Annals of Pde最新文献

We study the linear stability problem to gravitational and electromagnetic perturbations of the extremal, ( |Q|=M, ) Reissner–Nordström spacetime, as a solution to the Einstein–Maxwell equations. Our work uses and extends the framework [28, 32] of Giorgi, and contrary to the subextremal case we prove that instability results hold for a set of gauge invariant quantities along the event horizon ( {mathcal {H}}^+ ). In particular, for associated quantities shown to satisfy generalized Regge–Wheeler equations we prove decay, non-decay, and polynomial blow-up estimates asymptotically along ( {mathcal {H}}^+ ), the exact behavior depending on the number of translation invariant derivatives that we take. As a consequence, we show that for generic initial data, solutions to the generalized Teukolsky system of positive and negative spin satisfy both stability and instability results. It is worth mentioning that the negative spin solutions are significantly more unstable, with the extreme curvature component ( {underline{alpha }} ) not decaying asymptotically along the event horizon ( {mathcal {H}}^+, ) a result previously unknown in the literature.

The Obukhov–Corrsin theory of scalar turbulence [21, 54] advances quantitative predictions on passive-scalar advection in a turbulent regime and can be regarded as the analogue for passive scalars of Kolmogorov’s K41 theory of fully developed turbulence [47]. The scaling analysis of Obukhov and Corrsin from 1949 to 1951 identifies a critical regularity threshold for the advection-diffusion equation and predicts anomalous dissipation in the limit of vanishing diffusivity in the supercritical regime. In this paper we provide a fully rigorous mathematical validation of this prediction by constructing a velocity field and an initial datum such that the unique bounded solution of the advection-diffusion equation is bounded uniformly-in-diffusivity within any fixed supercritical Obukhov-Corrsin regularity regime while also exhibiting anomalous dissipation. Our approach relies on a fine quantitative analysis of the interaction between the spatial scale of the solution and the scale of the Brownian motion which represents diffusion in a stochastic Lagrangian setting. This provides a direct Lagrangian approach to anomalous dissipation which is fundamental in order to get detailed insight on the behavior of the solution. Exploiting further this approach, we also show that for a velocity field in (C^alpha ) of space and time (for an arbitrary (0 le alpha < 1)) neither vanishing diffusivity nor regularization by convolution provide a selection criterion for bounded solutions of the advection equation. This is motivated by the fundamental open problem of the selection of solutions of the Euler equations as vanishing-viscosity limit of solutions of the Navier-Stokes equations and provides a complete negative answer in the case of passive advection.

We study a popular kinetic model introduced by Saintillan and Shelley for the dynamics of suspensions of active elongated particles where the particles are described by a distribution in space and orientation. The uniform distribution of particles is the stationary state of incoherence which is known to exhibit a phase transition. We perform an extensive study of the linearised evolution around the incoherent state. We show (i) in the non-diffusive regime corresponding to spectral (neutral) stability that the suspensions experience a mixing phenomenon similar to Landau damping and we provide optimal pointwise in time decay rates in weak topology. Further, we show (ii) in the case of small rotational diffusion (nu ) that the mixing estimates persist up to time scale (nu ^{-1/2}) until the exponential decay at enhanced dissipation rate (nu ^{1/2}) takes over. The interesting feature is that the usual velocity variable of kinetic models is replaced by an orientation variable on the sphere. The associated orientation mixing leads to limited algebraic decay for macroscopic quantities. For the proof, we start with a general pointwise decay result for Volterra equations that may be of independent interest. While, in the non-diffusive case, explicit formulas on the sphere allow to conclude the desired decay, much more work is required in the diffusive case: here we prove mixing estimates for the advection-diffusion equation on the sphere by combining an optimized hypocoercive approach with the vector field method. One main point in this context is to identify good commuting vector fields for the advection-diffusion operator on the sphere. Our results in this direction may be useful to other models in collective dynamics, where an orientation variable is involved.

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.