Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userGlobal regularity of the three-dimensional fractional micropolar equations
92
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The global well-posedness of the smooth solution to the three-dimensional (3D) incompressible micropolar equations is a difficult open problem. This paper focuses on the 3D incompressible micropolar equations with fractional dissipations $( Delta)^{alpha}u$ and $(-Delta)^{beta}w$.Our objective is to establish the global regularity of the fractional micropolar equations with the minimal amount of dissipations. We prove that, if $alphageq frac{5}{4}$, $betageq 0$ and $alpha+betageqfrac{7}{4}$, the fractional 3D micropolar equations always possess a unique global classical solution for any sufficiently smooth data. In addition, we also obtain the global regularity of the 3D micropolar equations with the dissipations given by Fourier multipliers that are logarithmically weaker than the fractional Laplacian.
rate research
Read More
This article concerns with the global Holder regularity of weak solutions to a class of problems involving the fractional $(p,q)$-Laplacian, denoted by $(-Delta)^{s_1}_{p}+(-Delta)^{s_2}_{q}$, for $1<p,q<infty$ and $s_1,s_2in (0,1)$. We use a suitable Caccioppoli inequality and local boundedness result in order to prove the weak Harnack type inequality. Consequently, by employing a suitable iteration process, we establish the interior Holder regularity for local weak solutions, which need not be assumed bounded. The global Holder regularity result we prove expands and improves the regularity results of Giacomoni, Kumar and Sreenadh (arXiv: 2102.06080) to the subquadratic case (that is, $q<2$) and more general right hand side, which requires a different and new approach. Moreover, we establish a nonlocal Harnack type inequality for weak solutions, which is of independent interest.
We study the existence, uniqueness and regularity of solutions to the $n$-dimensional ($n=2,3$) Camassa-Holm equations with fractional Laplacian viscosity with smooth initial data. It is a coupled system between the Navier-Stokes equations with nonlocal viscosity and a Helmholtz equation. The main difficulty lies in establishing some a priori estimates for the fractional Laplacian viscosity. To achieve this, we need to explore suitable fractional-power Sobolev-type estimates, and bilinear estimates for fractional derivatives. Especially, for the critical case $displaystyle s=frac{n}{4}$ with $n=2,3$, we will make extra efforts for acquiring the expected estimates as obtained in the case $displaystyle frac{n}{4}<s<1$. By the aid of the fractional Leibniz rule and the nonlocal version of Ladyzhenskayas inequality, we prove the existence, uniqueness and regularity to the Camassa-Holm equations under study by the energy method and a bootstrap argument, which rely crucially on the fractional Laplacian viscosity. In particular, under the critical case $s=dfrac{n}{4}$, the nonlocal version of Ladyzhenskayas inequality is skillfully used, and the smallness of initial data in several Sobolev spaces is required to gain the desired results concernig existence, uniqueness and regularity.
We construct a global homeomorphism from any 3D Ricci limit space to a smooth manifold, that is locally bi-Holder. This extends the recent work of Miles Simon and the second author, and we build upon their techniques. A key step in our proof is the construction of local pyramid Ricci flows, existing on uniform regions of spacetime, that are inspired by Hochards partial Ricci flows.
In three previous papers by the two first authors, classes of initial data to the three dimensional, incompressible Navier-Stokes equations were presented, generating a global smooth solution although the norm of the initial data may be chosen arbitrarily large. The main feature of the initial data considered in the last paper is that it varies slowly in one direction, though in some sense it is ``well prepared (its norm is large but does not depend on the slow parameter). The aim of this article is to generalize the setting of that last paper to an ``ill prepared situation (the norm blows up as the small parameter goes to zero).The proof uses the special structure of the nonlinear term of the equation.
We consider suitable weak solutions of the incompressible Navier--Stokes equations in two cases: the 4D time-dependent case and the 6D stationary case. We prove that up to the boundary, the two-dimensional Hausdorff measure of the set of singular points is equal to zero in both cases.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
