The Hall-magnetohydrodynamics (Hall-MHD) equations, rigorously derived from kinetic models, are useful in describing many physical phenomena in geophysics and astrophysics. This paper studies the local well-posedness of classical solutions to the Hall-MHD equations with the magnetic diffusion given by a fractional Laplacian operator, $${(-\Delta)^\alpha}$$ ( - Δ ) α . Due to the presence of the Hall term in the Hall-MHD equations, standard energy estimates appear to indicate that we need $${\alpha\ge 1}$$ α ≥ 1 in order to obtain the local well-posedness. This paper breaks the barrier and shows that the fractional Hall-MHD equations are locally well-posed for any $${\alpha > \frac{1}{2}}$$ α > 1 2 . The approach here fully exploits the smoothing effects of the dissipation and establishes the local bounds for the Sobolev norms through the Besov space techniques. The method presented here may be applicable to similar situations involving other partial differential equations.

We introduce a continuous data assimilation (downscaling) algorithm for the two-dimensional Navier–Stokes equations employing coarse mesh measurements of only one component of the velocity field. This algorithm can be implemented with a variety of finitely many observables: low Fourier modes, nodal values, finite volume averages, or finite elements. We provide conditions on the spatial resolution of the observed data, under the assumption that the observed data is free of noise, which are sufficient to show that the solution of the algorithm approaches, at an exponential rate asymptotically in time, to the unique exact unknown reference solution, of the 2D Navier–Stokes equations, associated with the observed (finite dimensional projection of) velocity.

In this paper, we deal with the Cauchy problem of the two-dimensional magnetic Bénard problem with mixed partial viscosity. More precisely, the global well-posedness of 2D magnetic Bénard problem without thermal diffusivity and with vertical or horizontal magnetic diffusion is obtained. Moreover, the global regularity and some conditional regularity of strong solutions are obtained for 2D magnetic Bénard problem with mixed partial viscosity. The results extend the recent work (Appl Math Letter 26:627–630, 2013) on the global regularity of the magnetic Bénard problem with full dissipation and magnetic diffusion in two dimensions.

We investigate the stabilizing effects of the magnetic fields in the linearized magnetic Rayleigh–Taylor (RT) problem of a nonhomogeneous incompressible viscous magnetohydrodynamic fluid of zero resistivity in the presence of a uniform gravitational field in a three-dimensional bounded domain, in which the velocity of the fluid is non-slip on the boundary. By adapting a modified variational method and careful deriving a priori estimates, we establish a criterion for the instability/stability of the linearized problem around a magnetic RT equilibrium state. In the criterion, we find a new phenomenon that a sufficiently strong horizontal magnetic field has the same stabilizing effect as that of the vertical magnetic field on growth of the magnetic RT instability. In addition, we further study the corresponding compressible case, i.e., the Parker (or magnetic buoyancy) problem, for which the strength of a horizontal magnetic field decreases with height, and also show the stabilizing effect of a sufficiently large magnetic field.

In this paper we investigate the qualitative behaviour of the pressure function beneath an extreme Stokes wave over infinite depth. The presence of a stagnation point at the wave-crest of an extreme Stokes wave introduces a number of mathematical difficulties resulting in the irregularity of the free surface profile. It will be proven that the pressure decreases in the horizontal direction between a crest-line and subsequent trough-line, except along these lines themselves where the pressure is stationary with respect to the horizontal coordinate. In addition we will prove that the pressure strictly increases with depth throughout the fluid body.

By taking into account the $${\beta}$$ β -plane effects, we provide an exact nonlinear solution to the geophysical edge-wave problem within the Lagrangian framework. This solution describes trapped waves propagating eastward or westward along a sloping beach with the shoreline parallel to the Equator.

We establish the existence of small-amplitude uni- and bimodal steady periodic gravity waves with an affine vorticity distribution, using a bifurcation argument that differs slightly from earlier theory. The solutions describe waves with critical layers and an arbitrary number of crests and troughs in each minimal period. An important part of the analysis is a fairly complete description of the local geometry of the so-called kernel equation, and of the small-amplitude solutions. Finally, we investigate the asymptotic behavior of the bifurcating solutions.

It is shown both locally and globally that $${L_t^{\infty}(L_x^{3,q})}$$ L t ∞ ( L x 3 , q ) solutions to the three-dimensional Navier–Stokes equations are regular provided $${q\neq\infty}$$ q ≠ ∞ . Here $${L_x^{3,q}}$$ L x 3 , q , $${0 < q \leq\infty}$$ 0 < q ≤ ∞ , is an increasing scale of Lorentz spaces containing $${L^3_x}$$ L x 3 . Thus the result provides an improvement of a result by Escauriaza et al. (Uspekhi Mat Nauk 58:3–44, 2003; translation in Russ Math Surv 58, 211–250, 2003), which treated the case q = 3. A new local energy bound and a new $${\epsilon}$$ ϵ -regularity criterion are combined with the backward uniqueness theory of parabolic equations to obtain the result. A weak-strong uniqueness of Leray–Hopf weak solutions in $${L_t^{\infty}(L_x^{3,q})}$$ L t ∞ ( L x 3 , q ) , $${q\neq\infty}$$ q ≠ ∞ , is also obtained as a consequence.

New results are obtained for global regularity and long-time behavior of the solutions to the 2D Boussinesq equations for the flow of an incompressible fluid with positive viscosity and zero diffusivity in a smooth bounded domain. Our first result for global boundedness of the solution $${(u, \theta)}$$ ( u , θ ) in $${D(A)\times H^1}$$ D ( A ) × H 1 improves considerably the main result of the recent article (Hu et al. in J Math Phys 54(8):081507, 2013). Our second result on global boundedness of the solution $${(u, \theta)}$$ ( u , θ ) in $${V\times H^1}$$ V × H 1 for both bounded domain and the whole space $${\mathbb{R}^{2}}$$ R 2 is a new one. It has been open and also seems much more challenging than the first result. Global regularity of the solution $${(u, \theta)}$$ ( u , θ ) in $${D(A)\times H^{2}}$$ D ( A ) × H 2 is also proved.

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 $${{Q_{r}^{*}}(x_0,t_0)}$$ Q r ∗ ( x 0 , t 0 ) , then u is regular at (x 0, t 0).

In Benameur (Methods Appl 103:87–97, 2014), Benameur proved a blow-up result of the non regular solution of (NSE) in the Sobolev–Gevrey spaces. In this paper we improve this result, precisely we give an exponential type explosion in Sobolev–Gevrey spaces with less regularity on the initial condition. Fourier analysis and standard techniques are used.

In the present paper we study a singular perturbation problem for a Navier–Stokes–Korteweg model with Coriolis force. Namely, we perform the incompressible and fast rotation asymptotics simultaneously, while we keep the capillarity coefficient constant in order to capture surface tension effects in the limit. We consider here the case of variable rotation axis: we prove the convergence to a linear parabolic-type equation with variable coefficients. The proof of the result relies on compensated compactness arguments. Besides, we look for minimal regularity assumptions on the variations of the axis.

We study the Navier–Stokes equations of steady motion of a viscous incompressible fluid in $${\mathbb{R}^{3}}$$ R 3 . We prove that there are no nontrivial solution of these equations defined in the whole space $${\mathbb{R}^{3}}$$ R 3 for axially symmetric case with no swirl (the Liouville theorem). Also we prove the conditional Liouville type theorem for axial symmetric solutions to the Euler system.

Because pressure is determined globally for the incompressible Euler equations, a localized change to the initial velocity will have an immediate effect throughout space. For solutions to be physically meaningful, one would expect such effects to decrease with distance from the localized change, giving the solutions a type of stability. Indeed, this is the case for solutions having spatial decay, as can be easily shown. We consider the more difficult case of solutions lacking spatial decay, and show that such stability still holds, albeit in a somewhat weaker form.

We investigate the long-term behavior, as a certain regularization parameter vanishes, of the three-dimensional Navier–Stokes–Voigt model of a viscoelastic incompressible fluid. We prove the existence of global and exponential attractors of optimal regularity. We then derive explicit upper bounds for the dimension of these attractors in terms of the three-dimensional Grashof number and the regularization parameter. Finally, we also prove convergence of the (strong) global attractor of the 3D Navier–Stokes–Voigt model to the (weak) global attractor of the 3D Navier–Stokes equation. Our analysis improves and extends recent results obtained by Kalantarov and Titi (Chin Ann Math Ser B 30:697–714, 2009).

The second-grade fluid equations are a model for viscoelastic fluids, with two parameters: α > 0, corresponding to the elastic response, and $${\nu > 0}$$ ν > 0 , corresponding to viscosity. Formally setting these parameters to 0 reduces the equations to the incompressible Euler equations of ideal fluid flow. In this article we study the limits $${\alpha, \nu \to 0}$$ α , ν → 0 of solutions of the second-grade fluid system, in a smooth, bounded, two-dimensional domain with no-slip boundary conditions. This class of problems interpolates between the Euler-α model ( $${\nu = 0}$$ ν = 0 ), for which the authors recently proved convergence to the solution of the incompressible Euler equations, and the Navier-Stokes case (α = 0), for which the vanishing viscosity limit is an important open problem. We prove three results. First, we establish convergence of the solutions of the second-grade model to those of the Euler equations provided $${\nu = \mathcal{O}(\alpha^2)}$$ ν = O ( α 2 ) , as α → 0, extending the main result in (Lopes Filho et al., Physica D 292(293):51–61, 2015). Second, we prove equivalence between convergence (of the second-grade fluid equations to the Euler equations) and vanishing of the energy dissipation in a suitably thin region near the boundary, in the asymptotic regime $${\nu = \mathcal{O}(\alpha^{6/5})}$$ ν = O ( α 6 / 5 ) , $${\nu/\alpha^{2} \to \infty}$$ ν / α 2 → ∞ as α → 0. This amounts to a convergence criterion similar to the well-known Kato criterion for the vanishing viscosity limit of the Navier-Stokes equations to the Euler equations. Finally, we obtain an extension of Kato’s classical criterion to the second-grade fluid model, valid if $${\alpha = \mathcal{O}(\nu^{3/2})}$$ α = O ( ν 3 / 2 ) , as $${\nu \to 0}$$ ν → 0 . The proof of all these results relies on energy estimates and boundary correctors, following the original idea by Kato.

The goal of this article is to provide some essential results for the solution of a regularized viscoplastic frictional flow model adapted from the extensive mathematical analysis of the Bingham model. The Bingham model is a standard for the description of viscoplastic flows and it is widely used in many application areas. However, wet granular viscoplastic flows necessitate the introduction of additional non-linearities and coupling between velocity and stress fields. This article proposes a step toward a frictional coupling, characterized by a dependence of the yield stress to the pressure field. A regularized version of this viscoplastic frictional model is analysed in the framework of stationary flows. Existence, uniqueness and regularity are investigated, as well as finite-dimensional and algorithmic approximations. It is shown that the model can be solved and approximated as far as a frictional parameter is small enough. Getting similar results for the non-regularized model remains an issue. Numerical investigations are postponed to further works.

The weighted L q −L q (q = 1,∞) estimates for the Stokes flow are given in half spaces. Further large-time weighted decays for the second spatial derivatives of the Navier–Stokes equations are established, where the unboundedness of the projection operator $${P: L^q(\mathbb{R}^n_+) \rightarrow L^q_\sigma(\mathbb{R}^n_+)}$$ P : L q ( R + n ) → L σ q ( R + n ) (q = 1,∞) is overcome by employing a decomposition for the convection term. The main results in this article are motivated by the work in Bae (J Differ Equ 222:1–20, 2006; J Math Fluid Mech 10:503–530, 2008) and Bae and Jin (Proc R Soc Edinb Sect A 135:461–477, 2005).

This paper deals with the control of a differential turbulence model of the Ladyzhenskaya–Smagorinsky kind. In the equations we find local and nonlocal nonlinearities: the usual transport terms and a turbulent viscosity that depends on the global in space energy dissipated by the mean flow. We prove that the system is locally null-controllable, with distributed controls locally supported in space. The proof relies on rather well known arguments. However, some specific difficulties are found here because of the occurrence of nonlocal nonlinear terms. We also present an iterative algorithm of the quasi-Newton kind that provides a sequence of states and controls that converge towards a solution to the control problem. Finally, we give the details of a numerical approximation and we illustrate the behavior of the algorithm with a numerical experiment.

Basic properties of a reduced viscous compressible gas–liquid two-fluid model are explored. The model is composed of two conservation laws representing mass balance for gas and liquid coupled to two elliptic equations (Stokes system) for the two fluid velocities and obtained by ignoring acceleration terms in the full momentum equations. First, we present a result that shows existence and uniqueness of regular solutions for a fixed time T 0 > 0 which depends on the initial data and the constant viscosity coefficients. Moreover, T 0 can be large when the viscosity coefficients are large. However, for a fixed set of viscosity coefficients, we conjecture that the smooth solution might blow up, at least, as time tends to infinity. This result is backed up by considering a numerical example for a fixed set of viscosity coefficients demonstrating that for smooth and small initial data with no single-phase regions, the solution may tend to produce both single-phase regions and blow-up of mass gradients as time becomes large.