirichlet boundary conditions in two space dimensions, which arises in the KPZ model of growing interfaces. For $p>2$, solutions may develop gradient singularities on the boundary in finite time, and examples of single-point gradient blowup on the boundary are known, but the space-profile in the tangential direction has remained a completely open problem. In the parameter range $2<p\le 3$, for the case of a flat boundary and an isolated singularity at the origin, we give an answer to this question, obtaining the p…
The behavior near the extinction time is identified for non-negative solutions to the diffusive Hamilton-Jacobi equation with critical gradient absorption $\partial_t u - Δ_p u + |\nabla u|^{p-1} = 0$ in $(0, \infty) \times \mathbb{R}^N$, and fast diffusion $2N/(N + 1) < p < 2$. Given a non-negative and radially symmetric initial condition with a non-increasing profile which decays sufficiently fast as $|x| \to \infty$, it is shown that the corresponding solution $u$ to the above equation approaches a uniquely determined separate variable solution of the form $U(t, x) = (T_e - t)^{1/(2-p)} f_*…
We study the asymptotic behavior of solutions to the Dirichlet problem for Hamilton-Jacobi equations with large drift terms, where the drift terms are given by the Hamiltonian vector fields of Hamiltonian $H$. This is an attempt to understand the averaging effect for fully nonlinear degenerate elliptic equations. In this work, we restrict ourselves to the case of Hamilton-Jacobi equations. The second author has already established averaging results for Hamilton-Jacobi equations with convex Hamiltonians ($G$ below) under the classical formulation of the Dirichlet condition. Here we treat the Di…
We provide an example of a Hamilton-Jacobi equation in which stochastic homogenization does not occur. The Hamiltonian involved in this example satisfies the standard assumptions of the literature, except that it is not convex.
We discuss various kinds of representation formulas for the viscosity solutions of the contact type Hamilton-Jacobi equations by using the Herglotz' variational principle.
We study homogenization for a class of stationnary Hamilton-Jacobi equations in which the Hamiltonian is obtained by perturbing near the origin an otherwise periodic Hamiltonian. We prove that the limiting problem consists of a Hamilton-Jacobi equation outside the origin, with the same effective Hamiltonian as in periodic homogenization, supplemented at the origin with an effective Dirichlet condition that keeps track of the perturbation. Various comments and extensions are discussed.
We prove the comparison principle for viscosity sub and super solutions of degenerate nonlocal operators with general nonlocal gradient nonlinearities. The proofs apply to purely Hamilton-Jacobi equations of order $0<s<1$.
We consider the homogenization of monotone systems of viscous Hamilton-Jacobi equations with convex nonlinearities set in the stationary, ergodic setting. The primary focus of this paper is on collapsing systems which, as the microscopic scale tends to zero, average to a deterministic scalar Hamilton-Jacobi equation. However, our methods also apply to systems which do not collapse and, as the microscopic scale tends to zero, average to a deterministic system of Hamilton-Jacobi equations.
We establish that a viscosity solution to a multidimensional Hamilton-Jacobi equation with a convex non-degenerate hamiltonian and Bohr almost periodic initial data decays to its infimum as time $t\to+\infty$.
In this paper, we prove the stochastic homogenization of certain nonconvex Hamilton-Jacobi equations. The nonconvex Hamiltonians, which are generally uneven and inseparable, are generated by a sequence of quasiconvex Hamiltonians and a sequence of quasiconcave Hamiltonians through the min-max formula. We provide a monotonicity assumption on the contact values between those stably paired Hamiltonians so as to guarantee the stochastic homogenzation.
We present two new sharp regularity results (regularizing effect and propagation of regularity) for viscosity solutions of uniformly convex space homogeneous Hamilton-Jacobi equations. In turn, these estimates yield new intermittent stochastic regularization results for pathwise (stochastic) viscosity solutions of Hamilton-Jacobi equations with uniformly convex Hamiltonians and rough multiplicative time dependence. Finally, we use the intermittent estimates to study the long time behavior of the pathwise (stochastic) viscosity solutions of convex Hamilton-Jacobi equations.
This article is dedicated to the proof of C^α regularization effects of Hamilton- Jacobi equations. The proof is based on the De Giorgi method. The regularization is independent on the regularity of the Hamiltonian.
In recent years, there have been many contributions to the vanishing discount problem for Hamilton-Jacobi equations. In the case of the scalar equation, B. Ziliotto [Convergence of the solutions of the discounted Hamilton-Jacobi equation: a counterexample. J. Math. Pures Appl. (9) 128 (2019), 330-338] has shown an example of the Hamilton-Jacobi equation having non-convex Hamiltonian in the gradient variable, for which the full convergence of the solutions does not hold as the discount factor tends to zero. We give an example of the nonlinear monotone system of Hamilton-Jacobi equations having …
We prove that the solution of the discounted approximation of a degenerate viscous Hamilton--Jacobi equation with convex Hamiltonians converges to that of the associated ergodic problem. We characterize the limit in terms of stochastic Mather measures by naturally using the nonlinear adjoint method, and deriving a commutation lemma. This convergence result was first achieved by Davini, Fathi, Iturriaga, and Zavidovique for the first order Hamilton--Jacobi equation.
We study existence, uniqueness and regularity properties of classical solutions to viscous Hamilton-Jacobi equations with Caputo time-fractional derivative. Our study relies on a combination of a gradient bound for the time-fractional Hamilton-Jacobi equation obtained via nonlinear adjoint method and sharp estimates in Sobolev and Hölder spaces for the corresponding linear problem.
We study a cell problem arising in homogenization for a Hamilton-Jacobi equation whose Hamiltonian is not coercive. We introduce a generalized notion of effective Hamiltonians by approximating the equation and characterize the solvability of the cell problem in terms of the generalized effective Hamiltonian. Under some sufficient conditions, the result is applied to the associated homogenization problem. We also show that homogenization for non-coercive equations fails in general.
We provide Sobolev estimates for solutions of first order Hamilton-Jacobi equations with Hamiltonians which are superlinear in the gradient variable. We also show that the solutions are differentiable almost everywhere. The proof relies on an inverse Hölder inequality. Applications to mean field games are discussed.
Cagnetti, Gomes, Mitake and Tran (2013) introduced a new idea to study the large time behavior for degenerate viscous Hamilton--Jacobi equations. In this paper, we apply the method to study the large-time behavior of the solution to the obstacle problem for degenerate viscous Hamilton--Jacobi equations. We establish the convergence result under rather general assumptions.
In this paper, we will prove the random homogenization of general coercive non-convex Hamilton-Jacobi equations in one dimensional case. This extends the result of Armstrong, Tran and Yu when the Hamiltonian has a separable form $H(p,x,ω)=H(p)+V(x,ω)$ for any coercive $H(p)$.
Here, we study the selection problem for the vanishing discount approximation of non-convex, first-order Hamilton-Jacobi equations. While the selection problem is well understood for convex Hamiltonians, the selection problem for non-convex Hamiltonians has thus far not been studied. We begin our study by examining a generalized discounted Hamilton-Jacobi equation. Next, using an exponential transformation, we apply our methods to strictly quasi-convex and to some non-convex Hamilton-Jacobi equations. Finally, we examine a non-convex Hamiltonian with flat parts to which our results do not dire…
This paper is concerned with the asymptotic analysis of infinite systems of weakly coupled stationary Hamilton-Jacobi-Bellman equations as the discount factor tends to zero. With a specific Hamiltonian, we show the convergence of the solution and prove the solvability of the corresponding ergodic problem.
We consider a weakly coupled system of discounted Hamilton--Jacobi equations set on a closed Riemannian manifold. We prove that the corresponding solutions converge to a specific solution of the limit system as the discount factor goes to zero. The analysis is based on a generalization of the theory of Mather minimizing measures for Hamilton--Jacobi systems and on suitable random representation formulae for the discounted solutions.
We investigate asymptotic behaviors of a metric viscosity solution of a Hamilton-Jacobi equation defined on a general metric space in Gangbo-Świȩch sense. Our results include general stability and large time behavior of the solution.
We study Hamilton-Jacobi equations in [0, +$\infty$) of evolution type with nonlinear boundary conditions of Neumann type in the case where the Hamiltonian is non necessarily convex with respect to the gradient variable. In this paper, we give two main results. First, we prove a classification of boundary condition result for a nonconvex, coercive Hamiltonian, in the spirit of the flux-limited formulation for quasi-convex Hamilton-Jacobi equations on networks recently introduced by Imbert and Monneau. Second, we give a comparison principle for a nonconvex and noncoercive Hamiltonian where the …
We obtain space-time Hölder regularity estimates for solutions of first- and second-order Hamilton-Jacobi equations perturbed with an additive stochastic forcing term. The bounds depend only on the growth of the Hamiltonian in the gradient and on the regularity of the stochastic coefficients, in a way that is invariant with respect to a hyperbolic scaling.
In recent years there has been intense interest in the vanishing discount problem for Hamilton-Jacobi equations. In the case of the scalar equation, B. Ziliotto has recently given an example of the Hamilton-Jacobi equation having non-convex Hamiltonian in the gradient variable, for which the full convergence of the solutions does not hold as the discount factor tends to zero. We give here an explicit example of nonlinear monotone systems of Hamilton-Jacobi equations having convex Hamiltonians in the gradient variable, for which the full convergence of the solutions fails as the discount factor…
We present a simple new proof for the stochastic homogenization of quasiconvex (level-set convex) Hamilton-Jacobi equations set in stationary ergodic environments. Our approach, which is new even in the convex case, yields more information about the qualitative behavior of the effective nonlinearity.
We prove the existence and the uniqueness of strong solutions for the viscous Hamilton-Jacobi Equation with Neumann boundary condition and initial data a continious function. Then, we study the large time behavior of the solutions.
We study the representation formulae for the fundamental solutions and viscosity solutions of the Hamilton-Jacobi equations of contact type. We also obtain a vanishing contact structure result for relevant Cauchy problems which can be regarded as an extension to the vanishing discount problem.
We investigate a singular perturbation for Hamilton-Jacobi equations in an open subset of two dimensional Euclidean space, where the set is determined through a Hamiltonian function and the Hamilton-Jacobi equations are the dynamic programming equations for optimal control of the Hamiltonian flow of the Hamiltonian. We establish the convergence of solutions of the Hamilton-Jacobi equations and identify the limit of the solutions as solutions of systems of ordinary differential equations on a graph. The perturbation is singular in the sense that the domain degenerates to the graph in the limit …
We examine Hamilton-Jacobi equations driven by fully nonlinear degenerate elliptic operators in the presence of superlinear Hamiltonians. By exploring the Ishii-Jensen inequality, we prove that viscosity solutions are locally Lipschitz-continuous, with estimates depending on the structural conditions of the problem. We close the paper with an application of our findings to a two-phase free boundary problem.
This paper is concerned with the ergodic problem for viscous Hamilton-Jacobi equations having superlinear Hamiltonian, inward-pointing drift, and positive potential which vanishes at infinity. Assuming some radial symmetry of the drift and the potential outside a ball, we establish sharp estimates of the generalized principal eigenvalue with respect to a perturbation of the potential.
We prove that the Hamilton Jacobi equation for an arbitrary Hamiltonian $H$ (locally Lipschitz but not necessarily convex) and fractional diffusion of order one (critical) has classical $C^{1,α}$ solutions. The proof is achieved using a new Hölder estimate for solutions of advection diffusion equations of order one with bounded vector fields that are not necessarily divergence free.
In this paper a duality relation between the Mañé potential and Mather's action functional is derived in the context of convex and state-dependent Hamiltonians. The duality relation is used to obtain min-max representations of viscosity solutions of first order Hamilton-Jacobi equations. These min-max representations naturally suggest class\-es of subsolutions of Hamilton-Jacobi equations that arise in the theory of large deviations. The subsolutions, in turn, are good candidates for designing efficient rare-event simulation algorithms.
The large time behavior of solutions to Cauchy problem for viscous Hamilton-Jacobi equation is classified. The large time asymptotics are given by very singular self-similar solutions on one hand and by self-similar viscosity solutions on the other hand
Consider the diffusive Hamilton-Jacobi equation $$u_t-Δu=|\nabla u|^p+h(x)\ \ \text{ in } Ω\times(0,T)$$ with Dirichlet conditions, which arises in stochastic control problems as well as in KPZ type models. We study the question of the gradient blowup rate for classical solutions with $p>2$. We first consider the case of time-increasing solutions. For such solutions, the precise rate was obtained by Guo and Hu (2008) in one space dimension, but the higher dimensional case has remained an open question (except for radially symmetric solutions in a ball). Here, we partially answer this question …
In this paper we revisit the mild-solution approach to second-order semi-linear PDEs of Hamilton-Jacobi type in infinite-dimensional spaces. We show that a well-known result on existence of mild solutions in Hilbert spaces can be easily extended to non-autonomous Hamilton-Jacobi equations in Banach spaces. The main tool is the regularizing property of Ornstein-Uhlenbeck transition evolution operators for stochastic Cauchy problems in Banach spaces with time-dependent coefficients
This paper provides a counterexample about the asymptotic behavior of the solutions of a discounted Hamilton-Jacobi equation, as the discount factor vanishes. The Hamiltonian of the equation is a 1-dimensional continuous and coercive Hamiltonian.
We prove that if a sequence of pairs of smooth commuting Hamiltonians converge in the $C^0$ topology to a pair of smooth Hamiltonians, these commute. This allows us define the notion of commuting continuous Hamiltonians. As an application we extend some results of Barles and Tourin on multi-time Hamilton-Jacobi equations to a more general setting.
Capuzzo-Dolcetta and Ishii proved that the rate of periodic homogenization for coercive Hamilton-Jacobi equations is $O(\varepsilon^{1/3})$. We complement this result by constructing examples of coercive nonconvex Hamiltonians whose rate of periodic homogenization is $Ω(\varepsilon^{1/2})$.
The Hamilton-Jacobi equation on metric spaces has been studied by several authors; following the approach of Gangbo and Swiech, we show that the final value problem for the Hamilton-Jacobi equation has a unique solution even if we add a homological term to the Hamiltonian. In metric measure spaces which satisfy the $RCD(K,\infty)$ condition one can define a Laplacian which shares many properties with the ordinary Laplacian on $\R^n$; in particular, it is possible to formulate a viscous Hamilton-Jacobi equation. We show that, if the homological term is sufficiently regular, the viscous Hamilton…
We describe the family of initial conditions for Hamilton-Jacobi equations (HJE) corresponding to optimal control problems that can be retrieved by solving HJE backward in time.
We prove stochastic homogenization for a general class of coercive, nonconvex Hamilton-Jacobi equations in one space dimension. Some properties of the effective Hamiltonian arising in the nonconvex case are also discussed.
We give an example of a stochastic Hamilton-Jacobi equation $du = H(Du) dξ$ which has an infinite speed of propagation as soon as the driving signal $ξ$ is not of bounded variation.
