Well‐posedness and unique continuation property for the generalized Ostrovsky equation with low regularity

Zhang, Zaiyun; Huang, Jianhua

doi:10.1002/mma.3709

Cited by 9 publications

(1 citation statement)

References 61 publications

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“…Bourgain techniques, together with Theorem 1.1, can be used to prove a unique continuation result for other nonlinear models. See also [28] and [27].…”

Section: Introductionmentioning

confidence: 99%

Properties of the support of solutions of a class of nonlinear evolution equations

Bustamante

Urrea

2022

Mathematische Nachrichten

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In this work we consider equations of the form 84.0pt∂tubadbreak+Ptrue(∂xtrue)ugoodbreak+Gtrue(u,∂xu,⋯,∂xlutrue)=0,$$\begin{equation*}\hskip7pc \partial _t u+P\big (\partial _x\big ) u+G\big (u,\partial _xu,\dots ,\partial _x^l u\big )=0, \end{equation*}$$where P is any polynomial without constant term, and G is any polynomial without constant or linear terms. We prove that if u is a sufficiently smooth solution of the equation, such that prefixsuppufalse(0false),prefixsuppufalse(Tfalse)⊂(−∞,B]$\operatorname{supp}u(0),\operatorname{supp}u(T)\subset { (-\infty ,B ]}$ for some B>0$B>0$, then there exists R0>0$R_0>0$ such that prefixsuppu(t)⊂(−∞,R0]$\operatorname{supp}u(t)\subset (-\infty ,R_0]$ for every t∈false[0,Tfalse]$t\in [0,T]$. Then, as an example of the application of this result, we employ it to show a unique continuation principle for the Kawahara equation, 108.0pt∂tubadbreak+∂x5ugoodbreak+∂x3ugoodbreak+u∂xugoodbreak=0,$$\begin{equation*}\hskip9pc \partial _t u+\partial _x^5 u+\partial _x^3 u+u\partial _x u=0, \end{equation*}$$and for the generalized KdV hierarchy 72.0pt∂tubadbreak+false(−1false)k+1∂x2k+1ugoodbreak+Gtrue(u,∂xu,⋯,∂x2kutrue)=0.$$\begin{equation*}\hskip6pc \partial _t u+ (-1)^{k+1}\partial _x^{2k+1} u+G\big (u,\partial _x u,\dots , \partial _x^{2k}u\big ) =0. \end{equation*}$$

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“…Bourgain techniques, together with Theorem 1.1, can be used to prove a unique continuation result for other nonlinear models. See also [28] and [27].…”

Section: Introductionmentioning

confidence: 99%

Properties of the support of solutions of a class of nonlinear evolution equations

Bustamante

Urrea

2022

Mathematische Nachrichten

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Sharp global well‐posedness for the fractional BBM equation

Wang

Zhang

2018

Math Methods in App Sciences

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The Cauchy problem of the fractional Benjamin-Bona-Mahony equation with exponent ∈ (1, 2] on the real line is studied. First, some bilinear estimates for the fractional Benjamin-Bona-Mahony are proved, and the estimates are shown to be sharp by providing counter-examples. Second, the local well-posedness in H s with s ≥ max{0, 3 2 − } is proved by the contraction principle. Finally, the local solution is extended to a global one by the I-method. The well-posedness result turns out to be sharp and fills the gap between the results in Discrete Contin. Dyn. Syst, 23 (2009) 1253-1275 and that in Discrete Contin. Dyn. Syst., 30 (2011) 253-259.

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A trilinear estimate with application to the perturbed nonlinear Schrödinger equations with the Kerr law nonlinearity

et al. 2020

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Well‐posedness and unique continuation property for the generalized Ostrovsky equation with low regularity

Abstract: Communicated by Z. XinIn this paper, we investigate the initial value problem (IVP henceforth) associated with the generalized Ostrovsky equation as follows:

Cited by 9 publications

References 61 publications

Properties of the support of solutions of a class of nonlinear evolution equations

Properties of the support of solutions of a class of nonlinear evolution equations

Sharp global well‐posedness for the fractional BBM equation

A trilinear estimate with application to the perturbed nonlinear Schrödinger equations with the Kerr law nonlinearity

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