Journal of Mathematical Fluid Mechanics最新文献

We consider the Navier–Stokes Cauchy problem with an initial datum in a weighted Lebesgue space. The weight is a radial function increasing at infinity. Our study partially follows the ideas of the paper by Galdi and Maremonti (J Math Fluid Mech 25:7, 2023). The authors of the quoted paper consider a special study of stability of steady fluid motions. The results hold in 3D and for small data. Here, relatively to the perturbations of the rest state, we generalize the result. We study the nD Navier–Stokes Cauchy problem, (nge 3). We prove the existence (local) of a unique regular solution. Moreover, the solution enjoys a spatial asymptotic decay whose order of decay is connected to the weight.

In this paper, the sharp interface limit for compressible Navier–Stokes/Allen-Cahn system with relaxation is investigated, which is motivated by the Jin-Xin relaxation scheme ([Comm.Pure Appl.Math.,48,1995]). Given any entropy solution which consists of two different families of shocks interacting at some positive time for the immiscible two-phase compressible Euler equations, it is proved that such entropy solution is the singular limit for a family global strong solutions of the compressible Navier–Stokes/Allen-Cahn system with relaxation when the interface thickness of immiscible two-phase flow tends to zero. The weighted estimation and improved anti-derivative method are used in the proof. The results of this singular limit show that, the sharp interface limit of the compressible Navier–Stokes/Allen-Cahn system with relaxation is the immiscible two-phase compressible Euler equations with free interface between phases. Moreover, the interaction of shock waves belong to different families can pass through the two-phase flow interface and maintain the wave strength and wave speed without being affected by the interface for immiscible compressible two-phase flow.

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.