In this paper we present an exact and explicit solution to the geophysical governing equations in the Equatorial region, which represents internal oceanic waves in the presence of a constant underlying current.

We show here that capillary-gravity wave trains can propagate at the free surface of a rotational water flow of constant non-zero vorticity over a flat bed only if the flow is two-dimensional. Moreover, we also show that the vorticity must have only one non zero component which points in the horizontal direction orthogonal to the direction of wave propagation. This result is of relevance in the study of nonlinear resonances of wave trains. We perform such a study for three- and four wave interactions.

This paper presents existence theories for several families of axisymmetric solitary waves on the surface of an otherwise cylindrical ferrofluid jet surrounding a stationary metal rod. The ferrofluid, which is governed by a general (nonlinear) magnetisation law, is subject to an azimuthal magnetic field generated by an electric current flowing along the rod. The ferrohydrodynamic problem for axisymmetric travelling waves is formulated as an infinite-dimensional Hamiltonian system in which the axial direction is the time-like variable. A centre-manifold reduction technique is employed to reduce the system to a locally equivalent Hamiltonian system with a finite number of degrees of freedom, and homoclinic solutions to the reduced system, which correspond to solitary waves, are detected by dynamical-systems methods.

The purpose of this paper is to study boundary value problems of transmission type for the Navier-Stokes and Darcy-Forchheimer-Brinkman systems in two complementary Lipschitz domains on a compact Riemannian manifold of dimension m epsilon {2, 3} . We exploit a layer potential method combined with a fixed point theorem in order to show existence and uniqueness results when the given data are suitably small in L-2-based Sobolev spaces.

The abstract theory of critical spaces developed in Prüss and Wilke (J Evol Equ, 2017. doi:10.1007/s00028-017-0382-6), Prüss et al. (Critical spaces for quasilinear parabolic evolution equations and applications, 2017) is applied to the Navier–Stokes equations in bounded domains with Navier boundary conditions as well as no-slip conditions. Our approach unifies, simplifies and extends existing work in the $$L_p$$ Lp –$$L_q$$ Lq setting, considerably. As an essential step, it is shown that the strong and weak Stokes operators with Navier conditions admit an $$\mathcal {H}^\infty $$ H∞ -calculus with $$\mathcal {H}^\infty $$ H∞ -angle 0, and the real and complex interpolation spaces of these operators are identified.

Topology changes in multi-phase fluid flows are difficult to model within a traditional sharp interface theory. Diffuse interface models turn out to be an attractive alternative to model two-phase flows. Based on a Cahn–Hilliard–Navier–Stokes model introduced by Abels et al. (Math Models Methods Appl Sci 22(3):1150013, 2012), which uses a volume-averaged velocity, we derive a diffuse interface model in a Hele–Shaw geometry, which in the case of non-matched densities, simplifies an earlier model of Lee et al. (Phys Fluids 14(2):514–545, 2002). We recover the classical Hele–Shaw model as a sharp interface limit of the diffuse interface model. Furthermore, we show the existence of weak solutions and present several numerical computations including situations with rising bubbles and fingering instabilities.

We provide a self-contained proof of the solvability and regularity of a Hodge-type elliptic system, wherein the divergence and curl of a vector field u are prescribed in an open, bounded, Sobolev-class domain $${\Omega \subseteq \mathbb{R}^{\rm n}}$$ Ω ⊆ R n , and either the normal component $${{\bf u} \cdot {\bf N}}$$ u · N or the tangential components of the vector field $${{\bf u} \times {\bf N}}$$ u × N are prescribed on the boundary $${\partial \Omega}$$ ∂ Ω . For $${{\rm k} > {\rm n}/2}$$ k > n / 2 , we prove that u is in the Sobolev space $${H^{\rm k+1}(\Omega)}$$ H k + 1 ( Ω ) if $${\Omega}$$ Ω is an $${H^{\rm k+1}}$$ H k + 1 -domain, and the divergence, curl, and either the normal or tangential trace of u has sufficient regularity. The proof is based on a regularity theory for vector elliptic equations set on Sobolev-class domains and with Sobolev-class coefficients, and with a rather general set of Dirichlet and Neumann boundary conditions. The resulting regularity theory for the vector u is fundamental in the analysis of free-boundary and moving interface problems in fluid dynamics.

In this paper, we reconsider a circular cylinder horizontally floating on an unbounded reservoir in a gravitational field directed downwards, which was studied by Bhatnagar and Finn (Phys Fluids 18(4):047103, 2006). We follow their approach but with some modifications. We establish the relation between the total energy $$E_T$$ ET relative to the undisturbed state and the total force $$F_T$$ FT , that is, $$F_T = -\frac{dE_T}{dh}$$ FT=-dETdh , where h is the height of the center of the cylinder relative to the undisturbed fluid level. There is a monotone relation between h and the wetting angle $$\phi _0$$ ϕ0 . We study the number of equilibria, the floating configurations and their stability for all parameter values. We find that the system admits at most two equilibrium points for arbitrary contact angle $$\gamma $$ γ , the one with smaller $$\phi _0$$ ϕ0 is stable and the one with larger $$\phi _0$$ ϕ0 is unstable. Since the one-sided solution can be translated horizontally, the fluid interfaces may intersect. We show that the stable equilibrium point never lies in the intersection region, while the unstable equilibrium point may lie in the intersection region.

We provide the existence and asymptotic description of solitary wave solutions to a class of modified Green–Naghdi systems, modeling the propagation of long surface or internal waves. This class was recently proposed by Duchêne et al. (Stud Appl Math 137:356–415, 2016) in order to improve the frequency dispersion of the original Green–Naghdi system while maintaining the same precision. The solitary waves are constructed from the solutions of a constrained minimization problem. The main difficulties stem from the fact that the functional at stake involves low order non-local operators, intertwining multiplications and convolutions through Fourier multipliers.

This paper examines the global regularity problem on several 2D incompressible fluid models with partial dissipation. They are the surface quasi-geostrophic (SQG) equation, the 2D Euler equation and the 2D Boussinesq equations. These are well-known models in fluid mechanics and geophysics. The fundamental issue of whether or not they are globally well-posed has attracted enormous attention. The corresponding models with partial dissipation may arise in physical circumstances when the dissipation varies in different directions. We show that the SQG equation with either horizontal or vertical dissipation always has global solutions. This is in sharp contrast with the inviscid SQG equation for which the global regularity problem remains outstandingly open. Although the 2D Euler is globally well-posed for sufficiently smooth data, the associated equations with partial dissipation no longer conserve the vorticity and the global regularity is not trivial. We are able to prove the global regularity for two partially dissipated Euler equations. Several global bounds are also obtained for a partially dissipated Boussinesq system.

It is well known that the full Navier–Stokes–Fourier system does not possess a strong solution in three dimensions which causes problems in applications. However, when modeling the flow of a fluid in a thin long pipe, the influence of the cross section can be neglected and the flow is basically one-dimensional. This allows us to deal with strong solutions which are more convenient for numerical computations. The goal of this paper is to provide a rigorous justification of this approach. Namely, we prove that any suitable weak solution to the three-dimensional NSF system tends to a strong solution to the one-dimensional system as the thickness of the pipe tends to zero.

We consider the Bardina's model for turbulent incompressible flows in the whole space with a cut-off frequency of order . We show that for any fixed, the model has a unique regular solution defined for all [0, infinity].

The main objective of this article is to study the nonlinear stability and dynamic transitions of the basic (zonal) shear flows for the three-dimensional continuously stratified rotating Boussinesq model. The model equations are fundamental equations in geophysical fluid dynamics, and dynamics associated with their basic zonal shear flows play a crucial role in understanding many important geophysical fluid dynamical processes, such as the meridional overturning oceanic circulation and the geophysical baroclinic instability. In this paper, first we derive a threshold for the energy stability of the basic shear flow, and obtain a criterion for local nonlinear stability in terms of the critical horizontal wavenumbers and the system parameters such as the Froude number, the Rossby number, the Prandtl number and the strength of the shear flow. Next, we demonstrate that the system always undergoes a dynamic transition from the basic shear flow to either a spatiotemporal oscillatory pattern or circle of steady states, as the shear strength of the basic flow crosses a critical threshold. Also, we show that the dynamic transition can be either continuous or catastrophic, and is dictated by the sign of a transition number, fully characterizing the nonlinear interactions of different modes. Both the critical shear strength and the transition number are functions of the system parameters. A systematic numerical method is carried out to explore transition in different flow parameter regimes. In particular, our numerical investigations show the existence of a hypersurface which separates the parameter space into regions where the basic shear flow is stable and unstable. Numerical investigations also yield that the selection of horizontal wave indices is determined only by the aspect ratio of the box. We find that the system admits only critical eigenmodes with roll patterns aligned with the x-axis. Furthermore, numerically we encountered continuous transitions to multiple steady states, as well as continuous and catastrophic transitions to spatiotemporal oscillations.

We study the compressible and incompressible two-phase flows separated by a sharp interface with a phase transition and a surface tension. In particular, we consider the problem in $$\mathbb {R}^N$$ RN , and the Navier–Stokes–Korteweg equations is used in the upper domain and the Navier–Stokes equations is used in the lower domain. We prove the existence of $$\mathcal {R}$$ R -bounded solution operator families for a resolvent problem arising from its model problem. According to Göts and Shibata (Asymptot Anal 90(3–4):207–236, 2014), the regularity of $$\rho _+$$ ρ+ is $$W^1_q$$ Wq1 in space, but to solve the kinetic equation: $$\mathbf {u}_\Gamma \cdot \mathbf {n}_t = [[\rho \mathbf {u}]]\cdot \mathbf {n}_t /[[\rho ]]$$ uΓ·nt=[[ρu]]·nt/[[ρ]] on $$\Gamma _t$$ Γt we need $$W^{2-1/q}_q$$ Wq2-1/q regularity of $$\rho _+$$ ρ+ on $$\Gamma _t$$ Γt , which means the regularity loss. Since the regularity of $$\rho _+$$ ρ+ dominated by the Navier–Stokes–Korteweg equations is $$W^3_q$$ Wq3 in space, we eliminate the problem by using the Navier–Stokes–Korteweg equations instead of the compressible Navier–Stokes equations.

2D shallow water equations have degenerate viscosities proportional to surface height, which vanishes in many physical considerations, say, when the initial total mass, or energy are finite. Such a degeneracy is a highly challenging obstacle for development of well-posedness theory, even local-in-time theory remains open for a long time. In this paper, we will address this open problem with some new perspectives, independent of the celebrated BD-entropy (Bresch et al in Commun Math Phys 238:211–223, 2003, Commun Part Differ Eqs 28:843–868, 2003, Analysis and Simulation of Fluid Dynamics, 2007). After exploring some interesting structures of most models of 2D shallow water equations, we introduced a proper notion of solution class, called regular solutions, and identified a class of initial data with finite total mass and energy, and established the local-in-time well-posedness of this class of smooth solutions. The theory is applicable to most relatively physical shallow water models, broader than those with BD-entropy structures. We remark that our theory is on the local strong solutions, while the BD entropy is an essential tool for the global weak solutions. Later, a Beale-Kato-Majda type blow-up criterion is also established. This paper is mainly based on our early preprint (Li et al. in 2D compressible Navier–Stokes equations with degenerate viscosities and far field vacuum, preprint. arXiv:1407.8471 , 2014).

We consider the initial value problem to the Isobe–Kakinuma model for water waves and the structure of the model. The Isobe–Kakinuma model is the Euler–Lagrange equations for an approximate Lagrangian which is derived from Luke’s Lagrangian for water waves by approximating the velocity potential in the Lagrangian. The Isobe–Kakinuma model is a system of second order partial differential equations and is classified into a system of nonlinear dispersive equations. Since the hypersurface $$t=0$$ t=0 is characteristic for the Isobe–Kakinuma model, the initial data have to be restricted in an infinite dimensional manifold for the existence of the solution. Under this necessary condition and a sign condition, which corresponds to a generalized Rayleigh–Taylor sign condition for water waves, on the initial data, we show that the initial value problem is solvable locally in time in Sobolev spaces. We also discuss the linear dispersion relation to the model.

A two-dimensional water wave system is examined consisting of two discrete incompressible fluid domains separated by a free common interface. In a geophysical context this is a model of an internal wave, formed at a pycnocline or thermocline in the ocean. The system is considered as being bounded at the bottom and top by a flatbed and wave-free surface respectively. A current profile with depth-dependent currents in each domain is considered. The Hamiltonian of the system is determined and expressed in terms of canonical wave-related variables. Limiting behaviour is examined and compared to that of other known models. The linearised equations as well as long-wave approximations are presented.

We consider the Muskat Problem with surface tension in two dimensions over the real line, with H s initial data and allowing the two fluids to have different constant densities and viscosities. We take the angle between the interface and the horizontal, and derive an evolution equation for it. Via energy methods, it has been shown that a unique solution $${\theta}$$ θ exists locally and can be continued while $${||\theta||_{s}}$$ | | θ | | s remains bounded and the arc chord condition holds. We prove that when both fluids have the same viscosity and the initial data is sufficiently small, the energy estimate is dominated by second-order dissipative terms. As a result, the energy is non-increasing, and that the resulting solution $${\theta}$$ θ exists globally in time.

The main aim of the paper is to investigate the transitions of the thermohaline circulation in a spherical shell in a parameter regime which only allows transitions to multiple equilibria. We find that the first transition is either continuous (Type-I) or drastic (Type-II) depending on the sign of the transition number. The transition number depends on the system parameters and $$l_c$$ lc , which is the common degree of spherical harmonics of the first critical eigenmodes, and it can be written as a sum of terms describing the nonlinear interactions of various modes with the critical modes. We obtain the exact formulas of this transition number for $$l_c=1$$ lc=1 and $$l_c=2$$ lc=2 cases. Numerically, we find that the main contribution to the transition number is due to nonlinear interactions with modes having zero wave number and the contribution from the nonlinear interactions with higher frequency modes is negligible. In our numerical experiments we encountered both types of transition for $$\text {Le}1$$ Le>1 . In the continuous transition scenario, we rigorously prove that an attractor in the phase space bifurcates which is homeomorphic to the 2$$l_c$$ lc dimensional sphere and consists entirely of degenerate steady state solutions.

The paper is concerned with the regularity of weak solutions to the Navier–Stokes equations. The aim is to show a relaxed Prodi–Serrin condition for regularity. The most interesting aspect of the result is that no compatibility condition is required on the initial data $$v_\circ \in J^2(\Omega )$$ v∘∈J2(Ω) .