Annals of Pde最新文献

The spherically symmetric null hypersurfaces in a Schwarzschild spacetime are smooth away from the singularities and foliate the spacetime. We prove the existence of more general foliations by null hypersurfaces without the spherical symmetry condition. In fact we also relax the spherical symmetry of the ambient spacetime and prove a more general result: in a perturbed Schwarzschild spacetime (not necessary being vacuum), nearly round null hypersurfaces can be extended regularly to the past null infinity, thus there exist many foliations by regular null hypersurfaces in the exterior region of a perturbed Schwarzschild black hole. A significant point of the result is that the ambient spacetime metric is not required to be differentiable in all directions.

In this paper, we derive the early-time asymptotics for fixed-frequency solutions (phi _ell ) to the wave equation (Box _g phi _ell =0) on a fixed Schwarzschild background ((M>0)) arising from the no incoming radiation condition on ({mathscr {I}}^-) and polynomially decaying data, (rphi _ell sim t^{-1}) as (trightarrow -infty ), on either a timelike boundary of constant area radius (r>2M) (I) or an ingoing null hypersurface (II). In case (I), we show that the asymptotic expansion of (partial _v(rphi _ell )) along outgoing null hypersurfaces near spacelike infinity (i^0) contains logarithmic terms at order (r^{-3-ell }log r). In contrast, in case (II), we obtain that the asymptotic expansion of (partial _v(rphi _ell )) near spacelike infinity (i^0) contains logarithmic terms already at order (r^{-3}log r) (unless (ell =1)). These results suggest an alternative approach to the study of late-time asymptotics near future timelike infinity (i^+) that does not assume conformally smooth or compactly supported Cauchy data: In case (I), our results indicate a logarithmically modified Price’s law for each (ell )-mode. On the other hand, the data of case (II) lead to much stronger deviations from Price’s law. In particular, we conjecture that compactly supported scattering data on ({mathscr {H}}^-) and ({mathscr {I}}^-) lead to solutions that exhibit the same late-time asymptotics on ({mathscr {I}}^+) for each (ell ): (rphi _ell |_{{mathscr {I}}^+}sim u^{-2}) as (urightarrow infty ).

We consider the derivation of the defocusing cubic nonlinear Schrödinger equation (NLS) on ({mathbb {R}}^{3}) from quantum N-body dynamics. We reformat the hierarchy approach with Klainerman-Machedon theory and prove a bi-scattering theorem for the NLS to obtain convergence rate estimates under (H^{1}) regularity. The (H^{1}) convergence rate estimate we obtain is almost optimal for (H^{1}) datum, and immediately improves if we have any extra regularity on the limiting initial one-particle state.

We analyze the relativistic Euler equations of conservation laws of baryon number and momentum with a general pressure law. The existence of global-in-time bounded entropy solutions for the system is established by developing a compensated compactness framework. The proof relies on a careful analysis of the entropy and entropy-flux functions, which are represented by the fundamental solutions of the entropy and entropy-flux equations for the relativistic Euler equations. Based on a careful entropy analysis, we establish the compactness framework for sequences of both exact solutions and approximate solutions of the relativistic Euler equations. Then we construct approximate solutions via the vanishing viscosity method and employ our compactness framework to deduce the global-in-time existence of entropy solutions. The compactness of the solution operator is also established. Finally, we apply our techniques to establish the convergence of the Newtonian limit from the entropy solutions of the relativistic Euler equations to the classical Euler equations.

This paper is motivated by the non-linear stability problem for the expanding region of Kerr de Sitter cosmologies in the context of Einstein’s equations with positive cosmological constant. We show that under dynamically realistic assumptions the conformal Weyl curvature of the spacetime decays towards future null infinity. More precisely we establish decay estimates for Weyl fields which are (i) uniform (with respect to a global time function) (ii) optimal (with respect to the rate) and (iii) consistent with a global existence proof (in terms of regularity). The proof relies on a geometric positivity property of compatible currents which is a manifestation of the global redshift effect capturing the expansion of the spacetime.

In this paper we study the deterministic and stochastic homogenisation of free-discontinuity functionals under linear growth and coercivity conditions. The main novelty of our deterministic result is that we work under very general assumptions on the integrands which, in particular, are not required to be periodic in the space variable. Combining this result with the pointwise Subadditive Ergodic Theorem by Akcoglu and Krengel, we prove a stochastic homogenisation result, in the case of stationary random integrands. In particular, we characterise the limit integrands in terms of asymptotic cell formulas, as in the classical case of periodic homogenisation.

We construct mixing solutions to the incompressible porous media equation starting from Muskat type data in the partially unstable regime. In particular, we consider bubble and turned type interfaces with Sobolev regularity. As a by-product, we prove the continuation of the evolution of IPM after the Rayleigh–Taylor and smoothness breakdown exhibited in (Castro et al. in Arch Ration Mech Anal 208(3):805–909, 2013, Castro et al. in Ann Math. (2) 175(2):909–948, 2012). At each time slice the space is split into three evolving domains: two non-mixing zones and a mixing zone which is localized in a neighborhood of the unstable region. In this way, we show the compatibility between the classical Muskat problem and the convex integration method.

In this paper, we prove that the translating solitons of the mean curvature flow in (mathbb {R}^4) which arise as blow-up limit of embedded, mean convex mean curvature flow must have SO(2) symmetry.

We analyse an operator arising in the description of singular solutions to the two-dimensional Keller-Segel problem. It corresponds to the linearised operator in parabolic self-similar variables, close to a concentrated stationary state. This is a two-scale problem, with a vanishing thin transition zone near the origin. Via rigorous matched asymptotic expansions, we describe the eigenvalues and eigenfunctions precisely. We also show a stability result with respect to suitable perturbations, as well as a coercivity estimate for the non-radial part. These results are used as key arguments in a new rigorous proof of the existence and refined description of singular solutions for the Keller–Segel problem by the authors [8]. The present paper extends the result by Dejak, Lushnikov, Yu, Ovchinnikov and Sigal [11]. Two major difficulties arise in the analysis: this is a singular limit problem, and a degeneracy causes corrections not being polynomial but logarithmic with respect to the main parameter.

We consider the two-species Vlasov-Maxwell-Boltzmann (VMB) system with the scaling under which the moments of the fluctuations to the global Maxwellians formally converge to two-fluid incompressible Navier-Stokes-Fourier-Maxwell (NSFM) system with Ohm’s law. We prove the uniform estimates with respect to Knudsen number (varepsilon ) for the fluctuations by employing two types of micro-macro decompositions, and furthermore a hidden damping effect from the microscopic Ohm’s law. As consequences, the existence of the global-in-time classical solutions of VMB with all (varepsilon in (0,1]) is established. Moreover, the convergence of the fluctuations of the solutions of VMB to the classical solutions of NSFM with Ohm’s law is rigorously justified. This limit was justified in the recent breakthrough of Arsénio and Saint-Raymond (From the Vlasov-Maxwell-Boltzmann system to incompressible viscous electro-magneto-hydrodynamics. Vol. 1. EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2019) from renormalized solutions of VMB to dissipative solutions of incompressible viscous electro-magneto-hydrodynamics under the suitable scalings. In this sense, our result provides a classical solution analogue of the corresponding limit in Arsénio and Saint-Raymond (From the Vlasov-Maxwell-Boltzmann system to incompressible viscous electro-magneto-hydrodynamics. Vol. 1. EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2019) .