Journal of Mathematical Fluid Mechanics最新文献

We show that a convex combination of the Osgood and Nagumo conditions ensures the uniqueness of the solution to the boundary value problem for a second-order nonlinear differential equation on a semi-infinite interval. A typical example of such problem is a recently derived nonlinear model for the motion of arctic gyres.

We study by a combination of analytical and numerical methods multidimensional stability and transverse bifurcation of planar hydraulic shock and roll wave solutions of the inviscid Saint Venant equations for inclined shallow-water flow, both in the whole space and in a channel of finite width, obtaining complete stability diagrams across the full parameter range of existence. Technical advances include development of efficient multi-d Evans solvers, low- and high-frequency asymptotics, explicit/semi-explicit computation of stability boundaries, and rigorous treatment of channel flow with wall-type physical boundary. Notable behavioral phenomena are a novel essential transverse bifurcation of hydraulic shocks to invading planar periodic roll-wave or doubly-transverse periodic herringbone patterns, with associated metastable behavior driven by mixed roll- and herringbone-type waves initiating from localized perturbation of an unstable constant state; and Floquet-type transverse “flapping” bifurcation of roll wave patterns.

This paper concerns the initial-boundary-value problem of the compressible Navier-Stokes-Poisson equations subject to large and non-flat doping profile in 3D bounded domain, where the velocity admits slip boundary condition. The global existence of strong solutions and smooth solutions near a steady state for compressible NSP are established by using the energy estimates. In particular, an important feature is that the steady state (except velocity) and the doping profile are allowed to be large.

Several recent papers have addressed the modelling of tissue growth by multi-phase models where the velocity is related to the pressure by one of the physical laws (Stokes’, Brinkman’s or Darcy’s). While each of these models has been extensively studied, not so much is known about the connection between them. In the recent paper (David et al. in SIAM J. Math. Anal. 56(2):2090–2114, 2024), assuming the linear form of the pressure, the Authors connected two multi-phase models by an inviscid limit: the viscoelastic one (of Brinkman’s type) and the inviscid one (of Darcy’s type). Here, we prove that the same is true for a nonlinear, power-law pressure. The new ingredient is that we use the relation between the pressure p and the Brinkman potential W to deduce compactness in space of p from the compactness in space of W.

We study the stationary motion of an incompressible Navier–Stokes fluid past obstacles in (mathbb {R}^{3}), subject to the provided boundary velocity (u_{b}), external force (f = textrm{div} F), and nonzero constant vector (k {e_1}) at infinity. We first prove that the existence of at least one very weak solution u in (L^{3}(Omega ) + L^{4}(Omega )) for an arbitrary large (F in L^{3/2}(Omega ) + L^{2}(Omega )) provided that the flux of (u_{b}) on the boundary of each body is sufficiently small with respect to the viscosity (nu ). Moreover, we establish weak- and strong-regularity results for very weak solutions. Consequently, our existence and regularity results enable us to prove the existence of a weak solution satisfying (nabla u in L^{r}(Omega )) for a given (F in L^{r}(Omega )) with (3/2 le r le 2), and a strong solution satisfying (nabla ^{2} u in L^{s}(Omega )) for a given (f in L^{s}(Omega )) with (1 < s le 6/5), respectively.

Using a more geometric approach, we demonstrate that the solutions to the Navier–Stokes equations remain regular except on a set with a null Hausdorff measure of dimension 1. The proof primarily relies on a new compactness lemma and the monotonicity property of harmonic functions. The combination of linear and nonlinear approximation schemes makes the proof clear and transparent.

In this paper, we study the homogenization of 3D non-homogeneous incompressible Navier–Stokes system in perforated domains with holes of critical size. Under very mild assumptions concerning the shape of the obstacles and their mutual distance, we show that when (varepsilon rightarrow 0), the velocity and density converge to a solution of the non-homogeneous incompressible Navier–Stokes system with a friction term of Brinkman type.

In 1934, Leray (Acta Math 63:193–248, 1934) proved the existence of global-in-time weak solutions to the Navier–Stokes equations for any divergence-free initial data in (L^2(mathbb {R}^3)). In the 1980s, Giga (J Differ Equ 62(2):186–212, 1986) and Kato (Math Z 187(4):471–480, 1984) independently showed that there exist global-in-time mild solutions corresponding to small enough critical (L^3(mathbb {R}^3)) initial data. In 1990, Calderón (Trans Am Math Soc 318:179–200, 1990) filled the gap to show that there exist global-in-time weak solutions for all supercritical initial data in (L^p(mathbb {R}^3)) for (2< p<3) by utilising a splitting argument, blending the constructions of Leray and Giga-Kato. In this paper, we utilise a “Calderón-like” splitting to show the global-in-time existence of weak solutions to the Navier–Stokes equations corresponding to supercritical Besov space initial data (u_0 in dot{B}^{s}_{{q},{infty }}(mathbb {R}^3)) where (q>2) and (-1+frac{2}{q}<s<min left( -1+frac{3}{q},0 right) ), which fills a similar gap between Leray and known mild solution theory in the Besov space setting. We also use the Calderón-like splitting to investigate the structure of the singular set under a Type-I blow-up assumption in the Besov space setting, which is considerably rougher than in previous works.

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.