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<p>Considered herein is a Camassa-Holm-type equation with high-order nonlinearities for shallow water waves moving over a shear flow, which can be regarded as the generalization of the compressible hyper-elastic rod model in the material science. We first investigate the non-uniformly continuous dependence of the solutions of Cauchy problem to this equation on the circle in the framework of subcritical and critical Besov spaces. Our analysis relies upon the transport equations theory and the method that constructing approximate solutions. In addition, the results with respect to Hölder continuity of the solution map are stated provided that <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2422_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">\(H^{s}, B^{s}_{p,r}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mi>s</mi> </msup> <mo>,</mo> <msubsup> <mi>B</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>r</mi> </mrow> <mi>s</mi> </msubsup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2422_Article_IEq2.gif" Format="GIF" Height="27" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">\(B^{5/2}_{2,1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>B</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> <mrow> <mn>5</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msubsup> </math></EquationSource> </InlineEquation> are equipped with weaker topologies. Then, according to the generalized Ovsyannikov theorem, the local regularity and analyticity of the solutions in Sobolev-Gevrey spaces are established. Finally, the persistence properties of the solutions, which indicate that the solutions hold the same decay at infinity within the lifespan if the initial data decay exponentially and algebraically, are studied in the critical Besov spaces.</p>

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Some qualitative properties of the solutions of a Camassa-Holm-type equation with high-order nonlinearities for shallow water waves moving over a shear flow

  • 期刊论文

  • 发表日期：

    2025年1月27日

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  • Haiquan Wang

摘要

Considered herein is a Camassa-Holm-type equation with high-order nonlinearities for shallow water waves moving over a shear flow, which can be regarded as the generalization of the compressible hyper-elastic rod model in the material science. We first investigate the non-uniformly continuous dependence of the solutions of Cauchy problem to this equation on the circle in the framework of subcritical and critical Besov spaces. Our analysis relies upon the transport equations theory and the method that constructing approximate solutions. In addition, the results with respect to Hölder continuity of the solution map are stated provided that \(H^{s}, B^{s}_{p,r}\) H s , B p , r s and \(B^{5/2}_{2,1}\) B 2 , 1 5 / 2 are equipped with weaker topologies. Then, according to the generalized Ovsyannikov theorem, the local regularity and analyticity of the solutions in Sobolev-Gevrey spaces are established. Finally, the persistence properties of the solutions, which indicate that the solutions hold the same decay at infinity within the lifespan if the initial data decay exponentially and algebraically, are studied in the critical Besov spaces.