Annals of Pde最新文献

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.

We consider the Cauchy problem for an inviscid irrotational fluid on a domain with a free boundary governed by a fourth order linear elasticity equation. We first derive the Craig-Sulem-Zakharov formulation of the problem and then establish the existence of a global weak solution in two space dimensions for the fluid, in the general case without a damping term, for any initial data with finite energy.

We prove the local wellposedness of the Cauchy problems for the electron magnetohydrodynamics equations (E-MHD) without resistivity for possibly large perturbations of nonzero uniform magnetic fields. While the local wellposedness problem for (E-MHD) has been extensively studied in the presence of resistivity (which provides dissipative effects), this seems to be the first such result without resistivity. (E-MHD) is a fluid description of plasma in small scales where the motion of electrons relative to ions is significant. Mathematically, it is a quasilinear dispersive equation with nondegenerate but nonelliptic second-order principal term. Our result significantly improves upon the straightforward adaptation of the classical work of Kenig–Ponce–Rolvung–Vega on the quasilinear ultrahyperbolic Schrödinger equations, as the regularity and decay assumptions on the initial data are greatly weakened to the level analogous to the recent work of Marzuola–Metcalfe–Tataru in the case of elliptic principal term.

A key ingredient of our proof is a simple observation about the relationship between the size of a symbol and the operator norm of its quantization as a pseudodifferential operator when restricted to high frequencies. This allows us to localize the (non-classical) pseudodifferential renormalization operator considered by Kenig–Ponce–Rolvung–Vega, and produce instead a classical pseudodifferential renormalization operator. We furthermore incorporate the function space framework of Marzuola–Metcalfe–Tataru to the present case of nonelliptic principal term.

In this paper, we prove the existence of self-similar algebraic spiral solutions of the 2-D incompressible Euler equations for the initial vorticity of the form (|y|^{-frac1mu} mathring{omega}(theta)) with (mu > frac12) and (mathring{omega}in L^1({mathbb{T}})), satisfying m-fold symmetry ((mge 2)) and a dominant condition. As an important application, we prove the existence of weak solution when (mathring{omega}) is a Radon measure on ({mathbb{T}}) with m-fold symmetry, which is related to the vortex sheet solution.

We construct finite time blowup solutions to the parabolic-elliptic Keller-Segel system

and derive the final blowup profile

To our knowledge this provides a new blowup solution for the Keller-Segel system, rigorously answering a question by Brenner et al. in [Brenner, Nonlinearity 12, 1999].