发表机构
Indian Institute of Technology Roorkee（印度理工学院鲁尔基分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究在全空间及一般无界$\boldsymbol{\textrm{C}}^{1,1}$型域上构造了同时在索伯列夫与勒贝格空间收敛的无散向量值逼近，将其应用于建立不可压缩对流Brinkman-Forchheimer方程弱解的能量等式，还推广了Lions-Magenes引理。
AI 中文摘要
我们在全空间$\boldsymbol{\re}^d$（$d\boldsymbol{\neq}2$）以及一般的无界$\boldsymbol{\textrm{C}}^{1,1}$型区域上构造了无散的向量值逼近函数，这些逼近函数同时在索伯列夫空间与勒贝格空间中收敛。在全空间情形下，我们采用Bogovski\u012d算子来构造此类逼近，从而扩展了针对光滑有界域发展的逼近理论，以及由Fefferman、Hajduk和Robinson（《伦敦数学会会报》（第3辑），第125卷，2022年，第4期，759-777页）提出的同时逼近框架。作为应用，我们在$\boldsymbol{\re}^d$（$d\boldsymbol{\neq}2,3$）上建立了不可压缩对流Brinkman-Forchheimer（CBF）方程的Leray-Hopf弱解的能量等式，涵盖了临界与超临界 regimes。对于一般无界域，我们采用由Farwig、Kozono和Sohr（《数学学报》，第195卷，2005年，21-53页）开发的与Stokes算子相关的预解算子，以获得同时逼近结果。我们还建立了经典Lions-Magenes引理的推广形式，该引理具有独立的研究价值。最后，通过将该结果与针对无界域的同时逼近框架相结合，我们建立了一般无界域上CBF方程弱解的能量等式。
英文摘要
We construct divergence-free, vector-valued approximation functions on the whole space $\mathbb{R}^d$, $d\geq 2$, as well as on general unbounded domains of uniform $\mathrm{C}^{1,1}$-type. These approximations converge simultaneously in both Sobolev and Lebesgue spaces. In the whole-space setting, we employ Bogovski\uı\ operators to construct such approximations, thereby extending the approximation theory developed for smooth bounded domains and the simultaneous approximation framework introduced by \emph{Fefferman, Hajduk, and Robinson, {Proc. Lond. Math. Soc.} (3) \ {125} (2022), no.~4, 759-777}. As an application, we establish energy equality for Leray-Hopf weak solutions of the incompressible convective Brinkman-Forchheimer (CBF) equations on $\mathbb{R}^d$, $d\in\{2,3\}$ covering both the critical and supercritical regimes. For general unbounded domains, we employ the resolvent operator associated with the Stokes operator, developed by \emph{Farwig, Kozono and Sohr, {Acta Math.}, {195} (2005), 21-53}, to obtain simultaneous approximation results. We also establish a generalized version of the classical Lions-Magenes lemma, which is of independent interest. Finally, by combining this result with the simultaneous approximation framework for unbounded domains, we establish energy equality for weak solutions of the CBF equations on general unbounded domains.
