Fractal first-order partial differential equations. (English) Zbl 1111.35144
The authors study the following problems in the unknown \(u\), respectively for a Hamilton-Jacobi equation and for a scalar conservation law:
\[
\left\{ \begin{matrix} \partial_tu(t,x)+g_\lambda[u(t,\cdot)](x)=F(t,x,u(t,x),\nabla u(t,x)),\\ u(0,x)=u_0(x); \end{matrix} \right.
\]
\[ \left\{ \begin{matrix} \partial_tu(t,x)+\text{ div}(f(t,x.u(t,x)))+g_\lambda[u(t,\cdot)](x)=h(t,x,u(t,x)),\\ u(0,x)=u_0(x). \end{matrix} \right. \] Here, \(\lambda\in]0,2[\), and \(g_\lambda\) is the operator \[ g_\lambda[\varphi]={\mathcal F}^{-1}(| \cdot| ^\lambda{\mathcal F}(\varphi)), \] where \(\mathcal F\) is the Fourier transform. The authors give existence and uniqueness results for smooth solutions in the case \(\lambda\in]1,2[\) for both problems. For the general case \(\lambda\in]0,2[\), they prove existence and uniqueness of a viscosity solution for the Hamilton-Jacobi equation. They also study some asymptotic problems. The main ingredients for their results are a suitable integral representation of the operator \(g_\lambda\), and Duhamel’s formula.
\[ \left\{ \begin{matrix} \partial_tu(t,x)+\text{ div}(f(t,x.u(t,x)))+g_\lambda[u(t,\cdot)](x)=h(t,x,u(t,x)),\\ u(0,x)=u_0(x). \end{matrix} \right. \] Here, \(\lambda\in]0,2[\), and \(g_\lambda\) is the operator \[ g_\lambda[\varphi]={\mathcal F}^{-1}(| \cdot| ^\lambda{\mathcal F}(\varphi)), \] where \(\mathcal F\) is the Fourier transform. The authors give existence and uniqueness results for smooth solutions in the case \(\lambda\in]1,2[\) for both problems. For the general case \(\lambda\in]0,2[\), they prove existence and uniqueness of a viscosity solution for the Hamilton-Jacobi equation. They also study some asymptotic problems. The main ingredients for their results are a suitable integral representation of the operator \(g_\lambda\), and Duhamel’s formula.
Reviewer: Fabio Bagagiolo (Trento)
MSC:
| 35S10 | Initial value problems for PDEs with pseudodifferential operators |
| 49L25 | Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games |
| 35F25 | Initial value problems for nonlinear first-order PDEs |
| 35L65 | Hyperbolic conservation laws |
Keywords:
fractal operator; Hamilton-Jacobi equations; conservartion laws; viscosity solutions; existence; uniqueness; Duhamel’s formulaReferences:
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