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.

In this paper we use the short-wavelength instability approach to derive an instability threshold for exact trapped equatorial waves propagating eastwards in the presence of an underlying current.

We present an exact solution of the nonlinear governing equations for internal geophysical water waves propagating westward in the f-plane approximation near the equator. We show that the mass transport velocity induced by this internal equatorial wave is eastward.

We study a regularity criteria of the three-dimensional magnetohydrodynamics system. In particular, we obtain a regularity criteria that involves one velocity component and one current density component. Its immediate consequence is a regularity criteria in terms of one velocity component and two entries from the Jacobian matrix of the magnetic vector field, in contrast to many previous results that involve three entries all from the Jacobian matrix of the velocity vector field. Moreover, the conditions on the current density component is in the norm with scaling dimension zero; i.e. Serrin-class.

In this paper, we study the global well-posedness of the 2D compressible Navier–Stokes equations with large initial data and vacuum. It is proved that if the shear viscosity μ is a positive constant and the bulk viscosity λ is the power function of the density, that is, λ(ρ) = ρ β with β > 3, then the 2D compressible Navier–Stokes equations with the periodic boundary conditions on the torus $${\mathbb{T}^2}$$ T 2 admit a unique global classical solution (ρ, u) which may contain vacuums in an open set of $${\mathbb{T}^2}$$ T 2 . Note that the initial data can be arbitrarily large to contain vacuum states.

The purpose of this paper is to study boundary value problems of Robin type for the Brinkman system and a semilinear elliptic system, called the Darcy–Forchheimer–Brinkman system, on Lipschitz domains in Euclidean setting. In the first part of the paper, we exploit a layer potential analysis and a fixed point theorem to show the existence and uniqueness of the solution to the nonlinear Robin problem for the Darcy–Forchheimer–Brinkman system on a bounded Lipschitz domain in $${\mathbb{R}^n}$$ R n $${(n \in \{2,3\})}$$ ( n ∈ { 2 , 3 } ) with small data in L 2-based Sobolev spaces. In the second part, we show an existence result for the mixed Dirichlet–Robin problem for the same semilinear Darcy–Forchheimer-Brinkman system on a bounded creased Lipschitz domain in $${\mathbb{R}^3}$$ R 3 with small L 2-boundary data. We also study mixed Dirichlet–Robin problems and boundary value problems of mixed Dirichlet–Robin and transmission type for Brinkman systems on bounded creased Lipschitz domains in $${\mathbb{R}^n}$$ R n (n ≥ 3). Finally, we show the well-posedness of the Navier problem for the Brinkman system with boundary data in some L 2-based Sobolev spaces on a bounded Lipschitz domain in $${\mathbb{R}^3}$$ R 3 .

This paper studies the pullback asymptotic behaviors of solutions for a non-autonomous incompressible non-Newtonian fluid in two-dimensional bounded domains. The authors first prove the existence of smooth pullback attractors for the associated process, and then reveal their tempered behaviors in H 2 and H 4 norms as the initial time tends to −∞.

In 2000 Constantin showed that the incompressible Euler equations can be written in an “Eulerian–Lagrangian” form which involves the back-to-labels map (the inverse of the trajectory map for each fixed time). In the same paper a local existence result is proved in certain Hölder spaces $${C^{1,\mu}}$$ C 1 , μ . We review the Eulerian–Lagrangian formulation of the equations and prove that given initial data in H s for $${n \geq 2}$$ n ≥ 2 and $${s > \frac{n}{2}+1}$$ s > n 2 + 1 , a unique local-in-time solution exists on the n-torus that is continuous into H s and C 1 into H s-1. These solutions automatically have C 1 trajectories. The proof here is direct and does not appeal to results already known about the classical formulation. Moreover, these solutions are regular enough that the classical and Eulerian–Lagrangian formulations are equivalent, therefore what we present amounts to an alternative approach to some of the standard theory.

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 connection between the compressible flow of liquid crystals with low Mach number and the incompressible flow of liquid crystals is studied in a bounded domain. In particular, the convergence of weak solutions of the compressible flow of liquid crystals to the weak solutions of the incompressible flow of liquid crystals is proved when the Mach number approaches zero; that is, the incompressible limit is justified for weak solutions in a bounded domain.

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).

For two parallel plates held fixed and dipped in an idealized infinite liquid bath in a gravity field, we define the force profile to be the function $${F:(0,\infty)\to \mathbb{R}}$$ F : ( 0 , ∞ ) → R which assigns to each plate separation distance $${d\in (0,\infty)}$$ d ∈ ( 0 , ∞ ) the signed (horizontal) force density of mutual attraction between the plates (with repulsive forces corresponding to negative values). We show that precisely three nontrivial qualitative profiles are possible depending on the adhesion properties of the plates, namely the contact angles. Results of Finn, Lu, and Bhatnagar in three recent papers which treat this problem originally suggested by Laplace, show that the profiles are well-defined and depend, in a qualitative sense, only on the contact angle pair $${(\gamma_1,\gamma_2)}$$ ( γ 1 , γ 2 ) associated with the adhesion properties of the inner facing surfaces of the plates. We also describe symmetry relations satisfied by the quantitative force profiles with respect to $${(\gamma_1,\gamma_2)\in [0,\pi]\times[0,\pi]}$$ ( γ 1 , γ 2 ) ∈ [ 0 , π ] × [ 0 , π ] . Taken together, our discussion provides a unified overview of the recent progress on the problem and an alternative approach to several of the main results.

In this paper we investigate the issue of the inviscid limit for the compressible Navier–Stokes system in an impermeable fixed bounded domain. We consider two kinds of boundary conditions. The first one is the no-slip condition. In this case we extend the famous conditional result (Kato, T.: Remarks on zero viscosity limit for nonstationary Navier–Stokes flows with boundary. In: Seminar on nonlinear partial differential equations, vol. 2, pp. 85–98. Math. Sci. Res. Inst. Publ., Berkeley 1984) obtained by Kato in the homogeneous incompressible case. Kato proved that if the energy dissipation rate of the viscous flow in a boundary layer of width proportional to the viscosity vanishes then the solutions of the incompressible Navier–Stokes equations converge to some solutions of the incompressible Euler equations in the energy space. We provide here a natural extension of this result to the compressible case. The other case is the Navier condition which encodes that the fluid slips with some friction on the boundary. In this case we show that the convergence to the Euler equations holds true in the energy space, as least when the friction is not too large. In both cases we use in a crucial way some relative energy estimates proved recently by Feireisl et al. in J. Math. Fluid Mech. 14:717–730 (2012).

We give simple proofs that a weak solution u of the Navier-Stokes equations with H (1) initial data remains strong on the time interval [0, T] if it satisfies the Prodi-Serrin type condition u a L (s) (0, T;L (r,a)(Omega)) or if its L (s,a)(0, T;L (r,a)(Omega)) norm is sufficiently small, where 3 < r a parts per thousand currency sign a and (3/r) + (2/s) = 1.

The problem of the nonequivalence of the sets of equilibrium points and energy-Casimir extremal points, which occurs in the noncanonical Hamiltonian formulation of equations describing ideal fluid and plasma dynamics, is addressed in the context of the Euler equation for an incompressible inviscid fluid. The problem is traced to a Casimir deficit, where Casimir elements constitute the center of the Poisson algebra underlying the Hamiltonian formulation, and this leads to a study of singularities of the Poisson operator defining the Poisson bracket. The kernel of the Poisson operator, for this typical example of an infinite-dimensional Hamiltonian system for media in terms of Eulerian variables, is analyzed. For two-dimensional flows, a rigorously solvable system is formulated. The nonlinearity of the Euler equation makes the Poisson operator inhomogeneous on phase space (the function space of the state variable), and it is seen that this creates a singularity where the nullity of the Poisson operator (the “dimension” of the center) changes. The problem is an infinite-dimension generalization of the theory of singular differential equations. Singular Casimir elements stemming from this singularity are unearthed using a generalization of the functional derivative that occurs in the Poisson bracket.

We perform energy estimates for a sharp-interface model of two-dimensional, two-phase Darcy flow with surface tension. A proof of well-posedness of the initial value problem follows from these estimates. In general, the time of existence of these solutions will go to zero as the surface tension parameter vanishes. We then make two additional estimates, in the case that a stability condition is satisfied by the initial data: we make an additional energy estimate which is uniform in the surface tension parameter, and we make an estimate for the difference of two solutions with different values of the surface tension parameter. These additional estimates allow the zero surface tension limit to be taken, showing that solutions of the initial value problem in the absence of surface tension are the limit of solutions of the initial value problem with surface tension as surface tension vanishes.

We investigate the dynamics of a class of tumor growth models known as mixed models. The key characteristic of these type of tumor growth models is that the different populations of cells are continuously present everywhere in the tumor at all times. In this work we focus on the evolution of tumor growth in the presence of proliferating, quiescent and dead cells as well as a nutrient. The system is given by a multi-phase flow model and the tumor is described as a growing continuum Ω with boundary ∂Ω both of which evolve in time. Global-in-time weak solutions are obtained using an approach based on penalization of the boundary behavior, diffusion and viscosity in the weak formulation.

Motivated by Kolmogorov's theory of turbulence we present a unified approach to the regularity problems for the 3D Navier-Stokes and Euler equations. We introduce a dissipation wavenumber that separates low modes where the Euler dynamics is predominant from the high modes where the viscous forces take over. Then using an indifferent to the viscosity technique we obtain a new regularity criterion which is weaker than every Ladyzhenskaya-Prodi-Serrin condition in the viscous case, and reduces to the Beale-Kato-Majda criterion in the inviscid case. In the viscous case we prove that Leray-Hopf solutions are regular provided , which improves our previous condition. We also show that for all Leray-Hopf solutions. Finally, we prove that Leray-Hopf solutions are regular when the time-averaged spatial intermittency is small, i.e., close to Kolmogorov's regime.

Time-periodic solutions to the linearized Navier–Stokes system in the n-dimensional whole-space are investigated. For time-periodic data in L q -spaces, maximal regularity and corresponding a priori estimates for the associated time-periodic solutions are established. More specifically, a Banach space of time-periodic vector fields is identified with the property that the linearized Navier–Stokes operator maps this space homeomorphically onto the L q -space of time-periodic data.

The incompressible Navier–Stokes equations are considered in the two-dimensional strip $${\mathbb{R} \times [0,L]}$$ R × [ 0 , L ] , with periodic boundary conditions and no exterior forcing. If the initial velocity is bounded, it is shown that the solution remains uniformly bounded for all time, and that the vorticity distribution converges to zero as $${t \to \infty}$$ t → ∞ . This implies, after a transient period, the emergence of a laminar regime in which the solution rapidly converges to a shear flow described by the one-dimensional heat equation in an appropriate Galilean frame. The approach is constructive and provides explicit estimates on the size of the solution and the lifetime of the turbulent period in terms of the initial Reynolds number.