Journal of Mathematical Fluid Mechanics最新文献

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.

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.

In this paper, we consider the steadiness of symmetric solutions to two dispersive models in shallow water and hyperelastic mechanics, respectively. These models are derived previously in the two-dimensional setting and can be viewed as the generalization of the Camassa–Holm and Kadomtsev–Petviashvili equations. For these two models, we prove that the symmetry of classical solutions implies steadiness in the horizontal direction. We also confirm the connection between symmetry and steadiness for solutions in weak formulation, which covers in particular the peaked solutions.

We give a complete proof of the classical Benjamin and Lighthill conjecture for arbitrary two-dimensional steady water waves with vorticity. We show that the flow force constant of an arbitrary smooth solution is bounded by the flow force constants for the corresponding conjugate laminar flows. We prove these inequalities without any assumptions on the geometry of the surface profile and put no restrictions on the wave amplitude. Furthermore, we give a complete description of all cases when the equalities can occur. In particular, that excludes the existence of one-sided bores and multi-hump solitary waves. Our conclusions are new already for Stokes waves with a constant vorticity, while the case of equalities is new even in the classical setting of irrotational waves.

This paper considers the Navier–Stokes equations in the half plane with Euler-type initial conditions, i.e., initial conditions with a non-zero tangential component at the boundary. Under analyticity assumptions for the data, we prove that the solution exists for a short time independent of the viscosity. We construct the Navier–Stokes solution through a composite asymptotic expansion involving solutions of the Euler and Prandtl equations plus an error term. The norm of the error goes to zero with the square root of the viscosity. The Prandtl solution contains a singular term, which influences the regularity of the error. The error term is the sum of a first-order Euler correction, a first-order Prandtl correction, and a further error term. The use of an analytic setting is mainly due to the Prandtl equation.