Annals of Pde最新文献

In this paper, we seek to construct nontrivial global solutions to some quasilinear wave equations in three space dimensions. We first present a conditional result on the construction of nontrivial global solutions to a general system of quasilinear wave equations. Assuming that a global solution to the geometric reduced system exists and satisfies several well-chosen pointwise estimates, we find a matching exact global solution to the original wave equations. Such a conditional result is then applied to two types of equations which are of great interest. One is John’s counterexamples (Box u=u_t^2) or (Box u=u_t u_{tt}), and the other is the 3D compressible Euler equations with no vorticity. We explicitly construct global solutions to the corresponding geometric reduced systems and show that these global solutions satisfy the required pointwise bounds. As a result, there exists a large family of nontrivial global solutions to these two types of equations.

In this paper, we prove the first existence result of weak solutions to the 3D Euler equation with initial vorticity concentrated in a circle and velocity field in (C([0,T],L^{2^-})). The energy becomes finite and decreasing for positive times, with vorticity concentrated in a ring that thickens and moves in the direction of the symmetry axis. With our approach, there is no need to mollify the initial data or to rescale the time variable. We overcome the singularity of the initial data by applying convex integration within the appropriate time-weighted space.

We consider the Cauchy problem for the full free boundary Euler equations in 3d with an initial small velocity of size (O(varepsilon_0)), in a moving domain which is initially an (O(varepsilon_0)) perturbation of a flat interface. We assume that the initial vorticity is of size (O(varepsilon_1)) and prove a regularity result up to times of the order (varepsilon_1^{-1+}), independent of ({varepsilon _0}). A key part of our proof is a normal form type argument for the vorticity equation; this needs to be performed in the full three dimensional domain and is necessary to effectively remove the irrotational components from the quadratic stretching terms and uniformly control the vorticity. Another difficulty is to obtain sharp decay for the irrotational component of the velocity and the interface; to do this we perform a dispersive analysis on the boundary equations, which are forced by a singular contribution from the rotational component of the velocity. As a corollary of our result, when ({varepsilon _1}) goes to zero we recover the celebrated global regularity results of Wu (Invent. Math. 2012) and Germain, Masmoudi and Shatah (Ann. of Math. 2013) in the irrotational case.

Given a regular solution (mathbf{g}_0) of the Einstein-null dusts system without restriction on the number of dusts, we construct families of solutions ((mathbf{g}_lambda)_{lambdain(0,1]}) of the Einstein vacuum equations such that (mathbf{g}_lambda-mathbf{g}_0) and (partial(mathbf{g}_lambda-mathbf{g}_0)) converges respectively strongly and weakly to 0 when (lambdato0). Our construction, based on a multiphase geometric optics ansatz, thus extends the validity of the reverse Burnett conjecture without symmetry to a large class of massless kinetic spacetimes. In order to deal with the finite but arbitrary number of direction of oscillations we work in a generalised wave gauge and control precisely the self-interaction of each wave but also the interaction of waves propagating in different null directions, relying crucially on the non-linear structure of the Einstein vacuum equations. We also provide the construction of oscillating initial data solving the vacuum constraint equations and which are consistent with the spacetime ansatz.

In this paper, we prove a pseudolocality-type theorem for (mathcal L)-complete noncompact Ricci flow which may not have bounded sectional curvature; with the help of it we study the uniqueness of the Ricci flow on noncompact manifolds. In particular, we prove the strong uniqueness theorem for the (mathcal L)-complete Ricci flow on the Euclidean space. This partially answers a question proposed by B-L. Chen (J Differ Geom 82(2):363–382, 2009).

We prove finite-time vorticity blowup in the compressible Euler equations in (mathbb{R}^d) for any (d geq 3), starting from smooth, localized, and non-vacuous initial data. This is achieved by lifting the vorticity blowup result from (Chen arXiv preprint arXiv: 2407.06455, 2024) in (mathbb{R}^2) to (mathbb{R}^d) and utilizing the axisymmetry in (mathbb{R}^d). At the time of the first singularity, both vorticity blowup and implosion occur on a sphere (S^{d-2}). Additionally, the solution exhibits a non-radial implosion, accompanied by a stable swirl velocity that is sufficiently strong to initially dominate the non-radial components and to generate the vorticity blowup.

We introduce a novel mechanism that reveals finite time singularities within the 1D De Gregorio model and the 3D incompressible Euler equations. Remarkably, we do not construct our blow up using self-similar coordinates, but build it from infinitely many regions with vorticity, separated by vortex-free regions in between. It yields solutions of the 3D incompressible Euler equations in (mathbb{R}^3times [-T,0]) such that the velocity is in the space (C^{infty}(mathbb{R}^3 setminus {0})cap C^{1,alpha}cap L^2) where (0 < alpha ll 1) for times (tin (-T,0)) and is not (C^1) at time 0.

The Sadovskii vortex patch is a traveling wave for the two-dimensional incompressible Euler equations consisting of an odd symmetric pair of vortex patches touching the symmetry axis. Its existence was first suggested by numerical computations of Sadovskii in [J. Appl. Math. Mech., 1971], and has gained significant interest due to its relevance in inviscid limit of planar flows via Prandtl–Batchelor theory and as the asymptotic state for vortex ring dynamics. In this work, we prove the existence of a Sadovskii vortex patch, by solving the energy maximization problem under the exact impulse condition and an upper bound on the circulation.

In this paper, we study the interior (C^2) regularity problem for the Hessian quotient equation (left(frac{sigma_n}{sigma_k}right)(D^2u)=f). We give a complete answer to this longstanding problem: for (k=n-1,n-2), we establish an interior (C^2) estimate; for (kleq n-3), we show that interior (C^2) estimate fails by finding a singular solution.

We provide a complete local well-posedness theory in H^s based Sobolev spaces for the free boundary incompressible Euler equations with zero surface tension on a connected fluid domain. Our well-posedness theory includes: (i) Local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and the first proof of continuous dependence on the data, all in low regularity Sobolev spaces; (ii) Enhanced uniqueness: Our uniqueness result holds at the level of the Lipschitz norm of the velocity and the (C^{1,frac{1}{2}}) regularity of the free surface; (iii) Stability bounds: We construct a nonlinear functional which measures, in a suitable sense, the distance between two solutions (even when defined on different domains) and we show that this distance is propagated by the flow; (iv) Energy estimates: We prove refined, essentially scale invariant energy estimates for solutions, relying on a newly constructed family of elliptic estimates; (v) Continuation criterion: We give the first proof of a sharp continuation criterion in the physically relevant pointwise norms, at the level of scaling. In essence, we show that solutions can be continued as long as the velocity is in (L_T^1W^{1,infty}) and the free surface is in (L_T^1C^{1,frac{1}{2}}), which is at the same level as the Beale-Kato-Majda criterion for the boundaryless case; (vi) A novel proof of the construction of regular solutions. Our entire approach is in the Eulerian framework and can be adapted to work in more general fluid domains.