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In this article, we investigate the existence, uniqueness and exponential decay of asymptotically almost periodic solutions of the parabolic-elliptic Keller-Segel system on a real hyperbolic manifold. We prove the existence and uniqueness of such solutions in the linear equation case by using the dispersive and smoothing estimates of the heat semigroup. Then we pass to the well-posedness of the semi-linear equation case by using the results of linear equation and fixed point arguments. The exponential decay is proven by using Gronwall's inequality. [ABSTRACT FROM AUTHOR]

In this paper we establish the asymptotic stability of steady solutions for the Boussinesq systems in the framework of Cartesian product of critical weak-Morrey spaces on R n , where n ⩾ 3 . In our strategy, we first establish the continuity for the long time of the bilinear terms associated with the mild solutions of the Boussinesq systems, i.e., the bilinear estimates by using only the norm of the present spaces. As a direct consequence, we obtain the existence of global small mild solutions and asymptotic stability of steady solutions of the Boussinesq systems in the class of continuous functions from [ 0 , ∞) to the Cartesian product of critical weak-Morrey spaces. Our techniques consist interpolation of operators, duality, heat semigroup estimates, Hölder and Young inequalities in block spaces (based on Lorentz spaces) that are preduals of Morrey-Lorentz spaces. [ABSTRACT FROM AUTHOR]

We investigate a new diffuse-interface model that describes creeping two-phase flows (i.e., flows exhibiting a low Reynolds number), especially flows that permeate a porous medium. The system of equations consists of a Brinkman equation for the volume averaged velocity field and a convective Cahn–Hilliard equation with dynamic boundary conditions for the phase field, which describes the location of the two fluids within the domain. The dynamic boundary conditions are incorporated to model the interaction of the fluids with the wall of the container more precisely. In particular, they allow for a dynamic evolution of the contact angle between the interface separating the fluids and the boundary, and for a convection-induced motion of the corresponding contact line. For our model, we first prove the existence of global-in-time weak solutions in the case where regular potentials are used in the Cahn–Hilliard subsystem. In this case, we can further show the uniqueness of the weak solution under suitable additional assumptions. We further prove the existence of weak solutions in the case of singular potentials. Therefore, we regularize such singular potentials by a Moreau–Yosida approximation, such that the results for regular potentials can be applied, and eventually pass to the limit in this approximation scheme. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the two-dimensional (2D) incompressible magnetohydrodynamic (MHD) equations with velocity dissipation given by (- Δ) α and magnetic diffusion given by reducing about the square root of logarithmic diffusion from standard Laplacian diffusion. More precisely, we establish the global regularity of solutions to the system as long as the power α is a positive constant. In addition, we prove several global a priori bounds for the case α = 0 . Finally, for the case α = 0 , it is also shown that the control of the direction of the magnetic field in a suitable norm is enough to guarantee the global regularity. In particular, our results significantly improve previous works and take us one step closer to a complete resolution of the global regularity issue on the 2D resistive MHD equations, namely, the case when the MHD equations only have standard Laplacian magnetic diffusion. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and uniqueness of almost periodic solutions for the parabolic-elliptic Keller–Segel system on the whole space R n (n ⩾ 4) . We work in the framework of critical spaces such as on the weak-Lorentz space L n 2 , ∞ (R n) . Our method is based on the dispersive and smoothing estimates of the heat semigroup and fixed point arguments. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to establish regularity criteria of the 3D damped Boussinesq equations with zero thermal diffusion. It is shown that if two components of the velocity or the gradient of velocity belong to some Lorentz spaces in both time and spatial directions, then the weak solution is regular on [0, T]. [ABSTRACT FROM AUTHOR]

We analyze a diffuse interface model for multi-phase flows of N incompressible, viscous Newtonian fluids with different densities. In the case of a bounded and sufficiently smooth domain existence of weak solutions in two and three space dimensions and a singular free energy density is shown. Moreover, in two space dimensions global existence for sufficiently regular initial data is proven. In three space dimension, existence of strong solutions locally in time is shown as well as regularization for large times in the absence of exterior forces. Moreover, in both two and three dimensions, convergence to stationary solutions as time tends to infinity is proved. [ABSTRACT FROM AUTHOR]

We study the interaction of an incompressible fluid in two dimensions with an elastic structure yielding the moving boundary of the physical domain. The displacement of the structure is described by a linear viscoelastic beam equation. Our main result is the existence of a unique global strong solution. Previously, only the ideal case of a flat reference geometry was considered such that the structure can only move in vertical direction. We allow for a general geometric set-up, where the structure can even occupy the complete boundary. Our main tool—being of independent interest—is a maximal regularity estimate for the steady Stokes system in domains with minimal boundary regularity. In particular, we can control the velocity field in W 2 , 2 in terms of a forcing in L 2 provided the boundary belongs roughly to W 3 / 2 , 2. This is applied to the momentum equation in the moving domain (for a fixed time) with the material derivative as right-hand side. Since the moving boundary belongs a priori only to the class W 2 , 2 , known results do not apply here as they require a C 2 -boundary. [ABSTRACT FROM AUTHOR]

The question of whether the two-dimensional inviscid Boussinesq equations can develop a finite-time singularity from general initial data is a challenging open problem. In this paper, we obtain two new regularity criteria for the local-in-time smooth solution to the two-dimensional inviscid Boussinesq equations. Similar result is also valid for the nonlocal perturbation of the two-dimensional incompressible Euler equations. [ABSTRACT FROM AUTHOR]

We consider the three-dimensional inviscid Boussinesq–Voigt system which is a regularization model for the inviscid Boussinesq equations. We prove a regularity criterion for the weak solutions (in particular, for the time-derivative of the velocity), in L p -spaces, involving first derivatives of the pressure. [ABSTRACT FROM AUTHOR]

In the present paper, we study the generalized Cauchy problem of the incompressible micropolar fluid system in R 3. By using the Fourier localization argument and the Littlewood-Paley theory, we establish the local existence for large initial data and the global existence for small initial data in critical Fourier-Besov-Morrey spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study a dissipative systems modelling electrohydrodynamics in incompressible viscous fluids. The system consists of the Navier–Stokes equations coupled with a classical Poisson–Nernst–Planck equations. In the three-dimensional case, we establish a global regularity criteria in terms of the middle eigenvalue of the strain tensor in the framework of the anisotropic Lorentz spaces for local smooth solution. The proof relies on the identity for entropy growth introduced by Miller in the Arch. Ration. Mech. Anal. [16]. [ABSTRACT FROM AUTHOR]

We study the axisymmetric Navier-Stokes equations. In 2010, Loftus-Zhang used a refined test function and re-scaling scheme, and showed that | ω r (x , t) | + | ω z (r , t) | ⩽ C r 10 , 0 < r ⩽ 1 2. By employing the dimension reduction technique by Lei-Navas-Zhang, and analyzing ωr, ωz and ωθ/r on different hollow cylinders, we are able to improve it and obtain | ω r (x , t) | + | ω z (r , t) | ⩽ C | ln r | r 17 / 2 , 0 < r ⩽ 1 2. [ABSTRACT FROM AUTHOR]

This paper examines the stability of a 2D inviscid MHD system with anisotropic damping near a background magnetic field. It is well known that solutions of the incompressible Euler equations can grow rapidly in time and are thus unstable while solutions of the Euler equations with full damping are stable. Then naturally arises the question of whether solutions of the Euler equations with partial damping are stable. The main purpose of this paper is to give an affirmative answer to this question in the case when the fluid is coupled with the magnetic field through the MHD system with one-component damping. The result presented in this paper especially confirms the stabilizing effects of the magnetic field on the electrically conducting fluids, a phenomenon that has been observed in physical experiments and numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, we consider the regularity criteria of 3D incompressible Boussinesq equations. By using the Littlewood–Paley decomposition technique to establishing a regularity criterion in terms of the one-directional derivative of velocity in Besov spaces. Our result improves some previous works (Liu Q. A regularity criterion for the Navier–Stokes equations in terms of one-directional derivative of the velocity. Acta Appl Math. 2015;140(1):1–9; Gala S, Ragusa M A. On the regularity criterion for the Navier–Stokes equations in terms of one-directional derivative. Asian-Eur J Math. 2017;10(1):1750012). [ABSTRACT FROM AUTHOR]

In this paper, we study the Cauchy problem of the 2D micropolar Bénard system with partial viscosity, i.e., (1) μ 12 = μ 21 = γ 1 = 1 , μ 11 = μ 22 = γ 2 = ν 1 = ν 2 = 0 ; (2) μ 12 = μ 21 = γ 2 = 1 , μ 11 = μ 22 = γ 1 = ν 1 = ν 2 = 0 , where μ ij , γ i , ν i (i , j = 1 , 2) are the coefficients of dissipation, angular viscosity and thermal diffusivity, respectively. This work extends the result of Xu (Appl. Math. Lett. 108(2020)) on 2D micropolar Bénard system with full dissipation and angular viscosity. [ABSTRACT FROM AUTHOR]

In this paper, some new L p gradient estimates are justified for the three-dimensional compressible magnetohydrodynamic equations in the whole space R 3 . The key to derive the estimate ∥ ∇ u ∥ L 3 is the "div-curl" decomposition technique. For regular initial data with small energy, we prove the existence of global solutions belonging to a new class of functions in which the uniqueness can be shown to hold. [ABSTRACT FROM AUTHOR]

We consider the 3D simplified Bardina turbulence model with horizontal filtering, fractional dissipation, and the presence of a memory term incorporating hereditary effects. We analyze the regularity properties and the dissipative nature of the considered system and, in our main result, we show the existence of a global exponential attractor in a suitable phase space. [ABSTRACT FROM AUTHOR]

In this paper, we consider the initial-boundary value problem for the full compressible Navier–Stokes equations on the square domain. We show the global existence of the strong solution with vacuum if the coefficient of viscosity μ is suitably large. Moreover, an exponential decay rate of the strong solution is obtained. [ABSTRACT FROM AUTHOR]

We analyze the propagation of Lipschitz continuity of solutions to various linear and nonlinear drift-diffusion systems, with and without incompressibility constraints. Diffusion is assumed to be either fractional or classical. We derive conditions that guarantee the propagation of Lipschitz regularity by the incompressible NSE in the form of a non-local, one dimensional viscous Burgers-type inequality. We show the analogous inequality is always satisfied for the generalized viscous Burgers–Hilbert equation, in any spatial dimension, leading to global regularity. We also obtain a regularity criterion for the Navier–Stokes equation with fractional dissipation (- Δ) α , regardless of the power of the Laplacian α ∈ (0 , 1 ] , in terms of Hölder-type assumptions on the solution. If we drop the pressure term and analyze a linear drift-diffusion problem instead, we obtain regularity under a super-critical assumption on the drift when the power of diffusion is in (1/2, 1]. We also generalize previous known regularity results under subcritical assumptions on the drift when the power of diffusion is between (0, 1/2]. The technique we use builds upon the evolution of moduli of continuity as introduced by Kiselev, Nazarov, Volberg and Shterenberg. [ABSTRACT FROM AUTHOR]

We prove in this article for the first time for a multi-physical system: the 3D Boussinesq equations, the global well-posedness of the Voigt-regularized velocity-vorticity formulation model (VVVB), which attracts great interest from the numerical and computational study in fluid dynamics. We also obtain the higher-order regularity of the solution to the VVVB equations without regularizing the thermal fluctuation. Furthermore, we show that the velocity and vorticity of the VVVB equations converge to their counterparts of the 3D Boussinesq equations (BE), as the Voigt regularizing parameter approaches 0. Finally, we show the convergence of the curl of the velocity to the vorticity in the VVVB equations as the aforementioned parameter tends to 0. [ABSTRACT FROM AUTHOR]

This paper establishes the global regularity of classical solution to the 2D MHD system with only horizontal dissipation and horizontal magnetic diffusion in a strip domain T × R when the initial data are suitable small. To prove this, we combine the Littlewood–Paley decomposition with anisotropic inequalities to establish a crucial commutator estimate. We also analyze the asymptotic behavior of the solution. In addition, the global existence and uniqueness of classical solution is obtained for the 2D simplified tropical climate model with only horizontal dissipation. [ABSTRACT FROM AUTHOR]

We deal with an initial boundary value problem of nonhomogeneous Boussinesq equations for magnetohydrodynamics convection in two-dimensional domains. We prove that there is a unique global strong solution. Moreover, we show that the temperature converges exponentially to zero in H1 as time goes to infinity. In particular, the initial data can be arbitrarily large and vacuum is allowed. Our analysis relies on energy method and a lemma of Desjardins (Arch. Rational Mech. Anal. 137:135–158, 1997). [ABSTRACT FROM AUTHOR]

The regularity for the 3-D nematic liquid crystal equations is considered in this paper, it is proved that the Leray–Hopf weak solutions (u, d) is in fact smooth, if the velocity field u ∈ L ∞ (0 , T ; L x 3 , ∞ (R 3)) satisfies some addition local small condition r - 3 x ∈ B r (x 0) : | u (x , t 0) | > ε r - 1 ≤ ε , which is inspired by the papers [2, 35]. [ABSTRACT FROM AUTHOR]

In this paper, a logarithmically improved regularity criterion for the incompressible Boussinesq equations is established via partial horizontal derivatives of two velocity components. It is shown that if the partial horizontal derivatives of two velocity components satisfy ∫ 0 T ‖ ∂ 1 u 1 , ∂ 2 u 2 · , t ‖ M ˙ p , 3 r 2 2 - r 1 + ln e + ‖ u (· , t) ‖ L 4 d t < ∞ with 0 < r ≤ 1 and 2 ≤ p ≤ 3 r , then the weak solution (u , θ) to the 3D Boussinesq equations is regular on (0, T]. Compared to the result of Navier–Stokes equations, there is a logarithmic improvement involving u in the denominator. [ABSTRACT FROM AUTHOR]

As observed by Yamazaki, the third component b3 of the magnetic field can be estimated by the corresponding component u3 of the velocity field in Lλ (2 ⩽ λ ⩽ 6) norm. This leads him to establish regularity criterion involving u3, j3 or u3, ω3. Noticing that λ can be greater than 6 in this paper, we can improve previous results. [ABSTRACT FROM AUTHOR]

We establish the global well-posedness of a strong solution to the 3D tropical climate model with damping. We prove that there exists the global and unique solution for α, β, γ satisfying one of the following three conditions: (1) α, β ⩾ 4; (2) 7/2 ⩽ α < 4, β ⩾ (5α + 7)/(2α), γ ⩾ 7/(2α − 5); (3) 3 < α ⩽ 7/2, β, γ ⩾ 7/(2α − 5). [ABSTRACT FROM AUTHOR]

Tropical climate model derived by Frierson et al. (Commun Math Sci 2:591–626, 2004) and its modified versions have been investigated in a number of papers [see, e.g., Li and Titi (Discrete Contin Dyn Syst Series A 36(8):4495–4516, 2016), Wan (J Math Phys 57(2):021507, 2016), Ye (J Math Anal Appl 446:307–321, 2017) and more recently Dong et al. (Discrete Contin Dyn Syst Ser B 24(1):211–229, 2019)]. Here, we deal with the 2D tropical climate model with fractional dissipative terms in the equation of the barotropic mode u and in the equation of the first baroclinic mode v of the velocity, but without diffusion in the temperature equation, and we establish a regularity criterion for this system. [ABSTRACT FROM AUTHOR]

The author proposes a two-dimensional generalization of Constantin-Lax-Majda model. Some results about singular solutions are given. This model might be the first step toward the singular solutions of the Euler equations. Along the same line (vorticity formulation), the author presents some further model equations. He possibly models various aspects of difficulties related with the singular solutions of the Euler and Navier-Stokes equations. Some discussions on the possible connection between turbulence and the singular solutions of the Navier-Stokes equations are made. [ABSTRACT FROM AUTHOR]

This paper deals with the 2D incompressible MHD equations with density-dependent viscosity in bounded domain. The global existence of strong solutions are established in the vacuum cases, provided the assumption that ρ ¯ + ∥ b 0 ∥ L 4 is suitably small with large velocity. Furthermore, the exponential decay of the solution is also obtained. [ABSTRACT FROM AUTHOR]

In this paper, we consider the classical solutions to a model of two-dimensional incompressible inviscid Boussinesq–Bénard equations. Notice that, in the case when the source term of temperature equation in this model is the second component of velocity u 2 or no source term, there is no global-in-time existence result for the general initial data. Here, if the source term is only chosen as Δ u 2 , then we can obtain the global well-posedness, inviscid limit and some exponential decay estimates. Our key observation is the nice symmetrical structure hidden in the corresponding system, which plays an extremely important role in the global well-posedness studied here. [ABSTRACT FROM AUTHOR]

It is proved that if the solution of the Navier–Stokes system satisfies ∂ 3 u ∈ L p (0 , T ; L q (R 3)) , 2 p + 3 q = 22 13 + 3 13 q , 3 < q < 4 , or ∂ 3 u 3 ∈ L β (0 , T ; L α (R 3)) , 2 β + 3 α = 3 (65 α 2 - 78 α + 49 + 7 - α) 16 α , 3 + 17 4 ≤ α ≤ ∞ , then the solution is smooth on (0, T]. These two improve many previous results. [ABSTRACT FROM AUTHOR]

We consider the maximal regularity problem for a PDE of linear acoustics, named the Van Wijngaarden–Eringen equation, that models the propagation of linear acoustic waves in isothermal bubbly liquids, wherein the bubbles are of uniform radius. If the dimensionless bubble radius is greater than one, we prove that the inhomogeneous version of the Van Wijngaarden–Eringen equation, in a cylindrical domain, admits maximal regularity in Lebesgue spaces. Our methods are based on the theory of operator-valued Fourier multipliers. [ABSTRACT FROM AUTHOR]

We consider the regularity of weak solutions to the shear thinning fluids in R 3 . Let u be a weak solution in R 3 × (0 , T) and u ~ = (u 1 , u 2 , 0) . It is proved that u becomes a strong solution if u ~ ∈ L 5 p - 6 5 p - 8 (0 , T ; B M O (R 3)) , for 8 5 < p ≤ 2 . This is an improvement of the result given by Yang (Comput Math Appl 77:2854–2858). [ABSTRACT FROM AUTHOR]

We investigate regularity estimates for the stationary Navier–Stokes equations above a highly oscillating Lipschitz boundary with the no-slip boundary condition. Our main result is an improved Lipschitz regularity estimate at scales larger than the boundary layer thickness. We also obtain an improved C 1 , μ estimate and identify the building blocks of the regularity theory, dubbed 'Navier polynomials'. In the case when some structure is assumed on the oscillations of the boundary, for instance periodicity, these estimates can be seen as local error estimates. Although we handle the regularity of the nonlinear stationary Navier–Stokes equations, our results do not require any smallness assumption on the solutions. [ABSTRACT FROM AUTHOR]

We study the Cauchy problem for the 2 1 2 D 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 2 1 2 D magnetic Bénard fluid system without zero magnetic diffusion and thermal diffusivity. Furthermore, we derive two blow-up criteria of smooth solutions to the 2 1 2 D magnetic Bénard fluid system without dissipation. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Cauchy problem for the incompressible inhomogeneous nematic liquid crystal flows in R n with initial data in some critical Besov spaces. Under the assumptions that initial density is close to the equilibrium state and the smallness on the initial direction field and nonlinear smallness on the linearized velocity, we obtain the global existence of solutions by fully using the Fourier localization technique and the advantage of weighted function generated by heat kernel. The meanings of this kind of initial data are that we can provide an example of large highly oscillating data satisfying that nonlinear smallness assumption, despite each component of the initial data could be arbitrarily large. [ABSTRACT FROM AUTHOR]

This paper is concerned with the global well-posedness of the n-dimensional generalized incompressible Navier–Stokes equations with initial data in the critical Besov space. By fully using the advantage of suitable weighted functions and the Fourier localization technique, the global well-posedness is obtained without the hypothesis of smallness on initial data, but for some nonlinear smallness of the first iterate. We also give an example of initial data satisfying the nonlinear smallness condition, but whose norm is arbitrarily large. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with regularizing rates of the higher order derivatives of solutions for the initial value problem of a three-dimensional incompressible micropolar fluid system with initial data (u 0 , ω 0) ∈ a ˙ B p , ∞ − 1 + 3 / p (R 3) for 1 ≤ p < 6. More precisely, we prove that there exist two positive constants K 1 and K 2 such that for any β ∈ N 0 3 , the B ˙ q , ∞ s -norm of β-order derivatives of solutions obeys the following regularizing rates: ∥ (∂ x β u (⋅ , t) , ∂ x β ω (⋅ , t)) ∥ B ˙ q , ∞ s ≤ K 1 (K 2 | β |) | β | t − | β | / 2 − (1 / 2) (s + 1 − 3 / q) for some suitable choices of s and q. These estimates particularly imply the spatial analyticity and the temporal decay of global solutions. [ABSTRACT FROM AUTHOR]

This paper is devoted to the global regularity for the Cauchy problem of the two-dimensional Boussinesq equations with the temperature-dependent viscosity. We prove the global solutions for this system with any positive power of the fractional Laplacian for temperature under the assumption that the viscosity coefficient is sufficiently close to some positive constant. Our obtained result improves considerably the recent results in Abidi and Zhang (Adv Math 305:1202–1249, 2017) and Zhai et al. (J Differ Equ 267:364–387, 2019). In addition, a regularity criterion via the velocity is also obtained for this system without the above assumption on the viscosity coefficient. [ABSTRACT FROM AUTHOR]

This paper concerns improving Prodi-Serrin-Ladyzhenskaya type regularity criteria for the Navier-Stokes system, in the sense of multiplying certain negative powers of scaling invariant norms. [ABSTRACT FROM AUTHOR]

This work establishes a new logarithmical improved regularity criterion for the 3D Boussinesq equations in terms of one directional derivative of the pressure (i.e., ∂ 3 P ) on framework of the anisotropic Lebesgue spaces. More precisely, it is proved that if ∫ 0 T ‖ ∂ 3 P (· , t) ‖ L x 3 γ L x 1 x 2 α β 1 + ln (1 + ‖ θ (· , t) ‖ L 4) d t < ∞ , where 2 β + 1 γ + 2 α = k ∈ [ 2 , 3) and 3 k ≤ γ ≤ α < 1 k - 2 , for some T > 0 , then the corresponding solution (u , θ) to the 3D Boussinesq equations is regular on [0, T]. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the space-time regularity of solutions to (1) the three dimensional incompressible Navier–Stokes equations for initial data u 0 = (u 0 h , u 0 3) ∈ B ˙ p , r 3 p − 1 (R 3) with large initial vertical velocity component; and (2) the three dimensional incompressible magneto-hydrodynamic equations for initial datum u 0 = (u 0 h , u 0 3) ∈ B ˙ p , r 3 p − 1 (R 3) with large initial vertical velocity component and b 0 = (b 0 h , b 0 3) ∈ B ˙ p , r 3 p − 1 (R 3) with large initial vertical magnetic field component. [ABSTRACT FROM AUTHOR]

We review the developments of the regularity criteria for the Navier-Stokes equations, and make some further improvements. [ABSTRACT FROM AUTHOR]