Journal of Mathematical Fluid Mechanics最新文献

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.

The precipitating quasi-geostrophic equations go beyond the (dry) quasi-geostrophic equations by incorporating the effects of moisture. This means that both precipitation and phase changes between a water-vapour phase (outside a cloud) and a water-vapour-plus-liquid phase (inside a cloud) are taken into account. In the dry case, provided that a Laplace equation is inverted, the quasi-geostrophic equations may be formulated as a nonlocal transport equation for a single scalar variable (the potential vorticity). In the case of the precipitating quasi-geostrophic equations, inverting the Laplacian is replaced by a more challenging adversary known as potential-vorticity-and-moisture inversion. The PDE to invert is nonlinear and piecewise elliptic with jumps in its coefficients across the cloud edge. However, its global ellipticity is a priori unclear due to the dependence of the phase boundary on the unknown itself. This is a free boundary problem where the location of the cloud edge is one of the unknowns. Here we present the first rigorous analysis of this PDE, obtaining existence, uniqueness, and regularity results. In particular the regularity results are nearly sharp. This analysis rests on the discovery of a variational formulation of the inversion. This novel formulation is used to answer a key question for applications: which quantities jump across the interface and which quantities remain continuous? Most notably we show that the gradient of the unknown pressure, or equivalently the streamfunction, is Hölder continuous across the cloud edge.

We analyze a diffuse interface model for multi-phase flows of N incompressible, viscous Newtonian fluids with different densities. In the case of a bounded and sufficiently smooth domain existence of weak solutions in two and three space dimensions and a singular free energy density is shown. Moreover, in two space dimensions global existence for sufficiently regular initial data is proven. In three space dimension, existence of strong solutions locally in time is shown as well as regularization for large times in the absence of exterior forces. Moreover, in both two and three dimensions, convergence to stationary solutions as time tends to infinity is proved.

We construct by convex integration examples of energy dissipating solutions to the 2D Euler equations on ({mathbb {R}}^2) with vorticity in the Hardy space (H^p({mathbb {R}}^2)), for any (2/3<p<1).

In this paper, the Marker and Cell scheme based on a two-grid algorithm is proposed for the two-dimensional incompressible Darcy–Brinkman–Forchheimer equations in porous media. The motivation of the two-grid Marker and Cell algorithm is figuring out a nonlinear equation on a coarse grid with mesh size H and a linear equation on a fine grid with mesh size h. A small positive parameter (varepsilon ) is introduced. By using it, the non-differentiable nonlinear term can be transformed into the term which is twice continuously differentiable. The error estimates of the velocity and pressure in the (L^2) norms are obtained, which show (O(varepsilon +H^4+h^2)). Second-order accuracy for some terms of velocity in the (H^1) norms is also obtained. Several numerical experiments are provided to confirm the availability of this efficient second-order algorithm. Behavior of the fluid flow with different Brinkman number is considered.

This work is focused on an accelerated global-in-time solution strategy for the Oseen equations, which highly exploits the augmented Lagrangian methodology to improve the convergence behavior of the Schur complement iteration. The main idea of the solution strategy is to block the individual linear systems of equations at each time step into a single all-at-once saddle point problem. By elimination of all velocity unknowns, the resulting implicitly defined equation can then be solved using a global-in-time pressure Schur complement (PSC) iteration. To accelerate the convergence behavior of this iterative scheme, the augmented Lagrangian approach is exploited by modifying the momentum equation for all time steps in a strongly consistent manner. While the introduced discrete grad-div stabilization does not modify the solution of the discretized Oseen equations, the quality of customized PSC preconditioners drastically improves and, hence, guarantees a rapid convergence. This strategy comes at the cost that the involved auxiliary problem for the velocity field becomes ill conditioned so that standard iterative solution strategies are no longer efficient. Therefore, a highly specialized multigrid solver based on modified intergrid transfer operators and an additive block preconditioner is extended to solution of the all-at-once problem. The potential of the proposed overall solution strategy is discussed in several numerical studies as they occur in commonly used linearization techniques for the incompressible Navier–Stokes equations.

We address a system of equations modeling a compressible fluid interacting with an elastic body in dimension three. We prove the local existence and uniqueness of a strong solution when the initial velocity belongs to the space (H^{2+epsilon }) and the initial structure velocity is in (H^{1.5+epsilon }), where (epsilon in (0,1/2)).

The geodesics in the group of volume-preserving diffeomorphisms (volumorphisms) of a manifold M, for a Riemannian metric defined by the kinetic energy, can be used to model the movement of ideal fluids in that manifold. The existence of conjugate points along such geodesics reveal that these cease to be infinitesimally length-minimizing between their endpoints. In this work, we focus on the case of the torus (M={mathbb {T}}^2) and on geodesics corresponding to steady solutions of the Euler equation generated by stream functions (psi =-cos (mx)cos (ny)) for integers m and n, called Kolmogorov flows. We show the existence of conjugate points along these geodesics for all pairs of strictly positive integers (m, n), thereby completing the characterization of all pairs (m, n) such that the associated Kolmogorov flow generates a geodesic with conjugate points.