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Existence of viscosity solutions with the optimal regularity of a two-peakon Hamilton--Jacobi equation
- Publication Year :
- 2020
- Publisher :
- arXiv, 2020.
-
Abstract
- This work is devoted to the studies of a Hamilton--Jacobi equation with a quadratic and degenerate Hamiltonian, which comes from the dynamics of a multipeakon in the Camassa--Holm equation. It is given by a quadratic form with a singular positive semi-definite matrix. We increase the regularity of the value function considered in our previous paper, which is known to be the viscosity solution. We prove that for a two-peakon Hamiltonian such solutions are actually $1/2$-H\"{o}lder continuous in space and time-Lipschitz continuous. The time-Lipschitz regularity is proven in any dimension $N\geq 1$. Such a regularity is already known in the one-dimensional simplifications, moreover it is the best possible, as was shown in our previous papers.
- Subjects :
- General Mathematics
010102 general mathematics
Degenerate energy levels
01 natural sciences
Peakon
Hamilton–Jacobi equation
010101 applied mathematics
Viscosity
Quadratic equation
Mathematics - Analysis of PDEs
Quadratic form
FOS: Mathematics
0101 mathematics
Viscosity solution
Analysis
Hamiltonian (control theory)
Mathematics
Mathematical physics
Analysis of PDEs (math.AP)
Subjects
Details
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....3c464d525d9d6cadf8cb5206946ef46e
- Full Text :
- https://doi.org/10.48550/arxiv.2008.02065