Journal of Mathematical Fluid Mechanics最新文献

We examine a micro-scale model of superfluidity derived by Pitaevskii (Sov. Phys. JETP 8:282-287, 1959) which describes the interacting dynamics between superfluid He-4 and its normal fluid phase. This system consists of the nonlinear Schrödinger equation and the incompressible, inhomogeneous Navier-Stokes equations, coupled to each other via a bidirectional nonlinear relaxation mechanism. The coupling permits mass/momentum/energy transfer between the phases, and accounts for the conversion of superfluid into normal fluid. We prove the existence of global weak solutions in ({mathbb {T}}^3) for a power-type nonlinearity, beginning from small initial data. The main challenge is to control the inter-phase mass transfer in order to ensure the strict positivity of the normal fluid density, while obtaining time-independent a priori estimates.

We present and study a parallel grad-div stabilized finite element discretization algorithm based on entire-overlapping domain decomposition for the numerical simulation of Navier–Stokes equations. The algorithm is easy to implement on top of existing sequential software, in which each subproblem used to calculate a local solution in its designated subregion is actually a global problem with vast of degrees of freedom coming from its own subregion, and hence, can be solved independently with other subproblems. We derive error bounds of the approximate solution by employing the technical tool of local a priori estimate, and investigate the effect of grad-div stabilization term on the approximation solutions. Numerical comparisons, with both inf-sup stable and unstable mixed finite elements pairs for the velocity and pressure, show that our present algorithm has an amazing superiority to its counterpart without stabilization in the sense that accuracy of the approximate velocities could be improved by two orders of magnitude when the viscosity (nu ) is small. While compared with the usual standard serial grad-div stabilized finite element method, our algorithm saves lots of CPU time in computing a solution with comparable accuracy.

We consider the initial-boundary-value problem of the isentropic compressible Navier–Stokes–Poisson equations subject to large and non-flat doping profile in 3D bounded domain with slip boundary condition and vacuum. The global well-posedness of classical solution is established with small initial energy but possibly large oscillations and vacuum. The steady state (except velocity) and the doping profile are allowed to be of large variation.

We present the global existence of weak solutions to the Cucker–Smale–Stokes system in an infinitely long cylindrical domain with the specular boundary condition. The proposed system consists of the kinetic Cucker–Smale model and the Stokes system for flocking particles and an incompressible fluid, respectively, in an infinitely long cylindrical domain. It models the collective dynamics resulting from the fluid-particle-structure interactions. For this model, we provide the global existence of a weak solution and numerical simulations that exhibit collective behaviors of flocking particles.

The restricted ((N+1))-vortex problem is investigated in the plane with the first N identical vortices forming a relative equilibrium configuration of a regular N-polygon and the vorticity of the last vortex being zero. We characterize the global dynamics using the method of qualitative theory. It can be shown that the equilibrium points of the system are located at the vertices of three different regular N-polygons and the origin. The equilibrium points on one regular polygon are stable, whereas those on the other two regular polygons are unstable. The origin and singularities are also stable and surrounded by dense periodic orbits. For (N=3) or 4, there exist homoclinic and heteroclinic orbits; while for (Nge 5), the system’s orbits consist of equilibrium points, heteroclinic orbits, and periodic orbits. We numerically study the trajectories of the passive tracer (a particle with zero vorticity) under specific circumstances, which support our theoretical results.

We present recent results in study of a mathematical model of the sea-breeze flow, arising from a general model of the ’morning glory’ phenomena. Based on analysis of the Dirichlet spectrum of the corresponding Sturm–Liouville problem and application of the Fredholm alternative, we establish conditions of existence/uniqueness of solutions to the given problem.

We consider the initial value problem associated to the Poisson–Nernst–Planck–Navier–Stokes system which is first derived by Wang et al. (J Differ Equ 262:68–115, 2017) through an Energetic Variational Approach (EVA). Exploiting harmonic analysis tools (especially Littlewood–Paley theory), we first study the local and global well-posedness of the system in critical Besov spaces. Then, under a suitable condition involving only low-frequency of initial data, we use the Lyapunov-type inequality of the energy functionals to establish optimal time decay rates for the constructed global solutions. The proof crucially depends on a careful analysis for treating the extra effect of the distribution for the negative (positive) charge and non-standard product estimates, interpolation inequalities.

It is shown that both the Camassa–Holm and Novikov equations are ill-posed in (B_{p,r}^{1+1/p}(mathbb {R})) with ((p,r)in [1,infty ]times (1,infty ]) in Guo et al. (J Differ Equ 266:1698–1707, 2019) and well-posed in (B_{p,1}^{1+1/p}(mathbb {R})) with (pin [1,infty )) in Ye et al. (J Differ Equ 367: 729–748, 2023). Recently, the ill-posedness for the Camassa–Holm equation in (B^{1}_{infty ,1}(mathbb {R})) has been proved in Guo et al. (J Differ Equ 327: 127–144, 2022). In this paper, we shall solve the only left an endpoint case (r=1) for the Novikov equation. More precisely, we prove the ill-posedness for the Novikov equation in (B^{1}_{infty ,1}(mathbb {R})) by exhibiting the norm inflation phenomena.

In this paper, we prove the global existence and stability of the magnetic Bénard system with partial dissipation on perturbations near a background magnetic field in ({mathbb {T}}^d (d=2,3)). If there is no velocity dissipation, the stability result provides a significant example for the stabilizing effects of the magnetic field on electrically conducting fluids. In addition, we obtain an explicit large-time decay rate of the solutions.

We consider the stationary Navier–Stokes equations in a three-dimensional curved thin domain around a given closed surface under the slip boundary conditions. Our aim is to show that a solution to the bulk equations is approximated by a solution to limit equations on the surface appearing in the thin-film limit of the bulk equations. To this end, we take the average of the bulk solution in the thin direction and estimate the difference of the averaged bulk solution and the surface solution. Then we combine an obtained difference estimate on the surface with an estimate for the difference of the bulk solution and its average to get a difference estimate for the bulk and surface solutions in the thin domain, which shows that the bulk solution is approximated by the surface one when the thickness of the thin domain is sufficiently small.