Blow-up Criterion Of Smooth Solutions Research Articles

A Blowup Criteria of Smooth Solutions to the 3D Boussinesq Equations

A Blowup Criteria of Smooth Solutions to the 3D Boussinesq Equations

Logarithmically improved regularity criterion for the 3D Hall-MHD equations

In this work, we study the blow-up criterion of the smooth solutions of three-dimensional incompressible Hall-magnetohydrohynamics equations (in short, Hall-MHD). We obtain a logarithmically improved regularity criterion of smooth solutions in terms of the $$\dot{B}_{\infty ,\infty }^{0}$$ norm. We improve the blow-up criterion for smooth solutions established in Ye (Appl Anal 96:2669–2683, 2016).

Logarithmically improved blow-up criterion for smooth solutions to the Leray-$\alpha $-magnetohydrodynamic equations

In this paper, the Cauchy problem for the $3D$ Leray-$\alpha $-MHD model is investigated. We obtain the logarithmically improved blow-up criterion of smooth solutions for the Leray-$\alpha $-MHD model in terms of the magnetic field $B$ only in the framework of homogeneous Besov space with negative index.

Blow-up criteria of smooth solutions for magnetic Bénard system with partial viscosity

We study the magnetic Bénard system in two and three-dimensions. Firstly, we obtain a blow-up criterion of smooth solutions to the magnetic Bénard system without zero magnetic diffusion and thermal diffusivity in R2. Secondly, we derive a blow-up criterion of smooth solutions to the magnetic Bénard system without dissipation in Rn(n=2,3), and in what follows, we improve this result.

Local well-posedness for the quasi-geostrophic equations in Besov–Lorentz spaces

In this paper, we establish the local well-posedness for the quasi-geostrophic equations and obtain a blow-up criterion of smooth solutions in the Besov–Lorentz spaces.

Blow-up criteria for the 2½D magnetic Bénard fluid system with partial viscosity

ABSTRACTWe study the Cauchy problem for the 2D incompressible magnetic Bénard fluid system with partial viscosity and derive some blow-up criteria of smooth solutions. More precisely, we first obtain a blow-up criterion of smooth solutions to the 2D magnetic Bénard fluid system without zero magnetic diffusion and thermal diffusivity. Furthermore, we derive two blow-up criteria of smooth solutions to the 2D magnetic Bénard fluid system without dissipation.

Osgood type blow-up criterion for the 3D Boussinesq equations with partial viscosity

This paper is dedicated to studying the blow-up criterion of smooth solutions to the three-dimensional Boussinesq equations with partial viscosity. By means of the Littlewood-Paley decomposition, we give an improved logarithmic Sobolev inequality and through this, we obtain the corresponding blow-up criterion in a space larger than $\dot{B}^0_{\infty,\infty}$, which extends several previous works.

Brezis–Gallouet–Wainger Type Inequalities and Blow-Up Criteria for Navier–Stokes Equations in Unbounded Domains

We shall find the weakest norm that satisfies the Brezis–Gallouet–Wainger type inequality, under some conditions. As an application of the Brezis–Gallouet–Wainger type inequality, we shall establish Beale–Kato–Majda type blow-up criteria of smooth solutions to the 3-D Navier–Stokes equations in unbounded domains.

Blow-up criteria of smooth solutions to the three-dimensional micropolar fluid equations in Besov space

In this paper, we investigate the blow-up criteria of smooth solutions and the regularity of weak solutions to the micropolar fluid equations in three dimensions. We obtain that if $ \nabla_{h}u,\nabla_{h}\omega\in L^{1}(0,T;\dot{B}^{0}_{\infty,\infty})$ or $ \nabla_{h}u,\nabla_{h}\omega\in L^{\frac{8}{3}}(0,T;\dot{B}^{-1}_{\infty,\infty})$ then the solution $(u,\omega)$ can be extended smoothly beyond $t=T$.

Blow-up criteria of smooth solutions to the three-dimensional magneto-micropolar fluid equations

In this short article, the initial value problem for the 3D magneto-micropolar fluid equations is investigated. Some new blow-up criteria of smooth solutions in terms of the vorticity and the velocity in a homogenous Besov space are established, respectively.

Blowup criterion of smooth solutions for the incompressible chemotaxis-Euler equations

In this paper, we establish the blowup criterion of smooth solutions for the incompressible chemotaxis-Euler equations in with by and .

Blow-up criterion of smooth solutions for the Boussinesq equations

This paper is dedicated to the study the Boussinesq equations with fractional dissipation in n-dimensions (n≥2). We obtain a blow-up criterion for the local in time classical solution to the Boussinesq equations with arbitrarily small fractional powers of the Laplacian. Therefore our results significantly improve the previous works and are more closer to the resolution of the well-known regularity criterion issue on the inviscid Boussinesq equations.

On the well-posedness of the incompressible porous media equation in Triebel-Lizorkin spaces

In this paper, we prove the local well-posedness for the incompressible porous media equation in Triebel-Lizorkin spaces and obtain blow-up criterion of smooth solutions. The main tools we use are the Fourier localization technique and Bony’s paraproduct decomposition.MSC:76S05, 76D03.

On the blow-up criterion of smooth solutions for Hall-magnetohydrodynamics system with partial viscosity

In this paper we study the initial problem for the Hall-magnetohydrodynamics system with partial viscosity in $ R^n(n=2, 3)$. We obtain a Beale-Kato-Majda type blow up criterion of smooth solutions.

Blow-up criteria for smooth solutions to the generalized 3D MHD equations

In this paper, we focus on the generalized 3D magnetohydrodynamic equations. Two logarithmically blow-up criteria of smooth solutions are established. MSC:76D03, 76W05.

A logarithmically improved blow-up criterion for smooth solutions to the micropolar fluid equations in weak multiplier spaces

In this paper, we study the initial value problem for the three-dimensional micropolar fluid equations. A new logarithmically improved blow-up criterion for the three-dimensional micropolar fluid equations in a weak multiplier space is established. MSC:35K15, 35K45.

Blowup of Smooth Solutions for an Aggregation Equation

We study the blowup criterion of smooth solutions for an inviscid aggregation equation in <svg style="vertical-align:-0.1092pt;width:19.3125px;" id="M1" height="13.75" version="1.1" viewBox="0 0 19.3125 13.75" width="19.3125" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,13.55)"><path id="x211D" d="M751 18q-1 -4 -2 -13t-2 -13q-24 0 -35 1q-57 3 -92.5 23t-69.5 68q-21 30 -101 160q-13 22 -30.5 31t-54.5 9h-32v-159q0 -62 14 -77t74 -20v-28h-382v28q62 5 76 19.5t14 77.5v400q0 63 -14 77.5t-76 19.5v28h372q109 0 159 -34q67 -42 67 -131q0 -115 -127 -172&#xA;q27 -52 89 -144q49 -71 81 -104q31 -36 72 -47zM543 469q0 76 -42 112t-104 36q-41 0 -54 -10q-11 -7 -11 -44v-247h48q73 0 108 29q55 42 55 124zM290 131v390q0 62 -7.5 79.5t-30.5 17.5q-24 0 -31.5 -17.5t-7.5 -79.5v-390q0 -62 7.5 -79.5t31.5 -17.5q23 0 30.5 17.5&#xA;t7.5 79.5z" /></g> <g transform="matrix(.012,-0,0,-.012,12.663,5.388)"><path id="x1D45B" d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2&#xA;q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z" /></g> </svg>. By means of the losing estimates and the logarithmic Sobolev inequality, we establish an improved blowup criterion of smooth solutions.

A logarithmically improved blow-up criterion of smooth solutions for nematic liquid crystal flows with partial viscosity

In this paper we investigate the Cauchy problem for nematic liquid crystal flows with partial viscosity in R 3 . A logarithmically improved blow-up criterion of smooth solutions is established. The result is analogous to the celebrated Beale-Kato-Majda type criterion for the inviscid Euler equations of incompressible fluids.

A note on the blow-up criterion of smooth solutions to the 3D incompressible MHD equations

In this note, we will give a new proof of the blow-up criterion of smooth solutions to the 3D incompressible magneto-hydrodynamic equations by a simple application of Gagliardo-Nirenberg’ s inequality.

Remarks on the blow-up criterion for smooth solutions of the Boussinesq equations with zero diffusion

This article is concerned with the blow-up criterion for smooth solutions of three-dimensional Boussinesq equations with zero diffusion. It is shown that if the velocity field $u(x,t)$ satisfies \begin{eqnarray*} u\in L^p(0,T_1;B^r_{q,\infty}(R^3)),\quad \frac{2}{p}+\frac{3}{q}=1+r,\quad \frac{3}{1+r}< q \leq \infty, \quad -1 < r \leq 1, \end{eqnarray*} then the solution can be continually extended to the interval $(0,T)$ for some $T>T_1$.