Anisotropic Lebesgue Spaces Research Articles

Anisotropic Prodi–Serrin regularity criteria for the 3D Navier–Stokes equations involving the gradient of one velocity component

In this paper, we prove some optimal regularity criteria for the 3D incompressible Navier–Stokes equations involving the gradient of one velocity component in the framework of anisotropic Lebesgue spaces satisfying Prodi–Serrin condition. These results cover the previous ones established by Wolf (2015) and O (2023).

Boundedness of the solutions of a kind of nonlinear parabolic systems

We deal with nonlinear systems of parabolic type satisfying componentwise structural conditions. The nonlinear terms are Carathéodory maps having controlled growth with respect to the solution and the gradient and the data are in anisotropic Lebesgue spaces. Under these assumptions we obtain essential boundedness of the weak solutions.

On regularity criteria for MHD system in anisotropic Lebesgue spaces

<abstract><p>This paper concerns the regularity criteria of the three-dimensional magnetohydrodynamic (MHD) system in anisotropic Lebesgue spaces. Two regularity results were proved under additional assumptions on the horizontal components of the velocity field $ {{\bf{u}}} $ and the magnetic field $ {{\bf{B}}} $, or directions of Elsässer's variables $ {{\bf{u}}}\pm{{\bf{B}}} $.</p></abstract>

An optimal regularity criterion for 3D Navier-Stokes equations involving the gradient of one velocity component

In this paper, we provide an optimal regularity criterion for 3D Navier-Stokes equations involving the gradient of one velocity component in the framework of anisotropic Lebesgue spaces. More precisely, employing the anisotropic Littlewood-Paley theory, we prove that a weak solution u is regular if ∇u3 belongs to scaling invariant space L2(0,T;Lv∞Lh2), where h and v denote the horizontal and vertical components, respectively. This result verifies the limiting case of a previous result established by Guo, Caggio and Skalák (2017).

On Continuation Criteria for the Full Compressible Navier-Stokes Equations in Lorentz Spaces

In this paper, we derive several new sufficient conditions of the non-breakdown of strong solutions for both the 3D heat-conducting compressible Navier-Stokes system and nonhomogeneous incompressible Navier-Stokes equations. First, it is shown that there exists a positive constant ε such that the solution (ρ, u, θ) to the full compressible Navier-Stokes equations can be extended beyond t = T provided that one of the following two conditions holds: To the best of our knowledge, this is the first continuation theorem allowing the time direction to be in Lorentz spaces for the compressible fluid. Second, we establish some blow-up criteria in anisotropic Lebesgue spaces for the finite blow-up time T*: Third, without the condition on ρ in (0.1) and (0.3), the results also hold for the 3D non-homogeneous incompressible Navier-Stokes equations. The appearance of a vacuum in these systems could be allowed.

An indefinite quasilinear elliptic problem with weights in anisotropic spaces

In this paper we consider existence, nonexistence and multiplicity of solutions for a class of indefinite quasilinear elliptic problems in the upper half-space involving weights in anisotropic Lebesgue spaces. One of our basic tools consists in a Hardy type inequality proved in the present paper that allows us to establish Sobolev embeddings into Lebesgue spaces with weights in anisotropic Lebesgue spaces.

Regularity criteria via one directional derivative of the velocity in anisotropic Lebesgue spaces to the 3D Navier-Stokes equations

In this paper, we consider the regularity criterion for 3D incompressible Navier-Stokes equations in terms of one directional derivative of the velocity in anisotropic Lebesgue spaces. More precisely, it is proved that u becomes a regular solution if the ∂3u satisfies∫0T‖‖‖∂3u(t)‖Lx1p‖Lx2q‖Lx3rβ1+ln⁡(‖∂3u(t)‖L2+e)dt<∞, where 2β+1p+1q+1r=1 and 2<p,q,r≤∞,1−(1p+1q+1r)≥0.

On regularity of the 3D MHD equations based on one velocity component in anisotropic Lebesgue spaces

In this paper we establish a new regularity criterion for the 3D incompressible MHD equations via ∂3u3 and the magnetic field b. By considering different weights in spatial variables, we show in anisotropic Lebesgue spaces if ∂3u3 and b satisfy certain space–time integrable conditions, which are almost optimal from the scaling invariant point of view, then a weak solution (u,b) is actually regular. This result gives new insights into the understanding of regularity theory of weak solutions.

Fredholm Integral Equations with Partial Integrals in ℝ2

We consider partial integral in the classes of functions with the mixed Lp– and supnorms or in the anisotropic Lebesgue spaces. For second kind Fredholm integral equations with partial integrals we construct solutions in the form of the operator Neumann series by the method of successive approximations.

$$\varepsilon $$-Regularity criteria in anisotropic Lebesgue spaces and Leray’s self-similar solutions to the 3D Navier–Stokes equations

In this paper, we establish some $$\varepsilon $$ -regularity criteria in anisotropic Lebesgue spaces for suitable weak solutions to the 3D Navier–Stokes equations as follows: 0.1 $$\begin{aligned}&\limsup \limits _{\varrho \rightarrow 0} \varrho ^{1-\frac{2}{p}-\sum \limits ^{3}_{j=1}\frac{1}{q_{j}}} \Vert u\Vert _{L_{t}^{p}L^{\overrightarrow{q}}_{x}(Q(\varrho ))} \le \varepsilon , ~~\frac{2}{p}+\sum \limits ^{3}_{j=1}\frac{1}{q_{j}} {\le 2}~~~~~\text {with}~q_{j} > 1; \\&\quad \sup _{-1\le t\le 0}\Vert u\Vert _{L^{\overrightarrow{q}}(B(1))} \le \varepsilon ,~~\frac{1}{q_{1}}+\frac{1}{q_{2}}+\frac{1}{q_{3}}<2\quad \text {with}\, 1<q_{j}<\infty ; \\&\Vert u \Vert _{L_{t}^{p}L^{\overrightarrow{q}}_{x}(Q(1))} +\Vert \Pi \Vert _{L^{1}(Q(1))}\le \varepsilon , \quad \frac{2}{p}+\sum ^{3}_{j=1}\frac{1}{q_{j}}<2 ~~~\text {with}~~ 1<q_{j}<\infty , \end{aligned}$$ which extends the previous results in Caffarelli et al. (Commun Pure Appl Math 35:771–831, 1982), Choi and Vasseur (Ann Inst H Poincare Anal Non Lineaire 31:899–945, 2014), Gustafson et al. (Commun Math Phys 273:161–176, 2007), Guevara and Phuc Calc Var 56:68, 2017), He et al. (J Nonlinear Sci 29:2681–2698, 2019), Tian and Xin (Commun Anal Geom 7:221–257, 1999) and Wolf (Ann Univ Ferrara 61:149–171, 2015). As an application, in the spirit of Chae and Wolf (Arch Ration Mech Anal 225:549–572, 2017), we prove that there does not exist a nontrivial Leray’s backward self-similar solution with profiles in $$L^{\overrightarrow{p}}(\mathbb {R}^{3})$$ with $$\frac{1}{p_{1}}+\frac{1}{p_{2}}+\frac{1}{p_{3}}<2$$ . This generalizes the corresponding results of Chae and Wolf (Arch Ration Mech Anal 225:549–572, 2017), Guevara and Phuc (SIAM J Math Anal 50:541–556, 2017), Necas et al. (Acta Math 176, 283–294, 1996) and Tsai (Arch Ration Mech Anal 143(1):29–51, 1998).

Partial Integrals in Anisotropic Lebesgue Spaces. II: Multidimensional Case

We consider linear integral operators with multidimensional partial integrals in anisotropic Lebesgue spaces Lr(D) of functions defined in a finite parallelepiped D in the Euclidean spaces ℝn, where n ≥ 3. We obtain conditions on kernels of partial integrals and connections between multiindices r and p guaranteeing the boundedness of the partial integrals acting from Lr(D) to Lp(D).

Blowup Criterion via Only the Middle Eigenvalue of the Strain Tensor in Anisotropic Lebesgue Spaces to the 3D Double-Diffusive Convection Equations

In this paper, we consider the three-dimensional (3D) incompressible double-diffusive convection equations. By making use of a anisotropic Sobolev’s inequality, we obtain a blowup criterion for the strong solution to the 3D double-diffusive convection equations involving only the middle eigenvalue of the strain tensor in anisotropic Lebesgue spaces. This result improves a result established in a recent work by Miller (Arch Ration Mech Anal 235:99–139, 2019).

The anisotropic integrability logarithmic regularity criterion to the 3D micropolar fluid equations

The aim of this paper is to establish the regularity criterion of weak solutions to the 3D micropolar fluid equations by one directional derivative of the pressure in anisotropic Lebesgue spaces. We improve the regularity criterion for weak solutions previously given by Jia, Zhang and Dong in [ 21 ].

The global solutions of axisymmetric Navier–Stokes equations with anisotropic initial data

Consider the Cauchy problem of incompressible Navier–Stokes equations in three dimension with axisymmetric initial data. If the swirl component satisfies some certain regularity criteria in weighted spaces, the existence of the time-global solution has been proved in our previous work. However, it is more or less evident that the solution is anisotropic in vertical direction (i.e., the direction parallel to the axis of symmetry) and the horizontal directions (in the plane perpendicular to the axis of symmetry). In this paper, we are interested in constructing regularity criteria and time-global solution in anisotropic Lebesgue spaces.

Regularity Criteria of The Incompressible Navier-Stokes Equations via Only One Entry of Velocity Gradient

In this paper we establish regularity conditions for the three dimensional incompressible Navier-Stokes equations in terms of one entry of the velocity gradient tensor, say for example, $$ \partial _{3}u_{3}$$ . We show that if $$\partial _{3}u_{3}$$ satisfies certain integrable conditions with respect to time and space variables in anisotropic Lebesgue spaces, then a Leray-Hopf weak solution is actually regular. The anisotropic Lebesgue space helps us to almost reach the Prodi-Serrin level 2 in certain special case. Moreover, regularity conditions on non-diagonal element of gradient tensor $$\partial _1 u_3$$ are also established, which covers some previous literature.

On Regularity for the 3D MHD Equations via One Directional Derivative of the Pressure

This work establishes a new regularity criterion for the 3D incompressible MHD equations in term of one directional derivative of the pressure (i.e., \(\partial _{3}P\)) on framework of the anisotropic Lebesgue spaces. More precisely, it is proved that for \(T>0\), if \(\partial _{3}P\in L^{\beta }(0,T; L^{\alpha }(\mathbb {R}^{2}_{x_{1}x_{2}};L^{\gamma }(\mathbb {R}_{x_{3}})))\) with \(\frac{2}{\beta }+\frac{1}{\gamma }+\frac{2}{\alpha }=k\in [2,3)\) and \(\frac{3}{k}\le \gamma \le \alpha \le \frac{1}{k-2},\) then the corresponding solution (u, b) to the 3D MHD equations is regular on [0, T].

On 3D MHD equations with regularity in one directional derivative of the velocity

This work establishes a new regularity criterion for the 3D incompressible magneto-hydrodynamical (MHD) equations in terms of one directional derivative of the velocity (i.e., ∂3u) on framework of the anisotropic Lebesgue spaces. More precisely, it is proved that if ∂3u satisfies ∫0T‖‖∂3u(τ)‖Lx3α‖Lx1x2βqdτ<+∞,where2q+1α+2β=k∈[1,32] and 3k<α≤β≤1k−1,for some T>0, then the corresponding solution (u,b) to the 3D MHD equations is regular on (0,T)×R3.

Remarks on pressure regularity criterion for the 3D Boussinesq equations

This paper establishes two logarithmically improved regularity criteria for the Cauchy problem of the 3D Boussinesq equations in termsof the pressure on framework of the anisotropic Lebesgue spaces. More precisely, it is proved that if ∫0T‖‖‖P‖Lx1p‖Lx2q‖Lx3rβ1+ln(1+‖θ(⋅,t)‖L4)dt<∞,where2β+1p+1q+1r=2and2≤p,q,r≤∞,1−(1p+1q+1r)≥0,or ∫0T‖‖‖∇P‖Lx1p‖Lx2q‖Lx3rβ1+ln(1+‖θ(⋅,t)‖L4)dt<∞,where2β+1p+1q+1r=3and1≤p,q,r≤∞,1−(12p+12q+12r)≥0,for some T>0, then the corresponding weak solution (u,θ) to the 3D Boussinesq equations is regular on [0,T].

The application of anisotropic Troisi inequalities to the conditional regularity for the Navier–Stokes equations

We derive various anisotropic versions of the Troisi inequality for the study of the conditional regularity of solutions to the Navier–Stokes equations. We impose additional conditions on one or two non-diagonal entries of the velocity gradient, namely or and . Formulating our results in the frame of the anisotropic Lebesgue spaces we improve two recent results from the literature. Further we present two new criteria.

On the 3D Navier–Stokes equations with regularity in pressure

In this paper, we investigate the global regularity of Leray–Hopf weak solutions to the 3D Navier–Stokes equations in terms of pressure. In particular, we establish three Serrin-type regularity criteria in framework of the anisotropic Lebesgue spaces. Our result can be understood as a generalization of the notable works of Y. Zhou (2006) [23], C. Cao and E. Titi (2008) [5], and Y. Zhou and M. Pokorný (2010) [24].