Journal of Mathematical Fluid Mechanics最新文献

In this paper, we investigate a mathematical model of sea-breeze flow described by a second-order differential equation, which explains the morning glory phenomenon. Firstly, we establish the existence and uniqueness of solutions by applying the Fredholm alternative theorem. Then, we consider approximate solutions by using the Taylor expansion theorem. We also apply a Fourier analysis for computing the solution and present some numerical methods. Finally, by making appropriate assumptions for the forcing term, we transform the original equation into a Sturm–Liouville problem and analyze the corresponding eigenvalue problem.

We consider the Nernst-Planck-Stokes system on a bounded domain of ({{mathbb {R}}}^d), (d=2,3) with general nonequilibrium Dirichlet boundary conditions for the ionic concentrations. It is well known that, in a wide range of cases, equilibrium steady state solutions of the system, characterized by zero fluid flow, are asymptotically stable. In these regimes, the existence of a natural dissipative structure is critical in obtaining stability. This structure, in general, breaks down under nonequilibrium conditions, in which case, in the steady state, the fluid flow may be nontrivial. In this short paper, we show that, nonetheless, certain classes of very weak nonequilibrium steady states, with nonzero fluid flow, remain globally asymptotically stable.

Very recently, Barker (Proc. Amer. Math. Soc. Ser. B 11: 436-451, 2024), Barker and Wang (J Differ Equ 365:379–407, 2023), and, Beirao da Veiga and Yang (J. Geom. Anal. 35: no. 1, Paper No. 15, 14 pp, 2025) applied the higher integrability of weak solutions to study the singular set and energy equality of weak solutions to the incompressible Navier–Stokes equations with supercritical assumptions, respectively. In the spirit of this and, Leslie and Shvydkoy’s work (SIAM J Math Anal 50:870–890, 2018), we present some energy equality criteria for suitable weak solutions up to the first potential blow-up time in terms of pressure, its gradient or the direction of the velocity by (L^{p}) bound of the incompressible Navier–Stokes equations. Furthermore, along the same lines, we establish some (L^{p}) estimate of the isentropic compressible Navier–Stokes equations.

This paper focuses on the analysis of stratified steady periodic water waves that contain stagnation points. The initial step involves transforming the free-boundary problem into a quasilinear pseudodifferential equation through a conformal mapping technique, resulting in a periodic function of a single variable. By utilizing the theorems developed by Crandall and Rabinowitz, we establish the existence and formal stability of small-amplitude steady periodic capillary-gravity water waves in the presence of stratified linear flows. Notably, the stability of bifurcation solution curves is strongly influenced by the stratified nature of the system. Additionally, as the Bernoulli’s function (beta ) approaches critical values, we observe that the linearized problem exhibits a two-dimensional kernel. To address this new phenomenon, we perform the Lyapunov-Schmidt reduction, which enables us to establish the existence of two-mode water waves. Such wave is, generically, a combination of two different Fourier modes. As far as we know, the two-mode water waves in stratified flow are first constructed by us. Finally, we demonstrate the presence of internal stagnation points within these waves.

We study the nonhomogeneous boundary value problem for the steady-state Navier–Stokes equations under the slip boundary conditions in two-dimensional multiply-connected bounded domains. Employing the approach of Korobkov-Pileckas-Russo (Ann. Math. 181(2), 769-807, 2015), we prove that this problem has a solution if the friction coefficient is sufficiently large compared with the kinematic viscosity constant and the curvature of the boundary. No additional assumption (other than the necessary requirement of zero total flux through the boundary) is imposed on the boundary data. We also show that such an assumption on the friction coefficient is redundant for the existence of a solution in the case when the fluxes across each connected component of the boundary are sufficiently small, or the domain and the given data satisfy certain symmetry conditions. The crucial ingredient of our proof is the fact that the total head pressure corresponding to the solution to the steady Euler equations takes a constant value on each connected component of the boundary.

This paper is concerned with an initial-boundary value problem of full compressible magnetohydrodynamics (MHD) equations on 3D bounded domains subject to non-slip boundary condition for velocity, perfectly conducting boundary condition for magnetic field, and homogeneous Dirichlet boundary condition for temperature. The global well-posedness of strong solutions with initial vacuum is established and the exponential decay estimates of the solutions are obtained, provided the initial total energy is suitably small. More interestingly, it is shown that for (pin (3,6)), the (L^p)-norm of the gradient of density remains uniformly bounded for all (tge 0). This is in sharp contrast to that in (Chen et al. in Global well-posedness of full compressible magnetohydrodynamic system in 3D bounded domains with large oscillations and vacuum. arXiv:2208.04480, Li et al. in Global existence of classical solutions to full compressible Navier–Stokes equations with large oscillations and vacuum in 3D bounded domains. arXiv:2207.00441), where the exponential growth of the gradient of density in (L^p)-norm was explored.

In this paper, we consider the Onsager’s conjecture for the compressible Euler equations and elastodynamics in a torus or a bounded domain. Some energy conservation criteria in Onsager’s critical spaces ({underline{B}}^{alpha }_{p,VMO}) and Besov spaces (B^{alpha }_{p,infty }) for weak solutions in these systems are established, which extend the known corresponding results. A novel ingredient is the utilization of a test function in one single step rather than two steps in the case of incompressible models to capture the affect of the boundary.

We study the unique solvability of weak transmission problems in some unbounded domains containing at least one flat layer area, which is associated with the motion of two-phase fluids. In particular, we construct the solution to the transmission problem for the Laplace operator with non-homogeneous boundary conditions. As a direct consequence, the Helmholtz–Weyl decomposition for the two-phase problem is also proved.

We consider the movement of self-gravitating gas balls consisting of viscous barotropic fluids in the neighborhood of an equilibrium state. If this state fulfills a certain stability condition, we show that the solutions exist for all time. We allow perturbations that change the angular momentum.

We analyze vertically stratified three-dimensional oceanic flows under the assumption of constant vorticity. More precisely, these flows are governed by the f-plane approximation for the divergence-free incompressible Euler equations at arbitrary off-equatorial latitudes. A discontinuous stratification gives rise to a freely moving impermeable interface, which separates the two fluid layers of different constant densities; the fluid domain is bounded by a flat ocean bed and a free surface. It turns out that the constant vorticity assumption enforces almost trivial bounded solutions: the vertical fluid velocity vanishes everywhere; the horizontal velocity components are simple harmonic oscillators with Coriolis frequency f and independent of the spatial variables; the pressure is hydrostatic apart from sinusoidal oscillations in time; both the surface and interface are flat. To enable larger classes of solutions, we discuss a forcing method, which yields a characterization of steady stratified purely zonal currents with nonzero constant vorticity. Finally, we discuss the related viscous problem, which has no nontrivial bounded solutions.