Annals of Pde最新文献

In this paper, we establish the uniqueness and nonlinear stability of concentrated symmetric traveling vortex patch-pairs for the 2D Euler equation. We also prove the uniqueness of concentrated rotating polygons as well. The proofs are achieved by a combination of the local Pohozaev identity, a detailed description of asymptotic behaviors of the solutions and some symmetry properties obtained by the method of moving planes.

In this paper, we give the first rigorous justification of the Benjamin-Ono equation:

as an internal water wave model on the physical time scale. Here, ({varepsilon }) is a small parameter measuring the weak nonlinearity of the waves, (mu ) is the shallowness parameter, and (gamma in (0,1)) is the ratio between the densities of the two fluids. To be precise, we first prove the existence of a solution to the internal water wave equations for a two-layer fluid with surface tension, where one layer is of shallow depth and the other is of infinite depth. The existence time is of order ({mathcal {O}}(frac{1}{{varepsilon }})) for a small amount of surface tension such that ({varepsilon }^2 le textrm{bo}^{-1} ) where (textrm{bo}) is the Bond number. Then, we show that these solutions are close, on the same time scale, to the solutions of the BO equation with a precision of order ({mathcal {O}}(mu + textrm{bo}^{-1})). In addition, we provide the justification of new equations with improved dispersive properties, the Benjamin equation, and the Intermediate Long Wave (ILW) equation in the deep-water limit.

The long-time well-posedness of the two-layer fluid problem was first studied by Lannes [Arch. Ration. Mech. Anal., 208(2):481-567, 2013] in the case where both fluids have finite depth. Here, we adapt this work to the case where one of the fluid domains is of finite depth, and the other one is of infinite depth. The novelties of the proof are related to the geometry of the problem, where the difference in domains alters the functional setting for the Dirichlet-Neumann operators involved. In particular, we study the various compositions of these operators that require a refined symbolic analysis of the Dirichlet-Neumann operator on infinite depth and derive new pseudo-differential estimates that might be of independent interest.

The 2-D Peskin problem describes a 1-D closed elastic string immersed and moving in a 2-D Stokes flow that is induced by its own elastic force. The geometric shape of the string and its internal stretching configuration evolve in a coupled way, and they combined govern the dynamics of the system. In this paper, we show that certain geometric quantities of the moving string satisfy extremum principles and decay estimates. As a result, we can prove that the 2-D Peskin problem admits a unique global solution when the initial data satisfies a medium-size geometric condition on the string shape, while no assumption on the size of stretching is needed.

In this work, we construct distance like functions with integral Hessian bound on manifolds with small curvature concentration and use it to construct Ricci flows on manifolds with possibly unbounded curvature. As an application, we study the geometric structure of those manifolds without bounded curvature assumption. In particular, we show that manifolds with Ricci lower bound, non-negative scalar curvature, bounded entropy, Ahlfors n-regular and small curvature concentration are topologically Euclidean.

We prove an almost global existence result for space periodic solutions of the 1D gravity-capillary water waves equations with constant vorticity. The result holds for any value of gravity, vorticity and depth, a full measure set of surface tensions, and any small and smooth enough initial datum. The proof demands a novel approach—that we call paradifferential Hamiltonian Birkhoff normal form for quasi-linear PDEs—in presence of resonant wave interactions: the normal form is not integrable but it preserves the Sobolev norms thanks to its Hamiltonian nature. A major difficulty is that paradifferential calculus used to prove local well posedness (as the celebrated Alinhac good unknown) breaks the Hamiltonian structure. A major achievement of this paper is to correct (possibly) unbounded paradifferential transformations to symplectic maps, up to an arbitrary degree of homogeneity. Thanks to a deep cancellation, our symplectic correctors are smoothing perturbations of the identity. Thus we are able to preserve both the paradifferential structure and the Hamiltonian nature of the equations. Such Darboux procedure is written in an abstract functional setting applicable also in other contexts.

In this work we construct global unique solutions of the dissipative Surface quasi-geostrophic equation ((alpha )-SQG) that lose regularity instantly when there is super-critical fractional diffusion.

In this paper we study the regularity property of Hele-Shaw flow, where source and drift are present in the evolution. More specifically we consider Hölder continuous source and Lipschitz continuous drift. We show that if the free boundary of the solution is locally close to a Lipschitz graph, then it is indeed Lipschitz, given that the Lipschitz constant is small. When there is no drift, our result establishes (C^{1,gamma }) regularity of the free boundary by combining our result with the obstacle problem theory. In general, when the source and drift are both smooth, we prove that the solution is non-degenerate, indicating higher regularity of the free boundary.

In this work, we develop a wavelet-inspired, (L^3)-based convex integration framework for constructing weak solutions to the three-dimensional incompressible Euler equations. The main innovations include a new multi-scale building block, which we call an intermittent Mikado bundle; a wavelet-inspired inductive set-up which includes assumptions on spatial and temporal support, in addition to (L^p) and pointwise estimates for Eulerian and Lagrangian derivatives; and sharp decoupling lemmas, inverse divergence estimates, and space-frequency localization technology which is well-adapted to functions satisfying (L^p) estimates for p other than 1, 2, or (infty ). We develop these tools in the context of the Euler-Reynolds system, enabling us to give both a new proof of the intermittent Onsager theorem from Novack and Vicol (Invent Math 233(1):223–323, 2023) in this paper, and a proof of the (L^3)-based strong Onsager conjecture in the companion paper Giri et al. (The (L^3)-based strong Onsager theorem, arxiv).

In this paper, we consider the stability of the generalized Lagrangian mean curvature flow of graph case in the cotangent bundle, which is first defined by Smoczyk-Tsui-Wang (Smoczyk et al. J für die reine und angewandte Mathematik 750: 97–121, 2019). By new estimates of derivatives along the flow, we weaken the initial condition and remove the positive curvature condition in Smoczyk et al. (J für die reine und angewandte Mathematik 750: 97–121, 2019). More precisely, we prove that if the graph induced by a closed 1-form is a special Lagrangian submanifold in the cotangent bundle of a Riemannian manifold, then the generalized Lagrangian mean curvature flow is stable near it.

In this paper, we investigate the dynamics of solutions of the Muskat equation with initial interface consisting of multiple corners allowing for linear growth at infinity. Specifically, we prove that if the initial data contains a finite set of small corners then we can find a precise description of the solution showing how these corners desingularize and move at the same time. At the analytical level, we are solving a small data critical problem which requires renormalization. This is accomplished using a nonlinear change of variables which serves as a logarithmic correction and accurately describes the motion of the corners during the evolution.