Journal of Mathematical Fluid Mechanics最新文献

Consider a generalized Oseen evolution operator in 3D exterior domains, that is generated by a non-autonomous linearized system arising from time-dependent rigid motions. This was found by Hansel and Rhandi, and then the theory was developed by the second author, however, desired regularity properties such as estimate of the temporal derivative as well as the Hölder estimate have remained open. The present paper provides us with those properties together with weighted estimates of the evolution operator. The results are then applied to the Navier–Stokes initial value problem, so that a new theorem on existence of a unique strong (L^q)-solution locally in time is proved.

The Liouville theorem for smooth solutions with finite Dirichlet integrals and uniform vanishing conditions to high-dimension stationary Navier–Stokes equations was established as reported by Galdi (An introduction to the mathematical theory of the Navier–Stokes equations: Steady-state problems, Springer, New York, 2011). In this paper, we mainly concern with the Liouville type problem of weak solutions only with finite Dirichlet integral to the stationary Navier–Stokes equations on (mathbb {R}^d) with (dge 5). We first establish a Liouville type theorem under some restrictions on the high-frequency part tending to infinity of velocity fields. Then, we show the uniqueness of weak solutions to the stationary fractional Navier–Stokes equations with finite critical Dirichlet integral by establishing another Liouville type theorem.

The divergence constraint of the incompressible fluids usually causes the weak robustness of standard mixed finite element methods. Grad-div stabilization is a popular technique for improving the robustness. In this paper, we theoretically show that for magnetohydrodynamic flows at large Hartmann numbers, grad-div stabilization can improve the well-posedness and robust stability of the continuous problem, and remove the effect of Hartmann number on the finite element discrete errors. Besides, applying the backward Euler method and lagging the nonlinear term, we construct a linear grad-div stabilized finite element algorithm for magnetohydrodynamics flows at low magnetic Reynolds numbers. A complete theoretical analysis of its stability and convergency is provided. Some computational experiments illustrate the validness of our algorithm and its theoretical results and also the benefits of grad-div stabilization.

Consider the Cauchy problem of the Navier–Stokes equations in (mathbb {R}^n (n ge 2)) with the initial data (a in dot{B}^{-1+n/p}_{p, infty }) for (n< p < infty ). We establish the Gevrey type estimates for the error between the successive approximations ({u_j}_{j=0}^{infty }) and the strong solution u provided the convergence in the scaling invariant norm in (L^q(mathbb {R}^n)) with the time weight holds. It is also clarified that the convergence rate of the higher order approximation is at least the same as that of the lower order approximation. In addition, the approximation for the pressure is also established.

This paper is concerned with a compressible MHD equations describing the evolution of viscous non-resistive fluids in piecewise regular bounded Lipschitz domains. Under the general inflow-outflow boundary conditions, we prove existence of global-in-time weak solutions with finite energy initial data. The present result extends considerably the previous work by Li and Sun (J Differ Equ 267:3827–3851, 2019), where the homogeneous Dirichlet boundary condition for velocity field is treated. The proof leans on the specific mathematical structure of equations and the recently developed theory of open fluid systems. Furthermore, we establish the weak-strong uniqueness principle, namely a weak solution coincides with the strong solution on the lifespan of the latter provided they emanate from the same initial and boundary data. This basic property is expected to be useful in the study of convergence of numerical solutions.

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.