We present global Schauder type estimates in all variables and unique solvability results in kinetic Hölder spaces for kinetic Kolmogorov-Fokker-Planck (KFP) equations. The leading coefficients are Hölder continuous in the x, v variables and are merely measurable in the temporal variable. Our proof is inspired by Campanato's approach to Schauder estimates and does not rely on the estimates of the fundamental solution of the KFP operator.
In this paper we consider the following stochastic partial differential equation (SPDE) in the whole space:We prove the convergence of a Wong-Zakai type approximation scheme of the above equation in the space C θ ([0, T ], H γ p (R d )) in probability, for some θ ∈ (0, 1/2), γ ∈ (1, 2), and p > 2. We also prove a Stroock-Varadhan's type support theorem. To prove the results we combine V. Mackevičius's ideas from his papers on Wong-Zakai theorem and the support theorem for diffusion processes with N.V. Krylov's Lp-theory of SPDEs. dy n (t) = σ(y n (t)) dw n (t) − 1/2(σDσ)(y n (t)) dt, y n (0) = x 0 , dy(t) = σ(y(t)) dw(t), y(0) = x 0 .This fact can be used in practice to approximate certain SDEs.
<p style='text-indent:20px;'>We establish existence of finite energy weak solutions to the kinetic Fokker-Planck 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 <inline-formula><tex-math id="M1">\begin{document}$ S_p $\end{document}</tex-math></inline-formula> estimate of [<xref ref-type="bibr" rid="b7">7</xref>], we prove regularity in the kinetic Sobolev spaces <inline-formula><tex-math id="M2">\begin{document}$ S_p $\end{document}</tex-math></inline-formula> and anisotropic Hölder spaces for such weak solutions. Such <inline-formula><tex-math id="M3">\begin{document}$ S_p $\end{document}</tex-math></inline-formula> regularity leads to the uniqueness of weak solutions.</p>
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