The aim of this note is to prove sharp regularity estimates for solutions the continuity equation associated vector fields class $W^{1,p}$ with $p>1$. "logarithmic order" and measured by means suitable versions Gargliardo's seminorms.

In this paper, we employ a Schauder-type estimate method, as developed in \cite{CHN}, to establish critical well-posedness result for the Fractional Fokker-Planck Equation. This equation serves fundamental model kinetic theory and can be regarded semi-linear analogue of non-cutoff Boltzmann equation. We demonstrate that techniques introduced study are not only effective FFPE but also hold promise broader applications, particularly addressing Landau Our results contribute deeper understanding...

The energy equalities of compressible Navier-Stokes equations with general pressure law and degenerate viscosities are studied. By using a unified approach, we give sufficient conditions on the regularity weak solutions for these to hold. method proof is suitable case periodic as well homogeneous Dirichlet boundary conditions. In particular, by careful analysis condition, no layer assumptions required when dealing bounded domains boundary.

This article is devoted to the study of Cauchy problem for Muskat equation. We consider initial data belonging critical Sobolev space functions with three-half derivative in $L^2$, up a fractional logarithmic correction. As corollary, we obtain first local and global well-posedness results free surface which are not Lipschitz.

Abstract Two notions of “having a derivative logarithmic order” have been studied. They come from the study regularity flows and renormalized solutions for transport continuity equation associated to weakly differentiable drifts.

.In this paper we study the Peskin problem in two dimensions, which describes dynamics of a one-dimensional closed elastic structure immersed steady Stokes flow. We prove local well-posedness for arbitrary initial configuration \((C^2)^{\dot B^1_{\infty,\infty }}\) satisfying well-stretched condition, and global when is sufficiently close to an equilibrium \(\dot }\). Here closure \(C^2\) Besov space The global-in-time solution will converge exponentially as \(t\rightarrow +\infty\). This...

Abstract We characterize the existence of solutions to quasilinear Riccati-type equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>{</m:mo> <m:mtable columnspacing="0pt" displaystyle="true" rowspacing="0pt"> <m:mtr> <m:mtd columnalign="right"> <m:mo>-</m:mo> <m:mi>div</m:mi> <m:mo>⁡</m:mo> <m:mi mathvariant="script">𝒜</m:mi> </m:mrow> <m:mo>⁢</m:mo> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mo>∇</m:mo> <m:mi>u</m:mi> stretchy="false">)</m:mo>...

Abstract This paper is devoted to the study of flows associated non‐smooth vector fields. We prove well‐posedness regular Lagrangian fields B = ( 1 , …, d ) ∈ L (ℝ + ; ∞ )) satisfying b j BV and div for m ≥ 2 where are singular kernels in ℝ . Moreover, we also show that there exist an autonomous vector‐field Radon measures μ ijk distributional sense some k i 1, such field not unique. © 2021 Wiley Periodicals LLC.

We study a quite general family of nonlinear evolution equations diffusive type with nonlocal effects. More precisely, we porous medium fractional Laplacian pressure, and the problem is posed on bounded space domain. prove existence weak solutions suitable priori bounds regularity estimates.