Volume 49, Issue 3
A Nonhomogeneous Boundary Value Problem for the Boussinesq Equation on a Bounded Domain

Sheng-Hao Li, Ivonne Rivas & Bing-Yu Zhang

J. Math. Study, 49 (2016), pp. 238-258.

Published online: 2016-09

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  • Abstract

In this paper, we study the well-posedness of an initial-boundary-value problem (IBVP) for the Boussinesq equation on a bounded domain,

\begin{cases}    &u_{tt}-u_{xx}+(u^2)_{xx}+u_{xxxx}=0,\quad x\in (0,1), \;\;t>0,\\    &u(x,0)=\varphi(x),\;\;\; u_t(x,0)=ψ(x),\\    &u(0,t)=h_1(t),\;\;\;u(1,t)=h_2(t),\;\;\;u_{xx}(0,t)=h_3(t),\;\;\;u_{xx}(1,t)=h_4(t).\\   \end{cases} It is shown that the IBVP is locally well-posed in the space $H^s (0,1)$ for any $s\geq 0$ with the initial data $\varphi,$ $\psi$ lie in $H^s(0,1)$ and $ H^{s-2}(0,1)$, respectively, and the naturally compatible boundary data $h_1,$ $h_2$ in the space $H_{loc}^{(s+1)/2}(\mathbb{R}^+)$, and $h_3 $, $h_4$ in the the space of $H_{loc}^{(s-1)/2}(\mathbb{R}^+)$ with optimal regularity.

  • Keywords

Boussinesq equation, initial-boundary value problem, local well-posedness.

  • AMS Subject Headings

35Q53, 35B65

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

li2604@purdue.edu (Sheng-Hao Li)

ivonriv@gmail.com (Ivonne Rivas)

zhangb@ucmail.uc.edu (Bing-Yu Zhang)

  • BibTex
  • RIS
  • TXT
@Article{JMS-49-238, author = {Sheng-Hao and Li and li2604@purdue.edu and 13322 and Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA and Sheng-Hao Li and Ivonne and Rivas and ivonriv@gmail.com and 13323 and Department of Mathematics, Universidad del Valle, Cali, Colombia and Ivonne Rivas and Bing-Yu and Zhang and zhangb@ucmail.uc.edu and 13324 and Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45220, USA and College of Mathematics, Sichuan University, Chengdu, China and Bing-Yu Zhang}, title = {A Nonhomogeneous Boundary Value Problem for the Boussinesq Equation on a Bounded Domain}, journal = {Journal of Mathematical Study}, year = {2016}, volume = {49}, number = {3}, pages = {238--258}, abstract = {

In this paper, we study the well-posedness of an initial-boundary-value problem (IBVP) for the Boussinesq equation on a bounded domain,

\begin{cases}    &u_{tt}-u_{xx}+(u^2)_{xx}+u_{xxxx}=0,\quad x\in (0,1), \;\;t>0,\\    &u(x,0)=\varphi(x),\;\;\; u_t(x,0)=ψ(x),\\    &u(0,t)=h_1(t),\;\;\;u(1,t)=h_2(t),\;\;\;u_{xx}(0,t)=h_3(t),\;\;\;u_{xx}(1,t)=h_4(t).\\   \end{cases} It is shown that the IBVP is locally well-posed in the space $H^s (0,1)$ for any $s\geq 0$ with the initial data $\varphi,$ $\psi$ lie in $H^s(0,1)$ and $ H^{s-2}(0,1)$, respectively, and the naturally compatible boundary data $h_1,$ $h_2$ in the space $H_{loc}^{(s+1)/2}(\mathbb{R}^+)$, and $h_3 $, $h_4$ in the the space of $H_{loc}^{(s-1)/2}(\mathbb{R}^+)$ with optimal regularity.

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v49n3.16.03}, url = {http://global-sci.org/intro/article_detail/jms/1001.html} }
TY - JOUR T1 - A Nonhomogeneous Boundary Value Problem for the Boussinesq Equation on a Bounded Domain AU - Li , Sheng-Hao AU - Rivas , Ivonne AU - Zhang , Bing-Yu JO - Journal of Mathematical Study VL - 3 SP - 238 EP - 258 PY - 2016 DA - 2016/09 SN - 49 DO - http://doi.org/10.4208/jms.v49n3.16.03 UR - https://global-sci.org/intro/article_detail/jms/1001.html KW - Boussinesq equation, initial-boundary value problem, local well-posedness. AB -

In this paper, we study the well-posedness of an initial-boundary-value problem (IBVP) for the Boussinesq equation on a bounded domain,

\begin{cases}    &u_{tt}-u_{xx}+(u^2)_{xx}+u_{xxxx}=0,\quad x\in (0,1), \;\;t>0,\\    &u(x,0)=\varphi(x),\;\;\; u_t(x,0)=ψ(x),\\    &u(0,t)=h_1(t),\;\;\;u(1,t)=h_2(t),\;\;\;u_{xx}(0,t)=h_3(t),\;\;\;u_{xx}(1,t)=h_4(t).\\   \end{cases} It is shown that the IBVP is locally well-posed in the space $H^s (0,1)$ for any $s\geq 0$ with the initial data $\varphi,$ $\psi$ lie in $H^s(0,1)$ and $ H^{s-2}(0,1)$, respectively, and the naturally compatible boundary data $h_1,$ $h_2$ in the space $H_{loc}^{(s+1)/2}(\mathbb{R}^+)$, and $h_3 $, $h_4$ in the the space of $H_{loc}^{(s-1)/2}(\mathbb{R}^+)$ with optimal regularity.

Sheng-Hao Li, Ivonne Rivas & Bing-Yu Zhang. (2019). A Nonhomogeneous Boundary Value Problem for the Boussinesq Equation on a Bounded Domain. Journal of Mathematical Study. 49 (3). 238-258. doi:10.4208/jms.v49n3.16.03
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