大振幅浅水波模型的柯西问题研究  

On the Cauchy Problem for a Shallow Water Regime of Waves with Large Amplitude

作　　者：蔡森林 周寿明 陈容 Cai Senin;Zhou Shouming;Chen Rong(College of Mathematics Science,Chongqing Normal University,Chongqing 401331)

机构地区：[1]重庆师范大学数学科学学院,重庆市401331

出　　处：《数学物理学报（A辑）》2023年第4期1197-1220,共24页Acta Mathematica Scientia

基　　金：国家自然科学基金(11971082);重庆市自然科学基金项目(csts2020jcyj-jqX0022);重庆英才青年拔尖人才(cstc2021ycjh-bgzxm0130);重庆市教育委员会科学技术研究项目(KJZD-M202200501,KJZD-M201900501,KJQN202000518);重庆市留学人员回国创业创新支持计划(cx2022029)。

摘　　要：该文考虑单参数族浅水波方程的柯西问题,该模型是在参数δ<<1,ε=O(√δ)的范围内联合质量守恒方程对欧拉方程进行逼近展开得到的.首先考虑大振幅浅水波方程的解在Sobolev空间H^(s)(R),s>3/2中的局部适定性,这意味着初值到解的映射是存在且唯一的且连续依赖于初值.该文还进一步证明了初值到解映射的这种依赖关系在此Sobolev空间中是非一致连续的,但这种依赖关系在Sobolev空间H^(r)(0≤r<s)中是Holder连续的,并且Holder指数γ依赖于s和r,同时分析了该模型只会以波裂的形式发生爆破.最后,该文还研究了当初值属于加权空间L_(φ)^(p):=L^(p)(R,φ^(p)dx)时,方程的强解在空间变量趋于无穷远时的渐近行为.In this paper,we Considered herein the Cauchy problem for a one-parameter fam-ily shallow water wave equation which approximate the Euler's equations of motion and the equation of mass conservation in the regime ofδ<<1,ε=O(√δ).We first establish that this surface equation for shallow water waves of large amplitude is local well-posedness in Sobolev spaces Hs(R)with s>,which implies that the data-to-solution map is existence,uniqueness and continuous dependence on their initial data,we further show that this dependence is not uniformly continuous in these Sobolev spaces.Moreover,we obtain that the data-to-solution map for this shallow water wave equation is Holder continuous in the sense of H^(r)(R)-topology for all O≤r<s,and the Holder exponentγdepending on s and r.Then,the precise blow-up mechanism for the strong solutions is determined in the Sobolev space Hs with s>3/2.In addition,we also investigate the asymptotic behaviors of the strong solutions to this equation at infinity within its lifespan provided the initial data lie in weighted L_(φ)^(p):=L^(p)(R,φ^(p)dx)spaces.

关 键 词：浅水波 局部适定性 非一致连续性 HOLDER连续 爆破 持续性 

分 类 号：O175.29[理学—数学]