Break-down criterion for the water-wave equation

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• 作者：Chao Wang ; ZhiFei Zhang
• 关键词：water ; wave ; free surface ; blow ; up criterion
• 刊名：Science China Mathematics
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：60
• 期：1
• 页码：21-58
• 全文大小：464KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Applications of Mathematics;
• 出版者：Science China Press
• ISSN：1869-1862
• 卷排序：60

文摘

We study the break-down mechanism of smooth solution for the gravity water-wave equation of infinite depth. It is proved that if the mean curvature κ of the free surface Σ_t, the trace (V,B) of the velocity at the free surface, and the outer normal derivative \(\frac{{\partial P}}{{\partial n}}\) of the pressure P satisfy $$\begin{array}{*{20}c} {\mathop {\sup }\limits_{t \in [0,T]} \left\| {\kappa (t)} \right\|_{L^p \cap L^2 } + \int_0^T {\left\| {(\nabla V,\nabla B)(t)} \right\|_{L^\infty }^6 dt < + \infty ,} } \\ {\mathop {\inf }\limits_{(t,x,y) \in [0,T] \times \sum _t } - \frac{{\partial P}}{{\partial n}}(t,x,y) \geqslant c_0 ,} \\ \end{array} $$, for some p < 2d and c₀ < 0, then the solution can be extended after t = T.Keywordswater-wavefree surfaceblow-up criterionMSC(2010)35Q3135Q3535R35Download to read the full article textReferences1.Alazard T, Burq N, Zuily C. On the water-wave equations with surface tension. 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Comm Pure Appl Math, 2008, 61: 877–940MathSciNetCrossRefMATHGoogle ScholarCopyright information© Science China Press and Springer-Verlag Berlin Heidelberg 2016Authors and AffiliationsChao Wang1ZhiFei Zhang1Email author1.School of Mathematical SciencesPeking UniversityBeijingChina About this article CrossMark Publisher Name Science China Press Print ISSN 1674-7283 Online ISSN 1869-1862 About this journal Reprints and Permissions Article actions Export citation .RIS Papers Reference Manager RefWorks Zotero .ENW EndNote .BIB BibTeX JabRef Mendeley Share article Email Facebook Twitter LinkedIn Shareable link Article contents Article Abstract References Copyright information Authors and Affiliations About this article Cookies We use cookies to improve your experience with our site. More information Accept Over 10 million scientific documents at your fingertips