Orbital stability of smooth solitary waves for the Degasperis-Procesi equation
Orbital stability of smooth solitary waves for the Degasperis-Procesi equation
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DOI：
10.1090/proc/16087
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发表时间：
2020-02
影响因子：
1
通讯作者：
Ji Li;Yue Liu;Qi-liang Wu
中科院分区：
文献类型：
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作者：
Ji Li;Yue Liu;Qi-liang Wu
Degasperis-Procesi（DP）方程是一个可积的Camassa-Holm型模型，它是浅水波单向传播的渐近近似。这项工作建立了局部光滑孤立波对真实的线上DP方程的轨道稳定性，扩展了我们之前关于其谱稳定性的工作[J. Math. Pures Appl.（9）142（2020），pp. 298-314]。主要困难源于这样一个事实：自然能量空间是L 3 L^3的子空间，但DP方程的平移对称性产生了一个等效于L 2 L^2 -模的守恒量，导致L 3 L^3高阶非线性项在增广的汉密尔顿算子中。但通常的线性估计都是以DP方程的L2 L^2模表示的，不能直接用来控制L3 L^3高阶项。解决的方法是观察到，给定一个充分光滑的初始条件，满足一些温和的约束，扰动的L ∞ L^\infty轨道范数由它的L2 L^2轨道范数的一个函数上有界，从而在L2 <$L ∞ L^2\cap L^\infty空间中得到高阶控制和轨道稳定性.
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.