带一般外力的定常粘性Burgers方程周期解的存在性
Existence of Periodic Solutions to Steady Viscous Burgers Equation with a General Force
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中文总结 AI 辅助
本文针对带一般外力的定常粘性Burgers方程，提出新方法构造满足高阶收敛的周期解，扩展了此前特定外力下的结果，证明了其存在性。
中文摘要 AI 辅助
本文基于形式展开$u^\varepsilon(x)=u_0(x)+\varepsilon u_1(x)+\cdots+\varepsilon^n u_n(x)+\cdots$（其中$\varepsilon\geq0$代表粘性，$u_0(x)$是带外力$f(x)$的非粘性定常Burgers方程的解），构造带外力$f(x)$的定常粘性Burgers方程的周期解。本文关注关于粘性一致有界的解。在作者此前的工作中，从$u_0=-(2+\cos x)$出发，构造了带外力$f=u_0(x)u_{0x}=-2\sin x-\sin x\cos x$的定常粘性Burgers方程的周期解。本文将扩展此前工作的主要结果，从满足非粘性定常Burgers方程$u_0(x)u_{0x}=f$的一般$u_0$和$f$出发构造解。本文将证明存在依赖于$n$的$\varepsilon_0>0$，使得对任意$0<\varepsilon<\varepsilon_0$，带外力$f$的定常粘性Burgers方程存在周期解$u^\varepsilon(x)\in C^2([0,2\pi])$，满足$|u^\varepsilon(x)-u_0(x)-\varepsilon u_1(x)-\cdots-\varepsilon^n u_n(x)|\leq C\varepsilon^{n+1}$，其中$C>0$是依赖于$n$但与$\varepsilon$无关的常数。与Jauslin-Kreiss-Moser的结果相比，本文提出了一种构造带一般外力的定常粘性Burgers方程周期解的新方法；当粘性消失时，所构造的解以更优的收敛速率（直至高阶）趋近于非粘性Burgers方程的解。
英文摘要
In this paper, we will construct periodic solutions to the viscous steady Burgers equation with an external force $f(x)$, based on the following formal expansion $u^\varepsilon(x)=u_0(x)+\varepsilon u_1(x)+\cdots+\varepsilon^n u_n(x)+\cdots$, where $\varepsilon \ge 0$ represents the viscosity and $u_0(x)$ is a solution to non-viscous steady Burgers equation with the external force $f(x)$. We will focus on the solutions which are uniformly bounded with respect to the viscosity. In our previous work, starting from $u_0=-(2+\cos x)$, the authors constructed the periodic solutions to the viscous steady Burgers equation with the external force $f=u_0(x)u_{0x}=-2\sin x-\sin x\cos x$. In this paper, we will extend the main result obtained in our previous work and construct the solutions starting from general $u_0$ and $f$ satisfying the non-viscous steady Burgers equation $u_0(x)u_{0x}=f$. It will be shown that there exists a $\varepsilon_0>0$, which depends on $n$, such that for any $0<\varepsilon<\varepsilon_0$, the viscous steady Burgers equation with the external force $f$ has a periodic solution $u^\varepsilon(x) \in C^2([0,2π])$, satisfying $|u^\varepsilon(x)-u_0(x)-\varepsilon u_1(x)-\cdots-\varepsilon^n u_n(x)| \leq C\varepsilon^{n+1}$, where $C>0$ is a constant which depends on $n$, but is independent of $\varepsilon$. Compared with Jauslin-Kreiss-Moser's result, we present a new approach to construct periodic solutions to the viscous steady Burgers with a general external force. The constructed solutions will tend to the ones of the non-viscous Burgers equation with a sharper convergence rate (up to higher order) when the viscosity vanishes.
发表机构
- School of Mathematical Sciences, Capital Normal University（首都师范大学数学科学学院）
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