Morrey Campanato Space Research Articles

ON THE REGULARITY CRITERION ON ONE VELOCITY COMPONENT FOR THE MICROPOLAR FLUID EQUATIONS

In this paper, we establish a regularity criterion for micropolar fluid flows in terms of the one component of the velocity in critical Morrey-Campanato space. More precisely, we show that if ...<∞, where 0<r<9/10 then the weak solution (u,w) is regular.

Regularity criteria for the 3D tropical climate model in Morrey–Campanato space

Regularity criteria for the 3D tropical climate model in Morrey–Campanato space

Note on the weak–strong uniqueness criterion for the β-QG in Morrey–Campanato space

Consider the β-quasi-geostrophic equations with the initial data θ0∈L2(R2). Let θ and θ˜ be two weak solutions with the same initial value θ0. If θ satisfies the energy inequality and if∇θ∈Lr0,T;M.p,q(R2)with2p+αr=α+β−1and2α+β−1<p<3r,where M.p,q(R2) denotes the homogeneous Morrey–Campanato space, then we have θ=θ˜. Due to the embedding relation Lp(R2)⊂M.p,q(R2) with 1 < p < q < ∞, we see that our result is an improvement of the corresponding result given by Zhao and Liu (2013).

Fractional operators with singular drift: smoothing properties and Morrey–Campanato spaces

We investigate some smoothness properties for a linear transport-diffusion equation involving a class of non-degenerate Lévy type operators with singular drift. Our main argument is based on a duality method using the molecular decomposition of Hardy spaces through which we derive some Hölder continuity for the associated parabolic PDE. This property will be fulfilled as far as the singular drift belongs to a suitable Morrey–Campanato space for which the regularizing properties of the Lévy operator suffice to obtain global Hölder continuity.

A regularity criterion for 3D micropolar fluid flows in terms of one partial derivative of the velocity

In this work, we prove a regularity criterion for micropolar fluid flows in terms of the one partial derivative of the velocity in Morrey-Campanato space.

A logarithmically improved regularity criterion for the 3D MHD equations in Morrey-Campanato space

In this paper, we will establish a sufficient condition for the regularity criterion to the 3D MHD equation in terms of the derivative of the pressure in one direction. It is shown that if the partial derivative of the pressure $\partial _{3}\pi $ satisfies the logarithmical Serrin type condition $\partial _{3}\pi $ satisfies the logarithmical Serrin type condition \begin{equation*} \int_{0}^{T}\frac{\left\Vert \partial _{3}\pi (s)\right\Vert _{\overset{% \cdot }{\mathcal{M}}_{2,\frac{3}{r}}}^{\frac{2}{2-r}}}{1+\ln (1+\left\Vert b(s)\right\Vert _{L^{4}})}ds<\infty \text{ for }0<1, \end{equation*} then the solution $(u,b)$ remains smooth on $\left[ 0,T\right] $. Compared to the Navier-Stokes result, there is a logarithmic correction involving $b$ in the denominator.

Blow-up criteria of the 3D boussinesq equations

In this paper, we are concerned with the three-dimensional incompressible Boussinesq equations. We obtain a logarithmically improved blow-up criteria for the system in the Morrey-Campanato space.

Logarithmically improved criteria for the 3D nematic liquid crystal flows in the Morrey–Campanato space

We provide a sufficient condition for the regularity of solutions to the 3D nematic liquid crystal flow in the Morrey–Campanato space. More precisely, we prove that if the velocity u satisfies ∫0T‖u(⋅,t)‖Ṁp,3r21−r1+ln(e+‖u(⋅,t)‖L6)dt<∞with 0<r<1 and 2≤p≤3r, or the gradient of the velocity ∇u satisfies ∫0T‖∇u(⋅,t)‖Ṁp,3γ22−γ1+ln(e+‖u(⋅,t)‖L6)dt<∞with 0<γ≤1 and 2≤p≤3γ, then the solution (u,d) remains smooth on (0,T].

Regularity Criteria of the 3D Boussinesq Equations in the Morrey-Campanato Space

In this paper, we consider the three-dimensional (3D) incompressible Boussinesq equations. We obtain some regularity criteria for the three-dimensional (3D) Boussinesq equations in the Morrey-Campanato space.

Un modèle scalaire analogue aux équations de Navier–Stokes

We discuss on a new scalar model whose properties are similar to the NS equations (it preserves scaling, the antisymmetry of the bilinear term and is invariant by translations and dilations) but contains a singular integral operator. We construct a global-in-time weak solution when initial data is in a critical Morrey–Campanato space and show that it also satisfies a local energy inequality comparable to Schefferʼs one.

Remark on a regularity criterion in terms of pressure for the Navier-Stokes equations

In this note we establish a Serrin-type regularity criterion in terms of pressure for Leray weak solutions to the Navier-Stokes equation in R d \mathbb {R}^{d} . It is known that if a Leray weak solution u u belongs to L 2 1 − r ( ( 0 , T ) ; L d r ) \ \ for some \ \ 0 ≤ r ≤ 1 , \begin{equation} L^{\frac {2}{1-r}}\left ( \left ( 0,T\right ) ;L^{\frac {d}{r}}\right ) \text { \ \ for some \ \ }0\leq r\leq 1, \end{equation} then u u is regular. It is proved that if the pressure p p associated to a Leray weak solution u u belongs to L 2 2 − r ( ( 0 , T ) ; M . 2 , d r ( R d ) d ) , \begin{equation} L^{\frac {2}{2-r}}\left ( \left ( 0,T\right ) ;\overset {.}{\mathcal {M}}_{2,\frac { d}{r}}\left ( \mathbb {R}^{d}\right ) ^{d}\right ) , \end{equation} where M . 2 , d r ( R d ) \overset {.}{\mathcal {M}}_{2,\frac {d}{r}}\left ( \mathbb {R}^{d}\right ) is the critical Morrey-Campanato space (a definition is given in the text) for 0 &gt; r &gt; 1 0&gt;r&gt;1 , then the weak solution is actually regular. Since this space M . 2 , d r \overset {.}{\mathcal {M}}_{2,\frac {d}{r}} is wider than L d r L^{\frac {d}{r}} and X . r \overset {.}{X}_{r} , the above regularity criterion (0.2) is an improvement of Zhou’s result.

On regularity criteria for the three-dimensional micropolar fluid equations in the critical Morrey–Campanato space

In this paper, we establish a Serrin-type regularity criterion for the weak solutions to the micropolar fluid equations in R 3 . It is proved that, if the velocity field satisfies one of the following conditions: u ∈ L 2 1 − r ( 0 , T ; M ̇ 2 , 3 r ( R 3 ) ) or ∇ u ∈ L 2 2 − r ( 0 , T ; M ̇ 2 , 3 r ( R 3 ) ) , or the associated pressure satisfies one of the following conditions: p ∈ L 2 2 − r ( 0 , T ; M ̇ 2 , 3 r ( R 3 ) ) or ∇ p ∈ L 2 3 − r ( 0 , T ; M ̇ 2 , 3 r ( R 3 ) ) , then the solution remains smooth on [ 0 , T ] .

Regularity Criteria in Terms of the Pressure for the Navier–Stokes Equations in the Critical Morrey–Campanato Space

In this paper, we establish a Serrin-type regularity criterion in terms of the pressure for Leray weak solutions to the Navier–Stokes equation in ℝ^3 . It is proved that the solution is regular if the associate pressure satifies p \in L^{\frac 2{2-r}} ((0, T); \dot{\mathcal M}_{2,\frac 3 r} (ℝ^3)) \quad \text{or}\quad ∇p \in L^{\frac 2{3-r}} ((0, T); \dot{\mathcal M}_{2,\frac 3 r} (ℝ^3)) for 0 &lt; r &lt; 1 , where \dot{\mathcal M}_{2,\frac 3 r} is the critical Morrey–Campanto space. Regularity criteria for the 3D MHD equations are also given.

On the regularity criterion of axisymmetric weak solutions to the 3D Navier–Stokes equations

In this paper, we consider the regularity criterion of axisymmetric weak solutions to the Navier–Stokes equations in R 3 . Let u be an axisymmetric weak solution in R 3 × ( 0 , T ) , w = curl u , and w θ be the azimuthal component of w in the cylindrical coordinates. It is proved that u becomes a regular solution if w θ ∈ L 2 2 − s ( 0 , T ; M . 2 , 3 s ) , where M . 2 , 3 s is the critical Morrey–Campanato space.

Localized Morrey-Campanato spaces on metric measure spaces and applications to Schrüdinger operators

AbstractLet be a space of homogeneous type in the sense of Coifman and Weiss, and let be a collection of balls in . The authors introduce the localized atomic Hardy space the localized Morrey-Campanato space and the localized Morrey-Campanato-BLO (bounded lower oscillation) space with α ∊ ℝ and p ∊ (0, ∞) , and they establish their basic properties, including and several equivalent characterizations for In particular, the authors prove that when α &gt; 0 and p ∊ [1, ∞), then and when p ∈(0,1], then the dual space of is Let ρ be an admissible function modeled on the known auxiliary function determined by the Schrödinger operator. Denote the spaces and , respectively, by and when is determined by ρ. The authors then obtain the boundedness from of the radial and the Poisson semigroup maximal functions and the Littlewood-Paley g-function, which are defined via kernels modeled on the semigroup generated by the Schrödinger operator. These results apply in a wide range of settings, for instance, the Schrödinger operator or the degenerate Schrödinger operator on ℝd, or the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups.

Regularity criteria for the 3D magneto-micropolar fluid equations in the Morrey–Campanato space

In this paper, some improved regularity criteria for the 3D magneto-micropolar fluid equations are established in Morrey–Campanato spaces. It is proved that if the velocity field satisfies $$\quad u\in L^{\frac{2}{1-r}}\left(0,T;\overset{.}{\mathcal{M}}_{p,\frac{3}{r}}( \mathbb{R}^{3})\right)\quad\text{with} \;r\in \left( 0,1\right)\;\text{or}\;u\in C\left(0,T;\overset{.}{\mathcal{M}}_{p,\frac{3}{r}}(\mathbb{R} ^{3})\right)$$or the gradient field of velocity satisfies $$\nabla u\in L^{\frac{2}{2-r}}\left(0,T;\overset{.}{\mathcal{M}}_{p,\frac{3}{ r}}(\mathbb{R}^{3})\right)\text{ \ with \ }r\in \left( 0,2\right),$$then the solution remains smooth on [ 0, T] . By the embedding \({ L^{\frac{3}{r}} \subset \overset{.}{\mathcal{M}}_{p,\frac{3}{r}}}\) , we see that our result is an improvement of (Yuan in Acta Mathematica Scientia, to appear).

Unicité des solutions des équations de Navier–Stokes dans les espaces de Morrey–Campanato

In a recent work [P.G. Lemarié-Rieusset, Uniqueness for the Navier–Stokes problem: Remarks on a theorem of Jean-Yves Chemin, Nonlinearity 20 (2007) 1475–1490], P.G. Lemarié-Rieusset proved the uniqueness of solution to the Navier–Stokes equations in the space L p ( [ 0 , T ] , L q ( R d ) ) ∩ C ( [ 0 , T ] , B ∞ − 1 , ∞ ) provided that p > 2 and q > d . In this paper, we prove a local version of this result which covers the limit case q = d . Precisely, we prove the uniqueness of solution to the Navier–Stokes equations in the space L p ( [ 0 , T ] , M ˜ r , d ( R d ) ) ∩ C ( [ 0 , T ] , B ∞ − 1 , ∞ ) for every p > 2 and r > 2 where M ˜ r , d ( R d ) is the closure of the test functions in the Morrey–Campanato space M r , d ( R d ) . The prove of our result relies on an extension of the Comparison Theorem of P.G. Lemarié-Rieusset (Theorem 21.1 in [P.G. Lemarié-Rieusset, Recent developments in the Navier–Stokes problem, Chapman & Hall/CRC, 2002]). Moreover, this extension allows us to prove the uniqueness of solution to the Navier–Stokes equations in a functional space closed to the critical space C ( [ 0 , T ] , M 2 , d ( R d ) ) .

The Navier–Stokes equations in the critical Morrey–Campanato space

We shall discuss various points on solutions of the 3D Navier-Stokes equations from the point of view of Morrey-Campanato spaces (global solutions, strong-weak uniqueness, the role of real interpolation, regularity).