We consider the linear third order (in time) PDE known as the SMGTJ-equation, defined on a bounded domain, under the action of either Dirichlet or Neumann boundary control
Orbital stability of smooth solitary waves for the Degasperis-Procesi equation
The Degasperis-Procesi (DP) equation is an integrable Camassa-Holm-type model which is an asymptotic approximation for the unidirectional propagation of shallow water waves. This work establishes the orbital stability of localized smooth solitary waves to the DP equation on the real line, extending our previous work on their spectral stability [J. Math. Pures Appl. (9) 142 (2020), pp. 298–314]. The main difficulty stems from the fact that the natural energy space is a subspace of L 3 L^3 , but the translation symmetry for the DP equation gives rise to a conserved quantity equivalent to the L 2 L^2 -norm, resulting in L 3 L^3 higher-order nonlinear terms in the augmented Hamiltonian. But the usual coercivity estimate is in terms of L 2 L^2 norm for DP equation, which cannot be used to control the L 3 L^3 higher order term directly. The remedy is to observe that, given a sufficiently smooth initial condition satisfying some mild constraint, the L ∞ L^\infty orbital norm of the perturbation is bounded above by a function of its L 2 L^2 orbital norm, yielding the higher order control and the orbital stability in the L 2 ∩ L ∞ L^2\cap L^\infty space.
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- Award ID(s):
- 1815079
- NSF-PAR ID:
- 10382651
- Date Published:
- Journal Name:
- Proceedings of the American Mathematical Society
- Volume:
- 151
- Issue:
- 763
- ISSN:
- 0002-9939
- Page Range / eLocation ID:
- 151 to 160
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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\begin{document}$ g $\end{document} 1 ], after [41 ], when\begin{document}$ g \in L^2(0, T;L^2(\Gamma)) \equiv L^2(\Sigma) $\end{document} \begin{document}$ \gamma = 0 $\end{document} 19 ], [17 ], [24 ,Vol Ⅱ]. The interior or boundary regularity theory is however the same, whether\begin{document}$ \gamma = 0 $\end{document} \begin{document}$ 0 \neq \gamma \in L^{\infty}(\Omega) $\end{document} \begin{document}$ \gamma \neq 0 $\end{document} \begin{document}$ g $\end{document} \begin{document}$ L^2(\Sigma) $\end{document} 17 ]. Similarly for the Neumann boundary case, by invoking the corresponding results for the wave equation as in [22 ], [23 ], [37 ] for control smoother than\begin{document}$ L^2(0, T;L^2(\Gamma)) $\end{document} 44 ] for control less regular in space than\begin{document}$ L^2(\Gamma) $\end{document} 42 ], [24 ,Section 9.8.2]. -
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1090/proc/16087
