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

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

35Q53, 35B65

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 = {Li , Sheng-Hao and Rivas , Ivonne and Zhang , Bing-Yu}, 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 KW - initial-boundary value problem KW - 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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