Annals of Pde最新文献

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) .

We provide a self-contained proof of a trapped surface formation theorem, which simplifies the previous results by Christodoulou and by An–Luk. Our argument is based on a systematic approach for the scale-critical estimates in An–Luk and it connects Christodoulou’s short-pulse method and Klainerman–Rodnianski’s signature counting argument to the peeling properties previously studied in the small-data regime such as Klainerman–Nicolo. In particular this allows us to avoid elliptic estimates and geometric renormalizations, and gives us new technical improvements and simplifications.

This work is concerned with the mathematical analysis of the bulk rheology of random suspensions of rigid particles settling under gravity in viscous fluids. Each particle generates a fluid flow that in turn acts on other particles and hinders their settling. In an equilibrium perspective, for a given ensemble of particle positions, we analyze both the associated mean settling speed and the velocity fluctuations of individual particles. In the 1970s, Batchelor gave a proper definition of the mean settling speed, a 60-year-old open problem in physics, based on the appropriate renormalization of long-range particle contributions. In the 1980s, a celebrated formal calculation by Caflisch and Luke suggested that velocity fluctuations in dimension (d=3) should diverge with the size of the sedimentation tank, contradicting both intuition and experimental observations. The role of long-range self-organization of suspended particles in form of hyperuniformity was later put forward to explain additional screening of this divergence in steady-state observations. In the present contribution, we develop the first rigorous theory that allows to justify all these formal calculations of the physics literature.

We present a systematic approach to regularity theory of the multi-dimensional Euler alignment systems with topological diffusion introduced in [35]. While these systems exhibit flocking behavior emerging from purely local communication, bearing direct relevance to empirical field studies, global and even local well-posedness has proved to be a major challenge in multi-dimensional settings due to the presence of topological effects. In this paper we reveal two important classes of global smooth solutions—parallel shear flocks with incompressible velocity and stationary density profile, and nearly aligned flocks with close to constant velocity field but arbitrary density distribution. Existence of such classes is established via an efficient continuation criterion requiring control only on the Lipschitz norm of state quantities, which makes it accessible to the applications of fractional parabolic theory. The criterion presents a major improvement over the existing result of [28], and is proved with the use of quartic paraproduct estimates.

In this paper, we study the linear stability properties of perturbations around the homogeneous Couette flow for a 2D isentropic compressible fluid in the domain (mathbb {T}times mathbb {R}). In the inviscid case there is a generic Lyapunov type instability for the density and the irrotational component of the velocity field. More precisely, we prove that their (L^2) norm grows as (t^{1/2}) and this confirms previous observations in the physics literature. On the contrary, the solenoidal component of the velocity field experiences inviscid damping, namely it decays to zero even in the absence of viscosity. For a viscous compressible fluid, we show that the perturbations may have a transient growth of order (nu ^{-1/6}) (with (nu ^{-1}) being proportional to the Reynolds number) on a time-scale (nu ^{-1/3}), after which it decays exponentially fast. This phenomenon is also called enhanced dissipation and our result appears to be the first to detect this mechanism for a compressible flow, where an exponential decay for the density is not a priori trivial given the absence of dissipation in the continuity equation.