We consider the linear third order (in time) PDE known as the SMGTJequation, defined on a bounded domain, under the action of either Dirichlet or Neumann boundary control
We establish existence of finite energy weak solutions to the kinetic FokkerPlanck equation and the linear Landau equation near Maxwellian, in the presence of specular reflection boundary condition for general domains. Moreover, by using a method of reflection and the
 Award ID(s):
 2055244
 NSFPAR ID:
 10321247
 Date Published:
 Journal Name:
 Kinetic and Related Models
 Volume:
 15
 Issue:
 3
 ISSN:
 19375093
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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. Optimal interior and boundary regularity results were given in [\begin{document}$ g $\end{document} 1 ], after [41 ], when , which, moreover, in the canonical case\begin{document}$ g \in L^2(0, T;L^2(\Gamma)) \equiv L^2(\Sigma) $\end{document} , were expressed by the wellknown explicit representation formulae of the wave equation in terms of cosine/sine operators [\begin{document}$ \gamma = 0 $\end{document} 19 ], [17 ], [24 ,Vol Ⅱ]. The interior or boundary regularity theory is however the same, whether or\begin{document}$ \gamma = 0 $\end{document} , since\begin{document}$ 0 \neq \gamma \in L^{\infty}(\Omega) $\end{document} is responsible only for lower order terms. Here we exploit such cosine operator basedexplicit representation formulae to provide optimal interior and boundary regularity results with\begin{document}$ \gamma \neq 0 $\end{document} "smoother" than\begin{document}$ g $\end{document} , qualitatively by one unit, two units, etc. in the Dirichlet boundary case. To this end, we invoke the corresponding results for wave equations, as in [\begin{document}$ L^2(\Sigma) $\end{document} 17 ]. Similarly for the Neumann boundary case, by invoking the corresponding results for the wave equation as in [22 ], [23 ], [37 ] for control smoother than , and [\begin{document}$ L^2(0, T;L^2(\Gamma)) $\end{document} 44 ] for control less regular in space than . In addition, we provide optimal interior and boundary regularity results when the SMGTJ equation is subject to interior point control, by invoking the corresponding wave equations results [\begin{document}$ L^2(\Gamma) $\end{document} 42 ], [24 ,Section 9.8.2]. 
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