Journal of Mathematical Fluid Mechanics最新文献

DiPerna and Lions (Invent Math 98(3):511–547, 1989) established the existence and uniqueness results for weak solutions to linear transport equations with Sobolev velocity fields. Motivated by fluid mechanics, this paper provides mathematical analysis on two simple finite difference methods applied to linear transport equations on a bounded domain with divergence-free (unbounded) Sobolev velocity fields. The first method is based on a Lax-Friedrichs type explicit scheme with a generalized hyperbolic scale, where truncation of an unbounded velocity field and its measure estimate are implemented to ensure the monotonicity of the scheme; the method is (L^p)-strongly convergent in the class of DiPerna–Lions weak solutions. The second method is based on an implicit scheme with (L^2)-estimates, where the discrete Helmholtz–Hodge decomposition for discretized velocity fields plays an important role to ensure the divergence-free constraint in the discrete problem; the method is scale-free and (L^2)-strongly convergent in the class of DiPerna–Lions weak solutions. The key point for both of the methods is to obtain fine (L^2)-bounds of approximate solutions that tend to the norm of the exact solution given by DiPerna–Lions. Finally, the explicit scheme is applied to the case with smooth velocity fields from the viewpoint of the level-set method for sharp interfaces involving transport equations, where rigorous discrete approximation of level-sets and their geometric quantities is discussed.

We study an inverse problem of determining the shape of a bending wall with a given surface pressure distribution in the two-dimensional steady supersonic potential flow. The given pressure distribution on the wall surface is assumed to be a small perturbation of the pressure distribution corresponding to a bending convex wall and to have a bounded total variation. In this setting, we first give the background solution which only contains strong rarefaction waves generated by a bending convex wall. Then, we construct the approximate boundaries and corresponding approximate solutions of the inverse problem within a perturbation domain of this background solution. To achieve this, we employ a modified wave-front tracking algorithm. Finally, we show that the limit of approximate solutions provides a global entropy solution for the inverse problem, and the limit of approximate boundaries gives a boundary profile representing the shape of a bending wall that yields the given pressure distribution.

In Bénard-Rayleigh convection we consider the pattern defect in orthogonal domain walls connecting a set of convective rolls with another set of rolls orthogonal to the first set. This is understood as an heteroclinic orbit of a reversible system where the x - coordinate plays the role of time. This appears as a perturbation of the heteroclinic orbit proved to exist in a reduced 6-dimensional system studied by a variational method in Buffoni et al. (J Diff Equ, 2023, https://doi.org/10.1016/j.jde.2023.01.026), and analytically in Iooss (Heteroclinic for a 6-dimensional reversible system occuring in orthogonal domain walls in convection. Preprint, 2023). We then prove for a given amplitude (varepsilon ^2), and an imposed symmetry in coordinate y, the existence of a one-parameter family of heteroclinic connections between orthogonal sets of rolls, with wave numbers (different in general) which are linked with an adapted shift of rolls parallel to the wall.

In this article, the 3D Voigt-regularized Magnetohydrodynamic equations are considered, for which it is unknown if the uniqueness of weak solution exists. First, we prove that the uniform global attractor exists by constructing an evolutionary system. Then singular limits of this system are established. Namely, when a certain regularization parameter disappears, the convergence of global attractors is shown between the 3D autonomous Voigt-regularized Magnetohydrodynamic equations and Magnetohydrodynamic equations.

In this paper, we present an exact solution for the nonlinear governing equation coupled with relevant boundary conditions, which arise from the study of Saturn’s stratified circumpolar atmospheric flow. The solution is explicit in the Lagrangian framework by specifying its hypotrochoidal particle paths. An instability result of such nonlinear waves is also obtained by means of the short-wavelength instability approach.

In this work, we modify the weighted (L^p) bounds for elements of the Hilbert space (tilde{D}^{1,2}(Omega )). Using this bound, we derive the upper bound for the density, which is the key issue to global solution provided the shear viscosity is a positive constant and the bulk one is (lambda = rho ^{beta }) with (beta >4/3). Our results extend the earlier results due to Vaigant-Kazhikhov (Sib Math J 36:1283–1316, 1995) where they required that (beta >3), initial densities is strictly away from vacuum, and that the domain is bounded.

In this paper, we prove the norm inflation and get the ill-posedness for the modified Camassa-Holm equation in (B_{infty ,1}^0). Therefore we completed all well-posedness and ill-posedness problem for the modified Camassa-Holm equation in all critical spaces (B_{p,1}^frac{1}{p}) with (pin [1,infty ]).

Asymptotic expansion in far-field for the incompressible Navier–Stokes flow are established. It is well known that a velocity decays slowly in far-field. This property prevents classical procedure giving asymptotic expansions of solutions with high-order. In this paper, under natural settings and moment conditions on the initial vorticity, technique of renormalization together with Biot–Savart law derives an asymptotic expansion for velocity with high-order. Especially scalings and large-time behaviors of the expansions are clarified. By employing them, time evolution of velocity in far-field is drawn. As an appendix, asymptotic behavior of solutions as time variable tends to infinity is given. In this assertion, large-time behavior of velocity is discovered clearly.

We discuss in this short note the local-in-time strong well-posedness of the compressible Navier–Stokes system for non-Newtonian fluids on the three dimensional torus. We show that the result established recently by Kalousek, Mácha, and Nečasova in https://doi.org/10.1007/s00208-021-02301-8 can be extended to the case where vanishing density is allowed initially. Our proof builds on the framework developed by Cho, Choe, and Kim in https://doi.org/10.1016/j.matpur.2003.11.004 for compressible Navier–Stokes equations in the case of Newtonian fluids. To adapt their method, special attention is given to the elliptic regularity of a challenging nonlinear elliptic system. We show particular results in this direction, however, the main result of this paper is proven in the general case when elliptic (W^{2,p})-regularity is imposed as an assumption. Also, we give a finite time blow-up criterion.

The classical problem of removable singularities is considered for solutions to the stationary Navier–Stokes system in dimension (nge 3) and an old theorem of Shapiro (TAMS 187:335–363, 1974) is recovered and extended to solutions in a half ball vanishing on the flat boundary. Moreover, for (n=4) it is proved that there are not distributional solutions, smooth away from the singularity and such that (u(x)=O(|x|^{-1})).