Showing 1 to 10 of 68 matching Articles
Results per page:
Export (CSV)
By
Chiodaroli, Elisabetta; Krieger, Joachim; Lührmann, Jonas
We consider radially symmetric, energy critical wave maps from
$$(1+2)$$
dimensional Minkowski space into the unit sphere
$$\mathbb {S}^m$$
,
$$m \ge 1$$
, and prove global regularity and scattering for classical smooth data of finite energy. In addition, we establish a priori bounds on a suitable scattering norm of the radial wave maps and exhibit concentration compactness properties of sequences of radial wave maps with uniformly bounded energies. This extends and complements the beautiful classical work of Christodoulou and TahvildarZadeh (Duke Math J 71(1):31–69, 1993; Pure Appl Math 46(7):1041–1091, 1993) and Struwe (Math Z 242(3):407–414, 2002; Calc Var Partial Differ Equ 16(4):431–437, 2003) as well as of Nahas (Calc Var Partial Differ Equ 46(1–2):427–437, 2013) on radial wave maps in the case of the unit sphere as the target. The proof is based upon the concentration compactness/rigidity method of Kenig and Merle (Invent Math 166(3):645–675, 2006; Acta Math 201(2):147–212, 2008) and a “twisted” Bahouri–Gérard type profile decomposition (Am J Math 121(1):131–175, 1999), following the implementation of this strategy by the second author and Schlag (Concentration compactness for critical wave maps. EMS monographs in mathematics, European Mathematical Society (EMS), Zürich, 2012) for energy critical wave maps into the hyperbolic plane as well as by the last two authors (Ann PDE 1(1):1–208, 2015) for the energy critical Maxwell–Klein–Gordon equation.
more …
By
Lellis, Camillo; Spadaro, Emanuele; Spolaor, Luca
2 Citations
We construct a branched center manifold in a neighborhood of a singular point of a 2dimensional integral current which is almost minimizing in a suitable sense. Our construction is the first half of an argument which shows the discreteness of the singular set for the following three classes of 2dimensional currents: area minimizing in Riemannian manifolds, semicalibrated and spherical cross sections of 3dimensional area minimizing cones.
more …
By
Luk, Jonathan ; Oh, SungJin
This is the second and last paper of a twopart series in which we prove the
$$C^2$$
formulation of the strong cosmic censorship conjecture for the Einstein–Maxwell–(real)–scalar–field system in spherical symmetry for twoended asymptotically flat data. In the first paper, we showed that the maximal globally hyperbolic future development of an admissible asymptotially flat Cauchy initial data set is
$$C^2$$
futureinextendible provided that an
$$L^2$$
averaged (inverse) polynomial lower bound for the derivative of the scalar field holds along each horizon. In this paper, we show that this lower bound is indeed satisfied for solutions arising from a generic set of Cauchy initial data. Roughly speaking, the generic set is open with respect to a (weighted)
$$C^1$$
topology and is dense with respect to a (weighted)
$$C^\infty $$
topology. The proof of the theorem is based on extensions of the ideas in our previous work on the linear instability of Reissner–Nordström Cauchy horizon, as well as a new large data asymptotic stability result which gives good decay estimates for the difference of the radiation fields for small perturbations of an arbitrary solution.
more …
By
GlattHoltz, Nathan E.; Herzog, David P.; Mattingly, Jonathan C.
We establish the dual notions of scaling and saturation from geometric control theory in an infinitedimensional setting. This generalization is applied to the lowmode control problem in a number of concrete nonlinear partial differential equations. We also develop applications concerning associated classes of stochastic partial differential equations (SPDEs). In particular, we study the support properties of probability laws corresponding to these SPDEs as well as provide applications concerning the ergodic and mixing properties of invariant measures for these stochastic systems.
more …
By
HanKwan, Daniel; Léautaud, Matthieu
This work is devoted to the analysis of the linear Boltzmann equation on the torus, in the presence of a force deriving from a potential. The collision operator is allowed to be degenerate in the following two senses: (1) the associated collision kernel may vanish in a large subset of the phase space; (2) we do not assume that it is bounded below by a Maxwellian at infinity in velocity. We study how the association of transport and collision phenomena can lead to convergence to equilibrium, using concepts and ideas from control theory. We prove two main classes of results. On the one hand, we show that convergence towards an equilibrium is equivalent to an almost everywhere geometric control condition. The equilibria (which are not necessarily Maxwellians with our general assumptions on the collision kernel) are described in terms of the equivalence classes of an appropriate equivalence relation involving transport and collisions. On the other hand, we characterize the exponential convergence to equilibrium in terms of the Lebeau constant, which involves some averages of the collision frequency along the flow of the transport. We also explain how to handle the case of linear Boltzmann equations posed on the phase space associated to a compact Riemannian manifold without boundary.
more …
By
Alberti, Giovanni; Crippa, Gianluca; Mazzucato, Anna L.
We consider the Cauchy problem for the continuity equation in space dimension
$${d \ge 2}$$
. We construct a divergencefree velocity field uniformly bounded in all Sobolev spaces
$$W^{1,p}$$
, for
$$1 \le p<\infty $$
, and a smooth compactly supported initial datum such that the unique solution to the continuity equation with this initial datum and advecting field does not belong to any Sobolev space of positive fractional order at any positive time. We also construct velocity fields in
$$W^{r,p}$$
, with
$$r>1$$
, and solutions of the continuity equation with these velocities that exhibit some loss of regularity, as long as the Sobolev space
$$W^{r,p}$$
does not embed in the space of Lipschitz functions. Our constructions are based on examples of optimal mixers from the companion paper Exponential selfsimilar mixing by incompressible flows (Alberti et al. in J Am Math Soc 32(2):445–490, 2019), and have been announced in Exponential selfsimilar mixing and loss of regularity for continuity equations (Alberti et al. in Comptes Rendus Math Acad Sci Paris 352(11):901–906, 2014).
more …
By
Bodineau, Thierry; Gallagher, Isabelle; SaintRaymond, Laure
6 Citations
We derive the linear acoustic and Stokes–Fourier equations as the limiting dynamics of a system of N hard spheres of diameter
$${\varepsilon }$$
in two space dimensions, when
$$N\rightarrow \infty $$
,
$${\varepsilon }\rightarrow 0$$
,
$$N{\varepsilon }=\alpha \rightarrow \infty $$
, using the linearized Boltzmann equation as an intermediate step. Our proof is based on Lanford’s strategy (Time evolution of large classical systems, Springer, Berlin, 1975), and on the pruning procedure developed in Bodineau et al. (Invent Math 203:493–553, 2016) to improve the convergence time to all kinetic times with a quantitative control which allows us to reach also hydrodynamic time scales. The main novelty here is that uniform
$$L^2$$
a priori estimates combined with a subtle symmetry argument provide a weak version of chaos, in the form of a cumulant expansion describing the asymptotic decorrelation between the particles. A refined geometric analysis of recollisions is also required in order to discard the possibility of multiple recollisions.
more …
By
Ionescu, Alexandru D.; Klainerman, Sergiu
7 Citations
This paper is motivated by the problem of the nonlinear stability of the Kerr solution for axially symmetric perturbations. We consider a model problem concerning the axially symmetric perturbations of a wave map
$$\Phi $$
defined from a fixed Kerr solution
$${\mathcal K}(M,a)$$
,
$$0\le a \le M $$
, with values in the two dimensional hyperbolic space
$${\mathbb H}^2$$
. A particular such wave map is given by the complex Ernst potential associated to the axial Killing vectorfield
$$\mathbf{Z}$$
of
$${\mathcal K}(M,a)$$
. We conjecture that this stationary solution is stable, under small axially symmetric perturbations, in the domain of outer communication (DOC) of
$${\mathcal K}(M,a)$$
, for all
$$0\le a<M$$
and we provide preliminary support for its validity, by deriving convincing stability estimates for the linearized system.
more …
By
Guo, Yan; Nguyen, Toan T.
9 Citations
This paper concerns the validity of the Prandtl boundary layer theory in the inviscid limit for steady incompressible Navier–Stokes flows. The stationary flows, with small viscosity, are considered on
$$[0,L]\times \mathbb {R}_{+}$$
, with a noslip boundary condition over a moving plate at
$$y=0$$
. We establish the validity of the Prandtl boundary layer expansion and its error estimates.
more …
