Journal of Mathematical Fluid Mechanics最新文献

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.

DiPerna and Lions (Invent Math 98(3):511–547, 1989) established the existence and uniqueness results for weak solutions to linear transport equations with Sobolev velocity fields. Motivated by fluid mechanics, this paper provides mathematical analysis on two simple finite difference methods applied to linear transport equations on a bounded domain with divergence-free (unbounded) Sobolev velocity fields. The first method is based on a Lax-Friedrichs type explicit scheme with a generalized hyperbolic scale, where truncation of an unbounded velocity field and its measure estimate are implemented to ensure the monotonicity of the scheme; the method is (L^p)-strongly convergent in the class of DiPerna–Lions weak solutions. The second method is based on an implicit scheme with (L^2)-estimates, where the discrete Helmholtz–Hodge decomposition for discretized velocity fields plays an important role to ensure the divergence-free constraint in the discrete problem; the method is scale-free and (L^2)-strongly convergent in the class of DiPerna–Lions weak solutions. The key point for both of the methods is to obtain fine (L^2)-bounds of approximate solutions that tend to the norm of the exact solution given by DiPerna–Lions. Finally, the explicit scheme is applied to the case with smooth velocity fields from the viewpoint of the level-set method for sharp interfaces involving transport equations, where rigorous discrete approximation of level-sets and their geometric quantities is discussed.

We study an inverse problem of determining the shape of a bending wall with a given surface pressure distribution in the two-dimensional steady supersonic potential flow. The given pressure distribution on the wall surface is assumed to be a small perturbation of the pressure distribution corresponding to a bending convex wall and to have a bounded total variation. In this setting, we first give the background solution which only contains strong rarefaction waves generated by a bending convex wall. Then, we construct the approximate boundaries and corresponding approximate solutions of the inverse problem within a perturbation domain of this background solution. To achieve this, we employ a modified wave-front tracking algorithm. Finally, we show that the limit of approximate solutions provides a global entropy solution for the inverse problem, and the limit of approximate boundaries gives a boundary profile representing the shape of a bending wall that yields the given pressure distribution.

In Bénard-Rayleigh convection we consider the pattern defect in orthogonal domain walls connecting a set of convective rolls with another set of rolls orthogonal to the first set. This is understood as an heteroclinic orbit of a reversible system where the x - coordinate plays the role of time. This appears as a perturbation of the heteroclinic orbit proved to exist in a reduced 6-dimensional system studied by a variational method in Buffoni et al. (J Diff Equ, 2023, https://doi.org/10.1016/j.jde.2023.01.026), and analytically in Iooss (Heteroclinic for a 6-dimensional reversible system occuring in orthogonal domain walls in convection. Preprint, 2023). We then prove for a given amplitude (varepsilon ^2), and an imposed symmetry in coordinate y, the existence of a one-parameter family of heteroclinic connections between orthogonal sets of rolls, with wave numbers (different in general) which are linked with an adapted shift of rolls parallel to the wall.