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We prove the continuous dependence of the solution maps for the Euler equations in the (critical) Triebel-Lizorkin spaces, which was not shown in the previous works(cite{Ch02, Ch03, ChMiZh10}). The proof relies on the classical Bona-Smith method as cite{GuLiYi18}, where similar result was obtained in critical Besov spaces $B^1_{infty,1}$.
The subject is traces of Sobolev spaces with mixed Lebesgue norms on Euclidean space. Specifically, restrictions to the hyperplanes given by the first and last coordinates are applied to functions belonging to quasi-homogeneous, mixed-norm Lizorkin--
Including the previously untreated borderline cases, the trace spaces in the distributional sense of the Besov--Lizorkin--Triebel spaces are determined for the anisotropic (or quasi-homogeneous) version of these classes. The ranges of the trace are i
We characterize the Schauder and unconditional basis properties for the Haar system in the Triebel-Lizorkin spaces $F^s_{p,q}(Bbb R^d)$, at the endpoint cases $s=1$, $s=d/p-d$ and $p=infty$. Together with the earlier results in [10], [4], this comple
We study the Cauchy problem in $n$-dimensional space for the system of Navier-Stokes equations in critical mixed-norm Lebesgue spaces. Local well-posedness and global well-posedness of solutions are established in the class of critical mixed-norm Leb
Consider Yudovich solutions to the incompressible Euler equations with bounded initial vorticity in bounded planar domains or in $mathbb{R}^2$. We present a purely Lagrangian proof that the solution map is strongly continuous in $L^p$ for all $pin [1