Annals of Pde最新文献

In this paper, by considering 2d water waves with a pair of point vortices, we prove the existence of water waves with sign-changing Taylor sign coefficients. That is, the strong Taylor sign condition holds initially, while it breaks down at a later time. Such a phenomenon can be regarded as the transition between the stable and unstable regime in the sense of Rayleigh-Taylor of water waves. As a byproduct, we prove the wellposedness of 2d water waves in Gevrey-2 spaces.

We consider the focusing energy-critical nonlinear wave equation for radially symmetric initial data in space dimensions (D ge 4). This equation has a unique (up to sign and scale) nontrivial, finite energy stationary solution W, called the ground state. We prove that every finite energy solution with bounded energy norm resolves, continuously in time, into a finite superposition of asymptotically decoupled copies of the ground state and free radiation.

We construct non-unique Leray solutions of the forced Navier-Stokes equations in bounded domains via gluing methods. This demonstrates a certain locality and robustness of the non-uniqueness discovered by the authors in [1].

By establishing an invariant set (1.11) for the Prandtl equation in Crocco transformation, we prove the orbital and asymptotic stability of Blasius-like steady states against Oleinik’s monotone solutions.

We are concerned with strong axisymmetric solutions to the 3D incompressible Navier-Stokes equations. We show that if the weak (L^3) norm of a strong solution u on the time interval [0, T] is bounded by (A gg 1) then for each (kge 0 ) there exists (C_k>1) such that (Vert D^k u (t) Vert _{L^infty (mathbb {R}^3)} le t^{-(1+k)/2}exp exp A^{C_k}) for all (tin (0,T]).

We consider the wave equation on a manifold ((Omega ,g)) of dimension (dge 2) with smooth strictly convex boundary (partial Omega ne emptyset ), with Dirichlet boundary conditions. We construct a sharp local in time parametrix and then proceed to obtain dispersion estimates: our fixed time decay rate for the Green function exhibits a (t^{1/4}) loss with respect to the boundary less case. We precisely describe where and when these losses occur and relate them to swallowtail type singularities in the wave front set, proving that our decay is optimal. Moreover, we derive better than expected Strichartz estimates, balancing lossy long time estimates at a given incidence with short time ones with no loss: for (d=3), it heuristically means that, on average the decay loss is only (t^{1/6}).

In this paper, we consider the 2D incompressible Navier-Stokes equations on the torus. It is well known that for any (L^2) divergence-free initial data, there exists a global smooth solution that is unique in the class of (C_t L^2) weak solutions. We show that such uniqueness would fail in the class (C_t L^p) if ( p<2). The non-unique solutions we constructed are almost (L^2)-critical in the sense that (i) they are uniformly continuous in (L^p) for every (p<2); (ii) the kinetic energy agrees with any given smooth positive profile except on a set of arbitrarily small measure in time.

Given a conformally transversally anisotropic manifold (M, g), we consider the semilinear elliptic equation

We show that an a priori unknown smooth function q can be uniquely determined from the knowledge of the Dirichlet-to-Neumann map associated to the equation. This extends the previously known results of the works Feizmohammadi and Oksanen (J Differ Equ 269(6):4683–4719, 2020), Lassas et al. (J Math Pures Appl 145:44–82, 2021). Our proof is based on over-differentiating the equation: We linearize the equation to orders higher than the order two of the nonlinearity (qu^2), and introduce non-vanishing boundary traces for the linearizations. We study interactions of two or more products of the so-called Gaussian quasimode solutions to the linearized equation. We develop an asymptotic calculus to solve Laplace equations, which have these interactions as source terms.

This is a follow-up of [5] on the general covariant modulated (GCM) procedure in perturbations of Kerr. In this paper, we construct GCM hypersurfaces, which play a central role in extending GCM admissible spacetimes in [7] where decay estimates are derived in the context of nonlinear stability of Kerr family for (|a|ll m). As in [4], the central idea of the construction of GCM hypersurfaces is to concatenate a 1–parameter family of GCM spheres of [5] by solving an ODE system. The goal of this paper is to get rid of the symmetry restrictions in the GCM procedure introduced in [4] and thus remove an essential obstruction in extending the results to a full stability proof of the Kerr family.

This is the second and last paper of a series aimed at solving the local Cauchy problem for polarized ({mathbb {U}}(1)) symmetric solutions to the Einstein vacuum equations featuring the nonlinear interaction of three small amplitude impulsive gravitational waves. Such solutions are characterized by their three singular “wave-fronts” across which the curvature tensor is allowed to admit a delta singularity. Under polarized ({mathbb {U}}(1)) symmetry, the Einstein vacuum equations reduce to the Einstein–scalar field system in ((2+1)) dimensions. In this paper, we focus on the wave estimates for the scalar field in the reduced system. The scalar field terms are the most singular ones in the problem, with the scalar field only being Lipschitz initially. We use geometric commutators to prove energy estimates which reflect that the singularities are localized, and that the scalar field obeys additional fractional-derivative regularity, as well as regularity along appropriately defined “good directions”. The main challenge is to carry out all these estimates using only the low-regularity properties of the metric. Finally, we prove an anisotropic Sobolev embedding lemma, which when combined with our energy estimates shows that the scalar field is everywhere Lipschitz, and that it obeys additional (C^{1,theta }) estimates away from the most singular region.