Regularity Of Suitable Weak Solutions Research Articles

Partial Regularity of Suitable Weak Solutions of the Model Arising in Amorphous Molecular Beam Epitaxy

Partial Regularity of Suitable Weak Solutions of the Model Arising in Amorphous Molecular Beam Epitaxy

Partially regular weak solutions to the fractional Navier–Stokes equations with the critical dissipation

We show that there exist partially regular weak solutions of Navier–Stokes equations with fractional dissipation −Δs in the critical case of s=32, which satisfy certain local energy inequalities and whose singular sets have a locally finite two-dimensional parabolic Hausdorff measure. Actually, this problem had been studied by Chen and Wei [Discrete Contin. Dyn. Syst. 36(10), 5309–5322 (2016)]; in this paper, they established the partial regularity of suitable weak solutions for s=32. A point is that they admitted the existence of suitable weak solutions but did not give the proof. It should be noted that, when s=32, the existence of suitable weak solutions is not trivial due to the possible lack of compactness. To overcome this difficulty, we shall use a parabolic concentration-compactness theorem introduced by Wu [Arch. Ration. Mech. Anal. 239(3), 1771–1808 (2021)]. For the partial regularity theory, we will apply the idea of Chen and Wei.

On partial regularity criterion for the co-rotational Beris-Edwards system modeling nematic liquid crystal flow

We study the partial regularity of suitable weak solutions to the 3d co-rotational Beris-Edwards system modeling the hydrodynamical motion of nematic liquid crystal flows with the Landau-De Gennes bulk potential, and present an improved version of the Caffarelli-Kohn-Nirenberg criterion obtained recently by Du-Hu-Wang (2020) [9] in terms of the velocity gradient only. The result yields that the velocity field plays a more important role than the Q-tensor field in the partial regularity theory of the 3d co-rotational Beris-Edwards system.

Partial Regularity of Suitable Weak Solutions to the System of the Incompressible Shear-thinning Flow

Partial Regularity of Suitable Weak Solutions to the System of the Incompressible Shear-thinning Flow

On the partial regularity of suitable weak solutions in the non-Newtonian shear-thinning case

We study the partial regularity of suitable weak solutions to incompressible non-Newtonian fluids in the shear-thinning case p < 2. For the shear-thickening case p > 2 this problem was previously considered in 2002 by Guo and Zhu (J. Differ. Equ. 178 281--97). By partially appealing to some of their ideas, we show that in the p < 2 case the singular points are concentrated on a closed set whose one dimensional Hausdorff measure is zero.

$$\varepsilon $$-Regularity criteria for the 3D Navier–Stokes equations in Lorentz spaces

In this paper, we are concerned with the regularity of suitable weak solutions to the 3D Navier–Stokes equations in Lorentz spaces. We obtain $$\varepsilon $$ -regularity criteria in terms of either the velocity, the gradient of the velocity, the vorticity or deformation tensor in Lorentz spaces, which generalizes the corresponding results derived by Gustafson et al. (Commun Math Phys 273:161–176, 2007) in Lebesgue spaces. As applications, this allows us to extend recent results involving Leray’s blow up rate in time, and to show that the number of singular points of weak solutions belonging to $$ L^{p,\infty }(-1,0;L^{q,l}({\mathbb {R}}^{3})) $$ is finite, where the pair (p, q) satisfies $$ {2}/{p}+{3}/{q}=1$$ with $$3<q<\infty $$ and $$q\le l <\infty $$ .

Short Time Regularity of Navier–Stokes Flows with Locally L3 Initial Data and Applications

AbstractWe prove short time regularity of suitable weak solutions of 3D incompressible Navier–Stokes equations near a point where the initial data is locally in $L^3$. The result is applied to the regularity problems of solutions with uniformly small local $L^3$ norms and of forward discretely self-similar solutions.

On local regularity conditions for the Navier–Stokes equations

We address the interior regularity of suitable weak solutions of the 3D incompressible Navier–Stokes equations. We prove a partial regularity criterion based on only one component of the velocity. Furthermore, the uniform estimate obtained in this criterion allows us to prove a generalization of the Leray regularity condition. Finally, we extend the regularity due to Escauriaza–Serëgin–Šverák. Namely, we prove that if is a suitable weak solution and one of the velocity components belongs to , while the remaining two are in the larger Morrey space , and if the pressure satisfies , then the solution is regular.

On Wolf’s Regularity Criterion of Suitable Weak Solutions to the Navier–Stokes Equations

In this paper, we consider the local regularity of suitable weak solutions to the 3D incompressible Navier-Stokes equations. By means of the local pressure projection introduced by Wolf in [15,16], we present a $\varepsilon$-regularity criterion below of suitable weak solutions $$ \iint_{Q(1)}|u|^{20/7}dxdt\leq \varepsilon, $$ which gives an improvement of previous corresponding results obtained in Chae and Wolf [3, Arch. Ration. Mech. Anal., 225: 549-572, 2017], in Guevara and Phuc [6, Calc. Var., 56:68, 2017] and in Wolf [16, Ann. Univ. Ferrara, 61: 149-171, 2015].

On partial regularity of suitable weak solutions to the stationary fractional Navier–Stokes equations in dimension four and five

In this paper, we investigate the partial regularity of suitable weak solutions to the multi-dimensional stationary Navier–Stokes equations with fractional power of the Laplacian (−Δ) α (n/6 ≤ α < 1 and α ≠ 1/2). It is shown that the n + 2 − 6α (3 ≤ n ≤ 5) dimensional Hausdorff measure of the set of the possible singular points of suitable weak solutions to the system is zero, which extends a recent result of Tang and Yu [19] to four and five dimension. Moreover, the pressure in e-regularity criteria is an improvement of corresponding results in [1, 13, 18, 20].

Interior regularity criterion for incompressible Ericksen-Leslie system

An interior regularity criterion of suitable weak solutions is formulated for the Ericksen-Leslie system of liquid crystals. Such a criterion is point-wise, with respect to some appropriate norm of velocity u and the gradient of d, and it can be viewed as a sort of simply sufficient condition on the local regularity of suitable weak solutions.

A new local regularity criterion for suitable weak solutions of the Navier–Stokes equations in terms of the velocity gradient

We study the partial regularity of suitable weak solutions to the three dimensional incompressible Navier–Stokes equations. There have been several attempts to refine the Caffarelli–Kohn–Nirenberg criterion (1982). We present an improved version of the CKN criterion with a direct method, which also provides the quantitative relation in Seregin’s criterion (2007).

Partial regularity of suitable weak solutions to the multi-dimensional generalized magnetohydrodynamics equations

In this paper, we are concerned with the partial regularity of the suitable weak solutions to the fractional MHD equations in [Formula: see text] for [Formula: see text]. In comparison with the work of the 3D fractional Navier–Stokes equations obtained by Tang and Yu in [Partial regularity of suitable weak solutions to the fractional Navier–Stokes equations, Comm. Math. Phys. 334 (2015) 1455–1482], our results include their endpoint case [Formula: see text] and the external force belongs to a more general parabolic Morrey space. Moreover, we prove some interior regularity criteria just via the scaled mixed norm of the velocity for the suitable weak solutions to the fractional MHD equations.

An Anisotropic Partial Regularity Criterion for the Navier–Stokes Equations

In this paper, we address the partial regularity of suitable weak solutions of the incompressible Navier--Stokes equations. We prove an interior regularity criterion involving only one component of the velocity. Namely, if $(u,p)$ is a suitable weak solution and a certain scale-invariant quantity involving only $u_3$ is small on a space-time cylinder $Q_r(x_0,t_0)$, then $u$ is regular at $(x_0,t_0)$.

Anisotropic Regularity Conditions for the Suitable Weak Solutions to the 3D Navier–Stokes Equations

We are concerned with the problem, originated from Seregin (159–200, 2007), Seregin (J. Math. Sci. 143: 2961–2968, 2007), Seregin (Russ. Math. Surv. 62:149–168, 2007), what are minimal sufficiently conditions for the regularity of suitable weak solutions to the 3D Navier–Stokes equations. We prove some interior regularity criteria, in terms of either one component of the velocity with sufficiently small local scaled norm and the rest part with bounded local scaled norm, or horizontal part of the vorticity with sufficiently small local scaled norm and the vertical part with bounded local scaled norm. It is also shown that only the smallness on the local scaled L 2 norm of horizontal gradient without any other condition on the vertical gradient can still ensure the regularity of suitable weak solutions. All these conclusions improve pervious results on the local scaled norm type regularity conditions.

Partial regularity of solutions to the fractional Navier-Stokes equations

We study the partial regularity of suitable weak solutions to the Navier-Stokes equations with fractional dissipation $\sqrt{-\Delta}^s$ in the critical case of $s=\frac{3}{2}$. We show that the two dimensional Hausdorff measure of space-time singular set of these solutions is zero.

Local regularity criteria of the 3D Navier–Stokes and related equations

In this paper, we are concerned with the local regularity of suitable weak solutions to the 3D Navier–Stokes equations and its generalized case with fractional power of the Laplacian (−Δ)α (3/4≤α<1). In the first part, we derive some ϵ-regularity criteria in terms of the deformation tensor D(u) for the classical Navier–Stokes equations. In the second part, for the fractional case, we obtain some regularity conditions for suitable weak solutions including the velocity u, the gradient of the velocity ∇u, the rotation tensor Ω(u) and the deformation tensor D(u). This generalizes the results obtained by Gustafson et al. (2007).

On Partial Regularity for the Steady Hall Magnetohydrodynamics System

We study partial regularity of suitable weak solutions of the steady Hall magnetohydrodynamics equations in a domain $\Omega \subset \Bbb R^3$. In particular we prove that the set of possible singularities of the suitable weak solution has Hausdorff dimension at most one. Moreover, in the case $\Omega=\Bbb R^3$, we show that the set of possible singularities is compact.

Erratum to: Partial Regularity of Suitable Weak Solutions to the Fractional Navier–Stokes Equations

We wish to correct some notations and results contained in the original article. The main results (Theorems 1.2–1.3) remain unchanged but Lemma 2.1 should be slightly corrected.1 Let us start from the corrections for some notations and typos. (I). On Page 3 of original article, we defined the cylinders Qr (x0, t0),Q∗ r (x0, t0), cQr (x0, t0) and cQ∗ r (x0, t0). The center of these cylinders are at (x0, t0). In fact, we need (x0, t0) to be located on the top of these cylinders. Therefore the definitions of Qr (x0, t0),Q∗ r (x0, t0), cQr (x0, t0) and cQ∗ r (x0, t0) on page 3 of original article should be replaced by what is shown below:

Partial Regularity of Suitable Weak Solutions to the Fractional Navier–Stokes Equations

In this paper, we study the partial regularity of fractional Navier–Stokes equations in \({\mathbb{R}^3 \times (0, \infty)}\) with \({3/4 < s < 1}\) . We show that the suitable weak solution is regular away from a relatively closed singular set whose (5−4s)-dimentional Hausdorff measure is zero. The result is a generalization of the partial regularity for the classical Navier–Stokes system in Caffarelli et al. (Commun Pure Appl Math 35:771–831, 1982).