Abstract: We study the Navier–Stokes equations of steady motion of a viscous incompressible fluid in \({\mathbb{R}^{3}}\) We prove that there are no nontrivial solution of these equations defined in the whole space \({\mathbb{R}^{3}}\) 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

Abstract: 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).

Abstract: The purpose of this work is to analyze the mathematical model governing motion of n-component, heat conducting reactive mixture of compressible gases. We prove sequential stability of weak variational entropy solutions when the state equation essentially depends on the species concentration and the species diffusion fluxes depend on gradients of partial pressures. Of crucial importance for our analysis is the fact that viscosity coefficients vanish on vacuum and the source terms enjoy the admissibility condition dictated by the second law of thermodynamics.

Abstract: Instability of plane Poiseuille flow in viscous compressible gas is investigated. A condition for the Reynolds and Mach numbers is given in order for plane Poiseuille flow to be unstable. It turns out that plane Poiseuille flow is unstable for Reynolds numbers much less than the critical Reynolds number for the incompressible flow when the Mach number is suitably large. It is proved by the analytic perturbation theory that the linearized operator around plane Poiseuille flow has eigenvalues with positive real part when the instability condition for the Reynolds and Mach numbers is satisfied.

Abstract: 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.

Abstract: In the present paper we consider mild bounded ancient (backward) solutions to the Navier–Stokes equations in the half plane. We give two different definitions, prove their equivalence and prove smoothness up to the boundary. Such solutions appear as a result of rescaling around a singular point of the initial boundary value problem for the Navier–Stokes equations in the half-plane.

Abstract: We show that the incompressible 3D Navier–Stokes system in a \({{\mathscr{C}}^{1,1}}\) bounded domain or a bounded convex domain \({\Omega}\) with a non penetration condition \({ u\cdot u=0}\) at the boundary \({\partial\Omega}\) together with a time-dependent Robin boundary condition of the type \({ u\times{\rm curl}\,u=\beta(t) u}\) on \({\partial\Omega}\) admits a solution with enough regularity provided the initial condition is small enough in an appropriate functional space.

Abstract: As was found in Cheskidov and Kavlie (Pullback attractors for generalized evolutionary systems. DCDS-B 20(3), 749–779, 2015), the 3D Navier–Stokes equations with a translationally bounded force possesses pullback attractors \({\mathscr{A}_w(t)}\) in a weak sense. Moreover, those attractors consist of complete bounded trajectories. In this paper, we present a sufficient condition under which the pullback attractors are degenerate. That is, if the Grashof number is small enough, each section of the pullback attractor is a single point on a unique, complete, bounded, strong solution. We then apply our results to provide a new proof of the existence of a unique, strong, periodic solution to the 3D Navier–Stokes with a small, periodic forcing term.

Abstract: 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.

Abstract: The equations for micropolar Bingham fluid are considered and global existence of a weak solution for pressure driven flows is proved for a one-dimensional boundary-value problem with periodic boundary conditions. In contrast to the classical Bingham fluid, the micropolar Bingham fluid supports local micro-rotations and two types of plug zones. Our approach is different from that of Duvaut–Lions developed for the classical Bingham viscoplastic materials. We do not apply the variational inequality but make use an approximation of the generalized Bingham fluid by a Non-Newtonian fluid with a continuous constitutive law.

Abstract: We give a review of developments concerning the Gerstner wave solution to the incompressible water wave equations, including many recent contributions that have successfully extended the Gerstner wave theory to geophysical and stratified fluids. We also highlight aspects of the mass transport of Gerstner waves, which serves to contrast the Gerstner solution with linear and nonlinear irrotational theories.

Abstract: The paper deals with the motion of two immiscible viscous fluids in a container, one of the fluids being compressible while another one being incompressible. The interface between the fluids is an unknown closed surface where surface tension is neglected. We assume the compressible fluid to be barotropic, the pressure being given by an arbitrary smooth increasing function. This problem is considered in anisotropic Sobolev–Slobodetskiǐ spaces. We show that the L2-norms of the velocity and deviation of compressible fluid density from the mean value decay exponentially with respect to time. The proof is based on a local existence theorem (Denisova, Interfaces Free Bound 2:283–312, 2000) and on the idea of constructing a function of generalized energy, proposed by Padula (J Math Fluid Mech 1:62–77, 1999). In addition, we eliminate the restrictions for the viscosities which appeared in Denisova (Interfaces Free Bound 2:283–312, 2000).

Abstract: This article is devoted to the study of the critical dissipative surface quasi-geostrophic (SQG) equation in \({\mathbb{R}^2}\). For any initial data \({\theta_{0}}\) belonging to the space \({\Lambda^{s} ( H^{s}_{uloc}(\mathbb{R}^2)) \cap L^\infty(\mathbb{R}^2)}\), we show that the critical (SQG) equation has at least one global weak solution in time for all 1/4 ≤ s ≤ 1/2 and at least one local weak solution in time for all 0 < s < 1/4. The proof for the global existence is based on a new energy inequality which improves the one obtain in Lazar (Commun Math Phys 322:73–93, 2013) whereas the local existence uses more refined energy estimates based on Besov space techniques.

Abstract: 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 (R n) → L 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). Mathematics Subject Classification. 35Q35, 35B40, 75D05, 76D07.

Abstract: We construct global \({\dot{H}^1\cap \dot{H}^{-1}}\) solutions to a logarithmically modified 2D Euler vorticity equation. Our main tool is a new logarithm interpolation inequality which exploits the L∞−-conservation of the vorticity.

Abstract: We discuss the equations describing the dynamic of the heat transfer in a magnetic fluid flow under the action of an applied magnetic field. Instead of the usual heat transfer equation we use a generalization given by the Maxwell–Cattaneo law which is a system satisfied by the temperature and the heat flux. We prove a global existence of weak solutions to the system having a finite energy.

Abstract: We consider a non-standard eigenvalue problem arising in stability studies of 3-layer immiscible porous media and Hele-Shaw flows which contain the viscous profile of the middle layer as a coefficient in the eigenvalue problem. We characterize the eigenvalues and eigenfunctions of this eigenvalue problem. We then apply this characterization to an exponential viscous profile and numerically compute the associated eigenvalues and eigenfunctions. We provide an explicit sequence of numbers that give upper and lower bounds on the eigenvalues. We also discuss the limiting cases when either the length of the middle layer approaches zero or the exponential viscous profile approaches a constant viscosity profile.

Abstract: We find an explicit representation of the evolution of \({ t \mapsto \gamma_t = \{z (\zeta, t), \zeta \in \mathbb{C}, |\zeta| = 1 \} }\) of the contour \({ \gamma_t = \partial \omega_t }\) of fluid spots \({\omega_t = \{z (\zeta, t), |\zeta| 0}\) or \({t 0}\)) or a source (\({t < 0}\)) localized at point \({z(0, t)}\) described by trinomials $$z(\zeta, t) = a_1(t)\zeta + a_N(t) \zeta^N + a_M(t) \zeta^M,{\rm where} \quad M = 2N - 1,\quad{\rm and\,integer} \quad N \ge 2,$$ for the classical formulation of the problem when \({\omega_t }\) is within \({ \gamma_t }\) (inner Hele–Shaw problem), or by $$z(\zeta, t) = a_{-1}(t)\zeta^{-1} + a_N(t) \zeta^N + a_M(t)\zeta^M, {\rm where} \quad M = 2N + 1,\quad{\rm and\,integer} \quad N \ge 1,$$ for the outer Hele–Shaw problem when \({ \omega_t }\) is outside of \({\gamma_t}\). We obtained a sufficient condition for univalence of real trinomials, improving a result found by Ruscheweyh and Wirths (Ann Pol Math. 28:341–355, 1973). A sufficient condition is also found for functions used in the outer problem.

Abstract: In this paper using diffuse approximations the existence of a varifold solution to the two-phase Newtonian incompressible viscous flow problem is derived. On the free surface between the two phases we consider surface tension force. Also we prove that for axisymmetric, possibly with swirl, initial velocities and cylindrically symmetric initial volumes occupied by each fluid there exists a global in time axisymmetric, with swirl, solution.

Abstract: This paper is concerned with a system of heat equations with hysteresis and Navier–Stokes equations. In Tsuzuki (J Math Anal Appl 423:877–897, 2015) an existence result is obtained for the problem in a 2-dimensional domain with the Navier–Stokes equation in a weak sense. However the result does not include uniqueness for the problem due to the low regularity for solutions. This paper establishes existence and uniqueness in 2- and 3-dimensional domains with the Navier–Stokes equation in a stronger sense. Moreover this work decides required height of regularity for the initial data by introducing the fractional power of the Stokes operator.

Abstract: In this paper we prove the existence and uniqueness of path-wise strong solution to stochastic viscous flow in unbounded channels with multiple outlets using local monotonicity arguments. We devise a construction for solvability using a stochastic basic vector field.

Abstract: Bounding curves in the energy, enstrophy-plane are derived for the 3D Navier–Stokes equations under an assumption on coherence of the vorticity direction. The analysis in the critical case where the direction is Holder continuous with exponent r = 1/2 results in a curve with extraordinarily large maximal enstrophy (exponential in Grashof), in marked contrast to the subcritical case, r > 1/2 (algebraic in Grashof).

Abstract: We study the smoothing effects of diffusion in a 3D vector system of linear advection-diffusion PDEs known as the kinematic dynamo equations. We examine the case where both the divergence free drift velocity \({\mathbf{u}(x,t)}\) and the mild solution \({\mathbf{B}(x,t)}\) to the PDE obey the natural scaling of the equation. We consider initial data \({\mathbf{B}_0(x) \in L^3(\mathbb{R}^3)}\) and demonstrate how the regularization effects of diffusion are influenced by the profile of \({\mathbf{u}(x,t)}\).

Abstract: The two-dimensional Hele-Shaw problem for a fluid spot with free boundary can be solved using the Polubarinova–Galin equation. The main condition of its applicability is the smoothness of the spot boundary. In the sink-case, this problem is not well-posed and the boundary loses smoothness within finite time—the only exception being the disk centred on the sink. An extensive literature deals with the study of the Hele-Shaw problem with non-smooth boundary or with surface tension, but the problem remains open. In our work, we propose to study this flow from a control point of view, by introducing an analogue of multipoles (term taken from the theory of electromagnetic fields). This allows us to control the shape of the spot and to avoid non-smoothness phenomenon on its border. For any polynomial contours, we demonstrate how all the fluid can be extracted, while the border remains smooth until the very end. We find, in particular, sufficient conditions for controllability and a link between Richardson’s moments and Polubarinova–Galin equation.

Abstract: 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.

Abstract: We consider the equations of motion for an incompressible non-Newtonian fluid in a bounded Lipschitz domain $${G {\subset} \mathbb{R}^{d}}$$ during the time interval (0, T) together with a stochastic perturbation driven by a Brownian motion W. The balance of momentum reads as $${\rm {d}\mathbf{v}} = {\rm div}\mathbf{S} {\rm d}t-( abla \mathbf{v})\mathbf{v} {\rm d}t + abla\pi {\rm d}t + \mathbf{f}{\rm d}t + \Phi(\mathbf{v}) {\rm d}\mathbf{W}_{t},$$ where v is the velocity, $${\pi}$$ the pressure and f an external volume force. We assume the common power law model $${\mathbf{S}(\varepsilon(\mathbf{v}))=(1+|\varepsilon(\mathbf{v})|)^{p-2}\varepsilon(\mathbf{v})}$$ and show the existence of martingale weak solution provided $${p > \frac{2d+2}{d+2}}$$ . Our approach is based on the $${L^{\infty}}$$ -truncation and a harmonic pressure decomposition which are adapted to the stochastic setting.

Abstract: The incompressible Navier–Stokes equations are considered in the two-dimensional strip \({\mathbb{R} \times [0,L]}\), 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 \({t \to \infty}\). 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.

Abstract: By taking into account the $${\beta}$$ -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.

Abstract: A stability result for the compressible Navier–Stokes system with transport equation for entropy s is shown The proof comes as an outcome of the isentropic case and additional properties of the effective viscous flux We deal with the pressure term in the form \({\rho^{\gamma}e^{s}}\) with adiabatic index \({\gamma>3\slash 2}\); therefore the crucial renormalization method is restricted