具有线性恢复力的Boussinesq方程的适定性
Well-posedness of Boussinesq Equation with Linear Restoring Force
- Nazarbayev University（纳扎尔巴耶夫大学）
- School of Computing and Artificial Intelligence (SCAI)（计算与人工智能学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对带线性恢复力的n维Boussinesq方程，在修正Sobolev和Bourgain空间中建立了首个局部适定性结果，通过双线性和三线性估计处理二次和三次非线性。
AI中文摘要：
我们在修正的Sobolev空间$H_\omega^s$和Bourgain型空间$X_\omega^{s,b}$中，为带有线性恢复力的$n$维Boussinesq方程建立了局部适定性理论，这些空间适应于其修正的色散关系。恢复力引入了一个新的相位函数，其在低频区的行为与经典Boussinesq方程不同，进而影响了相关的能量泛函。我们分别证明了二次和三次非线性项的雙线性及三线性估计，这些估计在$s$和$\omega$的适当条件下蕴含局部适定性。据我们所知，这是该模型的第一个适定性结果。
英文摘要：
We establish a local well-posedness theory for the $n-$dimensional Boussinesq equation with a linear restoring force in a modified Sobolev spaces $H_ω^s$ and Bourgain-type spaces $X_ω^{s,b}$, which are adapted to its modified dispersion relation. The restoring force introduces a new phase function whose behavior in the low-frequencies is different from ones of the classical Boussinesq equation, that in turn, affects the associated energy functional. We prove bilinear and trilinear estimates for quadratic and cubic nonlinearities, respectively, which in turn imply local well-posedness under suitable conditions on $s$ and $ω$. To our knowledge, this is the first well-posedness result for this model.
