Journal of Mathematical Fluid Mechanics最新文献

The chemotaxis-Navier–Stokes system

modelling the behavior of aerobic bacteria in a fluid drop, is considered in a smoothly bounded domain (Omega subset mathbb R^2). For all (alpha > 0) and all sufficiently regular (Phi ), we construct global classical solutions and thereby extend recent results for the fluid-free analogue to the system coupled to a Navier–Stokes system. As a crucial new challenge, our analysis requires a priori estimates for u at a point in the proof when knowledge about n is essentially limited to the observation that the mass is conserved. To overcome this problem, we also prove new uniform-in-time (L^p) estimates for solutions to the inhomogeneous Navier–Stokes equations merely depending on the space-time (L^2) norm of the force term raised to an arbitrary small power.

In this paper, a kind of classic generalized Riemann problem for 2-dimensional isothermal Euler equations for compressible gas dynamics is considered. The problem is the gas ((u_{0}, v_{0}, r_{0} mid x mid ^{beta })) in the rectangular region expands into the vacuum. We construct the solution of the following form

where (rho ) and (u, v) denote the density and the velocity fields respectively, and (u_{0}, v_{0}, r_{0}>0) and (beta in (-1,0) cup (0,+infty )) are constants. The continuity of the self-similar solution depends on the value of (beta ). Under certain conditions, we get a weak solution with shock wave, which is necessarily generated initially and move apart along a plane. Furthermore, by the method of characteristic analysis, we explain the mechanism of the shock wave.

We are considering a steady-state turbulent Reynolds-Averaged Navier–Stokes (RANS) one-equation model, that couples the equation for the velocity-pressure mean field with the equation for the turbulent kinetic energy. Eddy viscosities vanish at the boundary, characterized by terms like (d(x, Gamma )^eta ) and (d(x, Gamma )^beta ), where (0< eta , beta < 1). We determine critical values (eta _c) and (beta _c) for which the system has a weak solution. This solution is obtained as the limit of viscous regularizations for (0< eta < eta _c) and (0< beta < beta _c).

Several fundamental problems on the 2D magnetohydrodynamic (MHD) equations with only magnetic diffusion (no velocity dissipation) remain open, especialy in the case when the spatial domain is the whole space ({mathbb {R}}^2). This paper establishes that, near a background magnetic field, any fractional dissipation in one direction in the velocity equation would allow us to establish the global existence and stability for perturbations near the background. The magnetic diffusion here is not required to be given by the standard Laplacian operator but any general fractional Laplacian with positive power.

By means of a unifying measure-theoretic approach, we establish lower bounds on the Hausdorff dimension of the space-time set which can support anomalous dissipation for weak solutions of fluid equations, both in the presence or absence of a physical boundary. Boundary dissipation, which can occur at both the time and the spatial boundary, is analyzed by suitably modifying the Duchon & Robert interior distributional approach. One implication of our results is that any bounded Euler solution (compressible or incompressible) arising as a zero viscosity limit of Navier–Stokes solutions cannot have anomalous dissipation supported on a set of dimension smaller than that of the space. This result is sharp, as demonstrated by entropy-producing shock solutions of compressible Euler (Drivas and Eyink in Commun Math Phys 359(2):733–763, 2018. https://doi.org/10.1007/s00220-017-3078-4; Majda in Am Math Soc 43(281):93, 1983. https://doi.org/10.1090/memo/0281) and by recent constructions of dissipative incompressible Euler solutions (Brue and De Lellis in Commun Math Phys 400(3):1507–1533, 2023. https://doi.org/10.1007/s00220-022-04626-0 624; Brue et al. in Commun Pure App Anal, 2023), as well as passive scalars (Colombo et al. in Ann PDE 9(2):21–48, 2023. https://doi.org/10.1007/s40818-023-00162-9; Drivas et al. in Arch Ration Mech Anal 243(3):1151–1180, 2022. https://doi.org/10.1007/s00205-021-01736-2). For (L^q_tL^r_x) suitable Leray–Hopf solutions of the (d-)dimensional Navier–Stokes equation we prove a bound of the dissipation in terms of the Parabolic Hausdorff measure (mathcal {P}^{s}), which gives (s=d-2) as soon as the solution lies in the Prodi–Serrin class. In the three-dimensional case, this matches with the Caffarelli–Kohn–Nirenberg partial regularity.

In this article, some elementary observations are is made regarding the behavior of solutions to the two-dimensional curl-free Burgers equation which suggests the distinguished role played by the scalar divergence field in determining the dynamics of the solution. These observations inspire a new divergence-based regularity condition for the two-dimensional Kuramoto–Sivashinsky equation (KSE) that provides conceptual clarity to the nature of the potential blow-up mechanism for this system. The relation of this regularity criterion to the Ladyzhenskaya–Prodi–Serrin-type criterion for the KSE is also established, thus providing the basis for the development of an alternative framework of regularity criterion for this equation based solely on the low-mode behavior of its solutions. The article concludes by applying these ideas to identify a conceptually simple modification of KSE that yields globally regular solutions, as well as providing a straightforward verification of this regularity criterion to establish global regularity of solutions to the 2D Burgers–Sivashinsky equation. The proofs are direct, elementary, and concise.

The present paper studies an asymptotic behavior of a spherically symmetric solution on the exterior domain of an unit ball for the compressible Navier–Stokes equation, describing a motion of viscous barotropic gas. Especially we study outflow problem, that is, the fluid blows out through boundary. Precisely we show an asymptotic stability of a spherically symmetric stationary solutions provided that an initial disturbance of the stationary solution is sufficiently small in the Sobolev space.

The problem of global-in-time regularity for the 3D Navier-Stokes equations, i.e., the question of whether a smooth flow can exhibit spontaneous formation of singularities, is a fundamental open problem in mathematical physics. Due to the super-criticality of the equations, the problem has been super-critical in the sense that there has been a ‘scaling gap’ between any regularity criterion and the corresponding a priori bound (regardless of the functional setup utilized). The purpose of this work is to present a mathematical framework-based on a suitably defined ‘scale of sparseness’ of the super-level sets of the positive and negative parts of the components of the higher-order spatial derivatives of the velocity field—in which the scaling gap between the regularity class and the corresponding a priori bound vanishes as the order of the derivative goes to infinity.

In this paper we study the compressible magnetohydrodynamics equations in three dimensions, which offer a good model for plasmas. Formation of singularity for (C^1)-solution in finite time is proved with axisymmetric initial data. The key observation is that the magnetic force term admits a good structure with axisymmetric assumption.