We prove that a solution to the 3D Navier–Stokes or magneto-hydrodynamics equations does not blow up at t = T provided $\displaystyle \limsup_{q \to \infty} \int_{\mathcal{T}_q}^T \|\Delta_q(\nabla \times u)\|_\infty \, dt$ is small enough, where u is the velocity, $\Delta_q$ is the Littlewood–Paley projection and $\mathcal T_q$ is a certain sequence such that $\mathcal T_q \to T$ as $q \to \infty$. This improves many existing regularity criteria.

This article is dedicated to investigating limit behaviours of invariant measures with respect to delay and system parameters of 3D Navier–Stokes–Voigt equations. Firstly, the well-posedness of such a system is obtained on arbitrary open sets that satisfy the Poincaré inequality, and then a unique minimal pullback attractor is attained by using the energy equation method and asymptotic compactness property. Furthermore, we construct a family of invariant Borel probability measures, which are supported on the pullback attractors. Specifically, when the external forcing terms are periodic in time, the periodic invariant measure can be obtained. Finally, as the delay approaches zero and system parameters tend to some numbers, the limit of the invariant measure sequences for this class of equations must be the invariant measure of the corresponding limit equations.

In this article, we focus on the Cauchy problem of the three-dimensional generalized incompressible micropolar system in critical Fourier–Besov–Morrey spaces. By using the Fourier localization argument and the Littlewood–Paley theory, we get the local well-posedness results and global well-posedness results with small initial data belonging to the critical Fourier–Besov–Morrey spaces.

We find solutions that describe the levelling of a thin fluid film, comprising a non-Newtonian power-law fluid, that coats a substrate and evolves under the influence of surface tension. We consider the evolution from periodic and localized initial conditions as separate cases. The particular (similarity) solutions in each of these two cases exhibit the generic property that the profiles are weakly singular (that is, higher-order derivatives do not exist) at points where the pressure gradient vanishes. Numerical simulations of the thin film equation, with either periodic or localized initial condition, are shown to approach the appropriate particular solution.

We consider heat or mass transport from a circular cylinder under a uniform crossflow at small Reynolds numbers, $\mathrm{Re}\ll 1$. This problem has been thwarted in the past by limitations inherent in the classical analyses of the singular flow problem, which have used asymptotic expansions in inverse powers of $\log \mathrm{Re}$. We here make use of the hybrid approximation of Kropinski, Ward & Keller [(1995) SIAM J. Appl. Math. 55, 1484], based upon a robust asymptotic expansion in powers of $\mathrm{Re}$. In that approximation, the “inner” streamfunction is provided by the product of a pre-factor $S$, a slowly varying function of $\mathrm{Re}$, with a $\mathrm{Re}$-independent “canonical” solution of a simple mathematical form. The pre-factor, in turn, is determined as an implicit function of $\log \mathrm{Re}$ via asymptotic matching with a numerical solution of the nonlinear single-scaled “outer” problem, where the cylinder appears as a point singularity. We exploit the hybrid approximation to analyse the transport problem in the limit of large Péclet number, $\mathrm{Pe}\gg 1$. In that limit, transport is restricted to a narrow boundary layer about the cylinder surface – a province contained within the inner region of the flow problem. With $S$ appearing as a parameter, a similarity solution is readily constructed for the boundary-layer problem. It provides the Nusselt number as $0.5799(S\,\mathrm{Pe})^{1/3}$. This asymptotic prediction is in remarkably close agreement with that of the numerical solution of the exact problem [Dennis, Hudson & Smith (1968) Phys. Fluids 11, 933] even for moderate $\mathrm{Re}$-values.

We give a comprehensive study of the 3D Navier–Stokes–Brinkman–Forchheimer equations in a bounded domain endowed with the Dirichlet boundary conditions and non-autonomous external forces. This study includes the questions related with the regularity of weak solutions, their dissipativity in higher energy spaces and the existence of the corresponding uniform attractors

In numerical linear stability investigations, the rates of change of the kinetic and thermal energy of the perturbation flow are often used to identify the dominant mechanisms by which kinetic or thermal energy is exchanged between the basic and the perturbation flow. Extending the conventional energy analysis for a single-phase Boussinesq fluid, the energy budgets of arbitrary infinitesimal perturbations to the basic two-phase liquid–gas flow are derived for an axisymmetric thermocapillary bridge when the material parameters in both phases depend on the temperature. This allows identifying individual transport terms and assessing their contributions to the instability if the basic flow and the critical mode are evaluated at criticality. The full closed-form energy budgets of linear modes have been derived for thermocapillary two-phase flow taking into account the temperature dependence of all thermophysical parameters. The influence of different approximations to the temperature dependence on the linear stability boundary of the axisymmetric flow in thermocapillary liquid bridges is tested regarding their accuracy. The general mechanism of symmetry breaking turns out to be very robust.

Vertically vibrating a liquid bath may allow a self-propelled wave-particle entity to move on its free surface. The horizontal dynamics of this walking droplet, under the constraint of an external drag force, can be described adequately by an integro-differential trajectory equation. For a sinusoidal wave field, this equation is equivalent to a closed three-dimensional system of nonlinear ODEs. We explicitly define a stability boundary for the system and a quantised criterion for its partial integrability in the meromorphic category.

This paper proves the energy equality for distributional solutions to fractional Navier-Stokes equations, which gives a new proof and covers the classical result of Galdi [Proc. Amer. Math. Soc. 147 (2019), 785–792].

In this paper, we considered the global strong solution to the 3D incompressible micropolar equations with fractional partial dissipation. Whether or not the classical solution to the 3D Navier–Stokes equations can develop finite-time singularity remains an outstanding open problem, so does the same issue on the 3D incompressible micropolar equations. We establish the global-in-time existence and uniqueness strong solutions to the 3D incompressible micropolar equations with fractional partial velocity dissipation and microrotation diffusion with the initial data $(\mathbf {u}_0,\ \mathbf {w}_0)\in H^1(\mathbb {R}^3)$.

This paper focuses on a 2D magnetohydrodynamic system with only horizontal dissipation in the domain $\Omega = \mathbb {T}\times \mathbb {R}$ with $\mathbb {T}=[0,\,1]$ being a periodic box. The goal here is to understand the stability problem on perturbations near the background magnetic field $(1,\,0)$. Due to the lack of vertical dissipation, this stability problem is difficult. This paper solves the desired stability problem by simultaneously exploiting two smoothing and stabilizing mechanisms: the enhanced dissipation due to the coupling between the velocity and the magnetic fields, and the strong Poincaré type inequalities for the oscillation part of the solution, namely the difference between the solution and its horizontal average. In addition, the oscillation part of the solution is shown to converge exponentially to zero in $H^{1}$ as $t\to \infty$. As a consequence, the solution converges to its horizontal average asymptotically.

We consider a spherical particle levitating above a liquid bath owing to the Leidenfrost effect, where the vapour of either the bath or sphere forms an insulating film whose pressure supports the sphere’s weight. Starting from a reduced formulation based on a lubrication-type approximation, we use matched asymptotics to describe the morphology of the vapour film assuming that the sphere is small relative to the capillary length (small Bond number) and that the densities of the bath and sphere are comparable. We find that this regime is comprised of two formally infinite sequences of distinguished limits which meet at an accumulation point, the limits being defined by the smallness of an intrinsic evaporation number relative to the Bond number. These sequences of limits reveal a surprisingly intricate evolution of the film morphology with increasing sphere size. Initially, the vapour film transitions from a featureless morphology, where the thickness profile is parabolic, to a neck–bubble morphology, which consists of a uniform pressure bubble bounded by a narrow and much thinner annular neck. Gravity effects then become important in the bubble leading to sequential formation of increasingly smaller neck–bubble pairs near the symmetry axis. This process terminates when the pairs closest to the symmetry axis become indistinguishable and merge. Subsequently, the inner section of that merger transitions into a uniform-thickness film that expands radially, gradually squishing increasingly larger neck–bubble pairs into a region of localised oscillations sandwiched between the uniform film and what remains of the bubble whose radial extent is presently comparable to the uniform film; the neck–bubble pairs farther from the axis remain essentially intact. Ultimately, the uniform film gobbles up the largest outermost bubble, whereby the morphology simplifies to a uniform film bounded by localised oscillations. Overall, the asymptotic analysis describes the continuous evolution of the vapour film from a neck–bubble morphology typical of a Leidenfrost drop levitating above a flat solid substrate to a uniform-film morphology which resembles that in the case of a large liquid drop levitating above a liquid bath.

Taking the consideration of two-dimensional stochastic Navier–Stokes equations with multiplicative Lévy noises, where the noises intensities are related to the viscosity, a large deviation principle is established by using the weak convergence method skillfully, when the viscosity converges to 0. Due to the appearance of the jumps, it is difficult to close the energy estimates and obtain the desired convergence. Hence, one cannot simply use the weak convergence approach. To overcome the difficulty, one introduces special norms for new arguments and more careful analysis.

In this article, we give a comprehensive characterization of $L^1$-summability for the Navier-Stokes flows in the half space, which is a long-standing problem. The main difficulties are that $L^q-L^r$ estimates for the Stokes flow don't work in this end-point case: $q=r=1$; the projection operator $P: L^1\longrightarrow L^1_\sigma$ is not bounded any more; useful information on the pressure function is missing, which arises in the net force exerted by the fluid on the noncompact boundary. In order to achieve our aims, by making full use of the special structure of the half space, we decompose the pressure function into two parts. Then the knotty problem of handling the pressure term can be transformed into establishing a crucial and new weighted $L^1$-estimate, which plays a fundamental role. In addition, we overcome the unboundedness of the projection $P$ by solving an elliptic problem with homogeneous Neumann boundary condition.

The classical model for studying one-phase Hele-Shaw flows is based on a highly nonlinear moving boundary problem with the fluid velocity related to pressure gradients via a Darcy-type law. In a standard configuration with the Hele-Shaw cell made up of two flat stationary plates, the pressure is harmonic. Therefore, conformal mapping techniques and boundary integral methods can be readily applied to study the key interfacial dynamics, including the Saffman–Taylor instability and viscous fingering patterns. As well as providing a brief review of these key issues, we present a flexible numerical scheme for studying both the standard and nonstandard Hele-Shaw flows. Our method consists of using a modified finite-difference stencil in conjunction with the level-set method to solve the governing equation for pressure on complicated domains and track the location of the moving boundary. Simulations show that our method is capable of reproducing the distinctive morphological features of the Saffman–Taylor instability on a uniform computational grid. By making straightforward adjustments, we show how our scheme can easily be adapted to solve for a wide variety of nonstandard configurations, including cases where the gap between the plates is linearly tapered, the plates are separated in time, and the entire Hele-Shaw cell is rotated at a given angular velocity.

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].

In this study, we continue our study of the Cauchy problem associated with the Brinkman equations [see (1.1) and (1.2) below] which model fluid flow in certain types of porous media. Here, we will consider the flow in the upper half-space

\[ \mathbb{R}_{+}^{3}=\left\{\left(x,y,z\right) \in\mathbb{R}^{3}\left\vert z\geqslant 0\right.\right\}, \]

$z=0$

Existence of non-negative weak solutions is shown for a full curvature thin-film model of a liquid thin film flowing down a vertical fibre. The proof is based on the application of a priori estimates derived for energy-entropy functionals. Long-time behaviour of these weak solutions is analysed and, under some additional constraints for the model parameters and initial values, convergence towards a travelling wave solution is obtained. Numerical studies of energy minimisers and travelling waves are presented to illustrate analytical results.

We show local higher integrability of derivative of a suitable weak solution to the surface growth model, provided a scale-invariant quantity is locally bounded. If additionally our scale-invariant quantity is small, we prove local smoothness of solutions.