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A non-local regularization of first order Hamilton-Jacobi equations. (English) Zbl 1073.35059
This paper is concerned with the non-local first-order Hamilton-Jacobi equation \[ \partial_t u+ H(t,x,u,\nabla u)+ g[u]= 0\quad\text{in }[0,\infty)\times \mathbb{R}^N,\tag{1} \]
\[ u(0,x)= u_0(x)\quad\text{for all }x\in\mathbb{R}^N,\tag{2} \] with \(u_0\in W^{1,\infty}(\mathbb{R}^N)\), where \(\nabla u\) denotes the gradient w.r.t. \(x\), and \(g[u]\) denotes the pseudodifferential operator defined by the symbol \(|\xi|^\lambda\), \(1< \lambda< 2\).
The main result asserts that there exists a solution of (1) with bounded Lipschitz continuous initial condition that is twice continuously differentiable in \(x\) and continuously differentiable in \(t\), i.e. is regular. Firstly, the viscosity solution theory is used to give a notion of merely continuous solution of (1) and to construct a bounded Lipschitz continuous one. Secondly, using Duhamel’s integral representation of (1), an “integral” solution that is \(C^2\) in \(x\) is constructed by a fixed point method and is proved that the “integral” solution is \(C^1\) in \(t\). It finally turns out to be a viscosity solution of (1) (with classical derivatives); the comparison result (which implies uniqueness) permits to conclude.
The author also proves that higher regularity (in fact \(C^\infty\) regularity in \((t,x)\)) can be obtained if the assumptions on \(H\) are strengthened. For \(\lambda= 2\), this method for proving regularity results is new. In the last section, thinking of the vanishing viscosity method, a vanishing Lévy operator is considered and an error estimate is given.

MSC:
35F25 Initial value problems for nonlinear first-order PDEs
35B65 Smoothness and regularity of solutions to PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35G25 Initial value problems for nonlinear higher-order PDEs
35K55 Nonlinear parabolic equations
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