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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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