We study the global existence of solutions to a two-component generalized Hunter–Saxton system in the periodic setting. We first prove a persistence result for the solutions. Then for some particular choices of the parameters (α, κ), we show the precise blow-up scenarios and the existence of global solutions to the generalized Hunter–Saxton system under proper assumptions on the initial data. This significantly improves recent results.

It remains unknown whether or not smooth solutions of the 3D incompressible MHD equations can develop finite-time singularities. One major difficulty is due to the fact that the dissipation given by the Laplacian operator is insufficient to control the nonlinearity and for this reason the 3D MHD equations are sometimes regarded as “supercritical”. This paper presents a global regularity result for the generalized MHD equations with a class of hyperdissipation. This result is inspired by a recent work of Terence Tao on a generalized Navier–Stokes equations (T. Tao, Global regularity for a logarithmically supercritical hyperdissipative Navier–Stokes equations, arXiv: 0906.3070v3 [math.AP] 20 June 2009), but the result for the MHD equations is not completely parallel to that for the Navier–Stokes equations. Besov space techniques are employed to establish the result for the MHD equations.

We introduce the notion of relative entropy for the weak solutions to the compressible Navier–Stokes system. In particular, we show that any finite energy weak solution satisfies a relative entropy inequality with respect to any couple of smooth functions satisfying relevant boundary conditions. As a corollary, we establish the weak-strong uniqueness property in the class of finite energy weak solutions, extending thus the classical result of Prodi and Serrin to the class of compressible fluid flows.

In this paper, logarithmically improved regularity criteria for the Navier–Stokes and the MHD equations are established in terms of both the vorticity field and the pressure.

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.

The aim of this paper is to present a short-wavelength stability analysis of an exact steady equatorial flow which does not vary in the azimuthal direction, but has an arbitrary variation with depth. We show that for some velocity profiles of the basic flow, this flow is locally stable to short-wavelength perturbations.

We consider the problem of solving numerically the stationary incompressible Navier–Stokes equations in an exterior domain in two dimensions. For numerical purposes we truncate the domain to a finite sub-domain, which leads to the problem of finding so called “artificial boundary conditions” to replace the boundary conditions at infinity. To solve this problem we construct – by combining results from dynamical systems theory with matched asymptotic expansion techniques based on the old ideas of Goldstein and Van Dyke – a smooth divergence free vector field depending explicitly on drag and lift and describing the solution to second and dominant third order, asymptotically at large distances from the body. The resulting expression appears to be new, even on a formal level. This improves the method introduced by the authors in a previous paper and generalizes it to non-symmetric flows. The numerical scheme determines the boundary conditions and the forces on the body in a self-consistent way as an integral part of the solution process. When compared with our previous paper where first order asymptotic expressions were used on the boundary, the inclusion of second and third order asymptotic terms further reduces the computational cost for determining lift and drag to a given precision by typically another order of magnitude.

The existence of global weak solutions is established to the magnetohydrodynamics (MHD) equations with Hall and ion-slip effects in a bounded domain, which coincide with the Hall-MHD equations with and without ion-slip effect for the complementary choices of the parameter $$\gamma =1$$ γ=1 and $$\gamma =0$$ γ=0 , respectively. It is also shown that a similar result holds in the whole space. Moreover, the local existence of a unique strong solution and the global well-posedness for small initial data are obtained to the Hall-MHD equations with and without ion-slip effect in both a cubic bounded domain with a flat boundary condition and the whole space. Furthermore, the vanishing viscosity limit to inviscid MHD equations is studied without the ion-slip effect ($$\gamma =0$$ γ=0 ) in the cubic bounded domain with the flat boundary condition and with the ion-slip effect ($$\gamma =1$$ γ=1 ) in the whole space, respectively.

We study 2d vortex sheets with unbounded support. First we show a version of the Biot–Savart law related to a class of objects including such vortex sheets. Next, we give a formula associating the kinetic energy of a very general class of flows with certain moments of their vorticities. It allows us to identify a class of vortex sheets of unbounded support being only $$\sigma $$ σ -finite measures (in particular including measures $$\omega $$ ω such that $$\omega (\mathbb {R}^2)=\infty $$ ω(R2)=∞ ), but with locally finite kinetic energy. One of such examples are celebrated Kaden approximations. We study them in details. In particular our estimates allow us to show that the kinetic energy of Kaden approximations in the neighbourhood of an origin is dissipated, actually we show that the energy is pushed out of any ball centered in the origin of the Kaden spiral. The latter result can be interpreted as an artificial viscosity in the center of a spiral.

In this paper, we study the stationary magnetohydrodynamics system in $$\mathbb {R}^2\times \mathbb {T}$$ R2×T . We prove trivialness of D-solutions (the velocity field u and the magnetic field h) when they are swirl-free. Meanwhile, this Liouville type theorem also holds provided u is swirl-free and h is axially symmetric, or both u and h are axially symmetric. Our method is also valid for certain related boundary value problems in the slab $$\mathbb {R}^2\times [-\pi ,\,\pi ]$$ R2×[-π,π] .

We study the asymptotic behavior of the forced linear Euler and nonlinear Navier–Stokes equations close to Couette flow on $$\mathbb {T}\times I$$ T×I . As our main result we show that for smooth time-periodic forcing linear inviscid damping persists, i.e. the velocity field (weakly) asymptotically converges. However, stability and scattering to the transport problem fail in $$H^{s}, s>-1$$ Hs,s>-1 . We further show that this behavior is consistent with the nonlinear Euler equations and that a similar result also holds for the nonlinear Navier–Stokes equations. Hence, these results provide an indication that nonlinear inviscid damping may still hold in Sobolev regularity in the above sense despite the Gevrey regularity instability results of Deng and Masmoudi (Long time instability of the Couette flow in low Gevrey spaces, 2018. arXiv:1803.01246).

The compressible Euler equations are the classical model in fluid dynamics. In this study, we investigate the life span of the projected 2-dimensional rotational C.sup.2C2 non-vacuum solutions of the Euler equations. By examining the corresponding projected 2-dimensional solutions, ([rho] (t,x.sub.1,x.sub.2),u.sub.1(t,x.sub.1,x.sub.2),u.sub.2(t,x.sub.1,x.sub.2),0), ([rho](t,x1,x2),u1(t,x1,x2),u2(t,x1,x2),0),in R.sup.3R3, we prove that there exist the corresponding blowup results for the rotational C.sup.2C2 solutions with a sufficiently large initial functional H(0)= .sub. R.sup.3 x u.sub.0dV. H(0)=a'R3x[right arrow]*u[right arrow]0dV.

We study the three-dimensional Navier–Stokes equations in the presence of the axisymmetric linear strain, where the strain rate depends on time in a specific manner. It is known that the system admits solutions which blow up in finite time and whose profiles are in a backward self-similar form of the familiar Burgers vortices. In this paper it is shown that the existing stability theory of the Burgers vortex leads to the stability of these blow-up solutions as well. The secondary blow-up is also observed when the strain rate is relatively weak.

We investigate the linear stability of inviscid columnar vortices with respect to finite energy perturbations. For a large class of vortex profiles, we show that the linearized evolution group has a sub-exponential growth in time, which means that the associated growth bound is equal to zero. This implies in particular that the spectrum of the linearized operator is entirely contained in the imaginary axis. This contribution complements the results of our previous work Gallay and Smets (Spectral stability of inviscid columnar vortices, 2018. arXiv:1805.05064), where spectral stability was established for the linearized operator in the enstrophy space.

We point out some criteria that imply regularity of axisymmetric solutions to Navier–Stokes equations. We show that boundedness of $$\Vert {v_{r}}/{\sqrt{r^3}}\Vert _{L_2({\mathbb {R}}^3\times (0,T))}$$ ‖vr/r3‖L2(R3×(0,T)) as well as boundedness of $$\Vert {\omega _{\varphi }}/{\sqrt{r}} \Vert _{L_2({\mathbb {R}}^3\times (0,T))}$$ ‖ωφ/r‖L2(R3×(0,T)) , where $$v_r$$ vr is the radial component of velocity and $$\omega _{\varphi }$$ ωφ is the angular component of vorticity, imply regularity of weak solutions.

This paper considers a family of non-diffusive active scalar equations where a viscosity type parameter enters the equations via the constitutive law that relates the drift velocity with the scalar field. The resulting operator is smooth when the viscosity is present but singular when the viscosity is zero. We obtain Gevrey-class local well-posedness results and convergence of solutions as the viscosity vanishes. We apply our results to two examples that are derived from physical systems: firstly a model for magnetostrophic turbulence in the Earth’s fluid core and secondly flow in a porous media with an “effective viscosity”.

The compressible Euler equations are the classical model in fluid dynamics. In this study, we investigate the life span of the projected 2-dimensional rotational $$C^{2}$$ C2 non-vacuum solutions of the Euler equations. By examining the corresponding projected 2-dimensional solutions, $$\begin{aligned} (\rho (t,x_{1},x_{2}),u_{1}(t,x_{1},x_{2}),u_{2}(t,x_{1},x_{2}),0), \end{aligned}$$ (ρ(t,x1,x2),u1(t,x1,x2),u2(t,x1,x2),0), in $$\mathbf {R}^{3}$$ R3 , we prove that there exist the corresponding blowup results for the rotational $$C^{2}$$ C2 solutions with a sufficiently large initial functional $$\begin{aligned} H(0)= {\displaystyle \int _{\mathbf {R}^{3}}} \vec {x}\cdot \vec {u}_{0}dV. \end{aligned}$$ H(0)=∫R3x→·u→0dV.

A class of Keller–Segel–Stokes systems generalizing the prototype $$\begin{aligned} \left\{ \begin{array}{l} n_t + u\cdot \nabla n = \Delta n - \nabla \cdot \left( n(n+1)^{-\alpha }\nabla c\right) , \\ c_t + u\cdot \nabla c = \Delta c-c+n, \\ u_t +\nabla P = \Delta u + n \nabla \phi + f(x,t), \quad \nabla \cdot u =0, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$ nt+u·∇n=Δn-∇·n(n+1)-α∇c,ct+u·∇c=Δc-c+n,ut+∇P=Δu+n∇ϕ+f(x,t),∇·u=0,(⋆) is considered in a bounded domain $$\Omega \subset \mathbb {R}^3$$ Ω⊂R3 , where $$\phi $$ ϕ and f are given sufficiently smooth functions such that f is bounded in $$\Omega \times (0,\infty )$$ Ω×(0,∞) . It is shown that under the condition that $$\begin{aligned} \alpha >\frac{1}{3}, \end{aligned}$$ α>13, for all sufficiently regular initial data a corresponding Neumann–Neumann–Dirichlet initial-boundary value problem possesses a global bounded classical solution. This extends previous findings asserting a similar conclusion only under the stronger assumption $$\alpha >\frac{1}{2}$$ α>12 . In view of known results on the existence of exploding solutions when $$\alpha <\frac{1}{3}$$ α<13 , this indicates that with regard to the occurrence of blow-up the criticality of the decay rate $$\frac{1}{3}$$ 13 , as previously found for the fluid-free counterpart of ($$\star $$ ⋆ ), remains essentially unaffected by fluid interaction of the type considered here.

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.