This paper studies a family of generalized surface quasigeostrophic (SQG) equations for an active scalar
This paper investigates the global existence of weak solutions for the incompressible
 Award ID(s):
 2106988
 NSFPAR ID:
 10447711
 Date Published:
 Journal Name:
 Discrete & Continuous Dynamical Systems  B
 Volume:
 27
 Issue:
 1
 ISSN:
 15313492
 Page Range / eLocation ID:
 469
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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on the whole plane whose velocities have been mildly regularized, for instance, logarithmically. The wellposedness of these regularized models in borderline Sobolev regularity have previously been studied by D. Chae and J. Wu when the velocity\begin{document}$ \theta $\end{document} is of lower singularity, i.e.,\begin{document}$ u $\end{document} , where\begin{document}$ u = \nabla^{\perp} \Lambda^{ \beta2}p( \Lambda) \theta $\end{document} is a logarithmic smoothing operator and\begin{document}$ p $\end{document} . We complete this study by considering the more singular regime\begin{document}$ \beta \in [0, 1] $\end{document} . The main tool is the identification of a suitable linearized system that preserves the underlying commutator structure for the original equation. We observe that this structure is ultimately crucial for obtaining continuity of the flow map. In particular, straightforward applications of previous methods for active transport equations fail to capture the more nuanced commutator structure of the equation in this more singular regime. The proposed linearized system nontrivially modifies the flux of the original system in such a way that it coincides with the original flux when evaluated along solutions of the original system. The requisite estimates are developed for this modified linear system to ensure its wellposedness.\begin{document}$ \beta\in(1, 2) $\end{document} 
We consider the linear third order (in time) PDE known as the SMGTJequation, defined on a bounded domain, under the action of either Dirichlet or Neumann boundary control
. Optimal interior and boundary regularity results were given in [\begin{document}$ 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) $\end{document} 42 ], [24 ,Section 9.8.2]. 
Any
conservative map\begin{document}$ C^d $\end{document} of the\begin{document}$ f $\end{document} dimensional unit ball\begin{document}$ d $\end{document} ,\begin{document}$ {\mathbb B}^d $\end{document} , can be realized by renormalized iteration of a\begin{document}$ d\geq 2 $\end{document} perturbation of identity: there exists a conservative diffeomorphism of\begin{document}$ C^d $\end{document} , arbitrarily close to identity in the\begin{document}$ {\mathbb B}^d $\end{document} topology, that has a periodic disc on which the return dynamics after a\begin{document}$ C^d $\end{document} change of coordinates is exactly\begin{document}$ C^d $\end{document} .\begin{document}$ f $\end{document} 
We establish an instantaneous smoothing property for decaying solutions on the halfline
of certain degenerate Hilbert spacevalued evolution equations arising in kinetic theory, including in particular the steady Boltzmann equation. Our results answer the two main open problems posed by Pogan and Zumbrun in their treatment of\begin{document}$ (0, +\infty) $\end{document} stable manifolds of such equations, showing that\begin{document}$ H^1 $\end{document} solutions that remain sufficiently small in\begin{document}$ L^2_{loc} $\end{document} (i) decay exponentially, and (ii) are\begin{document}$ L^\infty $\end{document} for\begin{document}$ C^\infty $\end{document} , hence lie eventually in the\begin{document}$ t>0 $\end{document} stable manifold constructed by Pogan and Zumbrun.\begin{document}$ H^1 $\end{document} 
For any finite horizon Sinai billiard map
on the twotorus, we find\begin{document}$ T $\end{document} such that for each\begin{document}$ t_*>1 $\end{document} there exists a unique equilibrium state\begin{document}$ t\in (0,t_*) $\end{document} for\begin{document}$ \mu_t $\end{document} , and\begin{document}$  t\log J^uT $\end{document} is\begin{document}$ \mu_t $\end{document} adapted. (In particular, the SRB measure is the unique equilibrium state for\begin{document}$ T $\end{document} .) We show that\begin{document}$  \log J^uT $\end{document} is exponentially mixing for Hölder observables, and the pressure function\begin{document}$ \mu_t $\end{document} is analytic on\begin{document}$ P(t) = \sup_\mu \{h_\mu \int t\log J^uT d \mu\} $\end{document} . In addition,\begin{document}$ (0,t_*) $\end{document} is strictly convex if and only if\begin{document}$ P(t) $\end{document} is not\begin{document}$ \log J^uT $\end{document} a.e. cohomologous to a constant, while, if there exist\begin{document}$ \mu_t $\end{document} with\begin{document}$ t_a\ne t_b $\end{document} , then\begin{document}$ \mu_{t_a} = \mu_{t_b} $\end{document} is affine on\begin{document}$ P(t) $\end{document} . An additional sparse recurrence condition gives\begin{document}$ (0,t_*) $\end{document} .\begin{document}$ \lim_{t\downarrow 0} P(t) = P(0) $\end{document}