Annals of Pde最新文献

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].

In this paper we prove integrated energy and pointwise decay estimates for solutions of the vacuum linearized Einstein equation on the domain of outer communication of the Kerr black hole spacetime. The estimates are valid for the full subextreme range of Kerr black holes, provided integrated energy estimates for the Teukolsky equation hold. For slowly rotating Kerr backgrounds, such estimates are known to hold, due to the work of one of the authors. The results in this paper thus provide the first stability results for linearized gravity on the Kerr background, in the slowly rotating case, and reduce the linearized stability problem for the full subextreme range to proving integrated energy estimates for the Teukolsky equation. This constitutes an essential step towards a proof of the black hole stability conjecture, i.e. the statement that the Kerr family is dynamically stable, one of the central open problems in general relativity.

We show a stability result for the Schwarzschild singularity (inside the black hole region) for the Einstein vacuum equations (EVE). The result is proven in the class of polarized axial symmetry, under perturbations of the Schwarzschild data induced on a hypersurface ({r=epsilon }), (epsilon<<2M). Our result is only partly a stability result, in that we show that while a (space-like) singularity persists under perturbations as above, the behavior of the metric approaching the singularity is much more involved than for the Schwarzschild solution. Indeed, we find that the solution displays asymptocially-velocity-term-dominated dynamics and approaches a different Kasner solution at each point of the singularity. These Kasner-type asymptotics are very far from isotropic, since (as in Schwarzschild) there are two contracting directions and one expanding one. Our proof relies on energy methods and on a new approach to the EVE in axial symmetry, which we believe has wider applicability: In this symmetry class and under a suitable geodesic gauge, the EVE can be studied as a free wave coupled to (nonlinear) ODEs, which couple the geometry of the projected, 2+1 space-time to the free wave. The fact that the nonlinear part of the Einstein equations is described by ODEs lies at the heart of how one can overcome a certain linear instability exhibited by the singularity.

The question of whether the hyper-dissipative (HD) Navier-Stokes (NS) system can exhibit spontaneous formation of singularities in the super-critical regime–the hyperviscous effects being represented by a fractional power of the Laplacian, say (beta ), confined to interval (bigl (1, frac{5}{4}bigr ))–has been a major open problem in the mathematical fluid dynamics since the foundational work of J.L. Lions in 1960s. In this work, an evidence of criticality of the Laplacian is presented, more precisely, a class of plausible blow-up scenarios is ruled out as soon as (beta ) is greater than one. While the framework is based on the ‘scale of sparseness’ of the super-level sets of the positive and negative parts of the components of the higher-order derivatives of the velocity previously introduced by the authors, a major novelty in the current work is classification of the HD flows near a potential spatiotemporal singularity in two main categories, ‘homogeneous’ (the case consistent with a near-steady behavior) and ‘non-homogenous’ (the case consistent with the formation and decay of turbulence). The main theorem states that in the non-homogeneous case any (beta ) greater than one prevents a singularity. In order to illustrate the impact of this result in a methodology-free setting, a two-parameter family of dynamically rescaled blow-up profiles is considered, and it is shown that as soon as (beta ) is greater than one, a new region in the parameter space is ruled out. More importantly, the region is a neighborhood (in the parameter space) of the self-similar profile, i.e., the approximately self-similar blow-up, a prime suspect in possible singularity formation, is ruled out for all HD NS models.

We prove the nonlinear inviscid damping for a class of monotone shear flows with non-constant background density for the two-dimensional ideal inhomogeneous fluids in (mathbb {T}times [0,1]) when the initial perturbation is in Gevrey-(frac{1}{s}) ((frac{1}{2}<s<1)) class with compact support.

We provide a geometric optics description in spaces of low regularity, (L^2) and (H^1), of the transport of oscillations in solutions to linear and some semilinear second-order hyperbolic boundary problems along rays that graze the boundary of a convex obstacle to arbitrarily high finite or infinite order. The fundamental motivating example is the case where the spacetime manifold is (M=(mathbb {R}^nsetminus mathcal {O})times mathbb {R}_t), where (mathcal {O}subset mathbb {R}^n) is an open convex obstacle with (C^infty ) boundary, and the governing hyperbolic operator is the wave operator (Box :=Delta -partial _t^2). The main theorem says that high frequency exact solutions are well approximated in spaces of low regularity by approximate solutions constructed from fairly explicit solutions to relatively simple profile equations. The theorem has two main assumptions. The first is that the grazing set, that is, the set of points on the spacetime boundary at which incoming characteristics meet the boundary tangentially, is a codimension two, (C^1) submanifold of spacetime. The second is that the reflected flow map, which sends points on the spacetime boundary forward in time to points on reflected and grazing rays, is injective and has appropriate regularity properties near the grazing set. Both assumptions are in general hard to verify, but we show that they are satisfied for the diffraction of incoming plane waves by a large class of strictly convex obstacles in all dimensions, involving grazing points of arbitrarily high finite or infinite order.

We prove fine higher regularity results of Calderón-Zygmund-type for equations involving nonlocal operators modelled on the fractional p-Laplacian with possibly discontinuous coefficients of VMO-type. We accomplish this by establishing precise pointwise bounds in terms of certain fractional sharp maximal functions. This approach is new already in the linear setting and enables us to deduce sharp regularity results also in borderline cases.

We study the properties of spacelike singularities in spherically symmetric spacetimes obeying the Einstein equations, in the presence of matter. Building upon previous work of An–Zhang [4], we consider matter described by a scalar field, both in the presence of an electromagnetic field and without. We prove that, if a spacelike singularity obeying several reasonable assumptions is formed, then the Hawking mass, the Kretschmann scalar, and the matter fields have inverse polynomial blow-up rates near the singularity that may be described precisely. Furthermore, one may view the resulting spacetime in the context of the BKL heuristics regarding spacelike singularities in relativistic cosmology. In particular, near any point p on the singular boundary in our spherically symmetric spacetime, we obtain a leading order BKL-type expansion, including a description of Kasner exponents associated to p. This confirms heuristics of Buonanno–Damour–Veneziano [14]. As a result, we provide a rigorous description of a detailed, quantitative correspondence between Kasner-like singularities most often associated to the cosmological setting, and the singularities observed in (spherically symmetric) gravitational collapse. Moreover, we outline a program concerning the study of the stability and instability of spacelike singularities in the latter picture, both outside of spherical symmetry and within (where the electromagnetic field acts as a proxy for angular momentum).