Abstract: We study the stability of the Kolmogorov flows which are stationary solutions to the two-dimensional Navier–Stokes equations in the presence of the shear external force. We establish the linear stability estimate when the viscosity coefficient \(\nu \) is sufficiently small, where the enhanced dissipation is rigorously verified in the time scale \(O(\nu ^{-\frac{1}{2}})\) for solutions to the linearized problem, which has been numerically conjectured and is much shorter than the usual viscous time scale \(O(\nu ^{-1})\) . Our approach is based on the detailed analysis for the resolvent problem. We also provide the abstract framework which is applicable to the resolvent estimate for the Kolmogorov flows. PubDate: 2019-09-09

Abstract: Abstract We prove in this paper that the Schwarzschild family of black holes are linearly stable as a family of solutions to the system of equations that result from expressing the Einstein vacuum equations in a generalised wave gauge. In particular we improve on our recent work (Johnson in The linear stability of the Schwarzschild solution to gravitational perturbations in the generalised wave gauge, arXiv: 1803.04012, 2018) by modifying the generalised wave gauge employed therein so as to establish asymptotic flatness of the associated linearised system. The result thus complements the seminal work (Dafermos et al. in The linear stability of the Schwarzschild solution to gravitational perturbations, arXiv:1601.06467, 2016) of Dafermos–Holzegel–Rodnianski in a similar vein as to how the work (Lindblad and Rodnianski. in Ann Math 171:1401–1477, 2010) of Lindblad–Rodnianski complemented that of Christodoulou–Klainerman (Christodoulou and Klainerman in The global nonlinear stability of the Minkowski space. Princeton Mathematical Series, vol 41. Princeton University Press, Princeton, 1993) in establishing the nonlinear stability of the Minkowski space. PubDate: 2019-09-07

Abstract: Abstract In this paper, we prove the existence of smooth initial data for the 2D free boundary incompressible Navier-Stokes equations, for which the smoothness of the interface breaks down in finite time into a splash singularity. PubDate: 2019-05-11

Abstract: Abstract Consider the spatially inhomogeneous Landau equation with moderately soft potentials (i.e. with \(\gamma \in (-\,2,0)\) ) on the whole space \({\mathbb {R}}^3\) . We prove that if the initial data \(f_{\mathrm {in}}\) are close to the vacuum solution \(f_{\mathrm {vac}} \equiv 0\) in an appropriate norm, then the solution f remains regular globally in time. This is the first stability of vacuum result for a binary collisional model featuring a long-range interaction. Moreover, we prove that the solutions in the near-vacuum regime approach solutions to the linear transport equation as \(t\rightarrow +\,\infty \) . Furthermore, in general, solutions do not approach a traveling global Maxwellian as \(t \rightarrow +\,\infty \) . Our proof relies on robust decay estimates captured using weighted energy estimates and the maximum principle for weighted quantities. Importantly, we also make use of a null structure in the nonlinearity of the Landau equation which suppresses the most slowly-decaying interactions. PubDate: 2019-05-08

Abstract: Abstract In this paper we prove global well-posedness and modified scattering for the massive Maxwell–Klein–Gordon equation in the Coulomb gauge on \(\mathbb {R}^{1+d}\) \((d \ge 4)\) for data with small critical Sobolev norm. This extends to the general case \( m^2 > 0 \) the results of Krieger–Sterbenz–Tataru ( \(d=4,5 \) ) and Rodnianski–Tao ( \( d \ge 6 \) ), who considered the case \( m=0\) . We proceed by generalizing the global parametrix construction for the covariant wave operator and the functional framework from the massless case to the Klein–Gordon setting. The equation exhibits a trilinear cancelation structure identified by Machedon–Sterbenz. To treat it one needs sharp \(L^{2}\) null form bounds, which we prove by estimating renormalized solutions in null frames spaces similar to the ones considered by Bejenaru–Herr. To overcome logarithmic divergences we rely on an embedding property of \( \Box ^{-1} \) in conjunction with endpoint Strichartz estimates in Lorentz spaces. PubDate: 2019-04-02

Abstract: Abstract We consider the Cauchy problem for the continuity equation in space dimension \({d \ge 2}\) . We construct a divergence-free 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 self-similar mixing by incompressible flows (Alberti et al. in J Am Math Soc 32(2):445–490, 2019), and have been announced in Exponential self-similar mixing and loss of regularity for continuity equations (Alberti et al. in Comptes Rendus Math Acad Sci Paris 352(11):901–906, 2014). PubDate: 2019-03-28

Abstract: Abstract We show the local in time well-posedness of the Prandtl equations for data with Gevrey 2 regularity in x and Sobolev regularity in y. The main novelty of our result is that we do not make any assumption on the structure of the initial data: no monotonicity or hypothesis on the critical points. Moreover, our general result is optimal in terms of regularity, in view of the ill-posedness result of Gérard-Varet and Dormy (J Am Math Soc 23(2):591–609, 2010). PubDate: 2019-03-19

Abstract: Abstract We are concerned with the global well-posedness and large time asymptotic behavior of strong and classical solutions to the Cauchy problem of the Navier–Stokes equations for viscous compressible barotropic flows in two or three spatial dimensions with vacuum as far field density. For strong and classical solutions, some a priori decay with rates (in large time) for both the pressure and the spatial gradient of the velocity field are obtained provided that the initial total energy is suitably small. Moreover, by using these key decay rates and some analysis on the expansion rates of the essential support of the density, we establish the global existence and uniqueness of classical solutions (which may be of possibly large oscillations) in two spatial dimensions, provided the smooth initial data are of small total energy. In addition, the initial density can even have compact support. This, in particular, yields the global regularity and uniqueness of the re-normalized weak solutions of Lions–Feireisl to the two-dimensional compressible barotropic flows for all adiabatic number \(\gamma >1\) provided that the initial total energy is small. PubDate: 2019-03-15

Abstract: This is the second and last paper of a two-part 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 two-ended 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 \(C^2\) -future-inextendible 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. PubDate: 2019-03-06

Abstract: Abstract We consider the exit event from a metastable state for the overdamped Langevin dynamics \(dX_t = -\nabla f(X_t) dt + \sqrt{h} dB_t\) . Using tools from semiclassical analysis, we prove that, starting from the quasi stationary distribution within the state, the exit event can be modeled using a jump Markov process parametrized with the Eyring–Kramers formula, in the small temperature regime \(h \rightarrow 0\) . We provide in particular sharp asymptotic estimates on the exit distribution which demonstrate the importance of the prefactors in the Eyring–Kramers formula. Numerical experiments indicate that the geometric assumptions we need to perform our analysis are likely to be necessary. These results also hold starting from deterministic initial conditions within the well which are sufficiently low in energy. From a modelling viewpoint, this gives a rigorous justification of the transition state theory and the Eyring–Kramers formula, which are used to relate the overdamped Langevin dynamics (a continuous state space Markov dynamics) to kinetic Monte Carlo or Markov state models (discrete state space Markov dynamics). From a theoretical viewpoint, our analysis paves a new route to study the exit event from a metastable state for a stochastic process. PubDate: 2019-03-04

Abstract: Abstract Coherent vortices are often observed to persist for long times in turbulent 2D flows even at very high Reynolds numbers and are observed in experiments and computer simulations to potentially be asymptotically stable in a weak sense for the 2D Euler equations. We consider the incompressible 2D Euler equations linearized around a radially symmetric, strictly monotone decreasing vorticity distribution. For sufficiently regular data, we prove the inviscid damping of the \(\theta \) -dependent radial and angular velocity fields with the optimal rates \(\left\ u^r(t)\right\ \lesssim \langle t \rangle ^{-1}\) and \(\left\ u^\theta (t)\right\ \lesssim \langle t \rangle ^{-2}\) in the appropriate radially weighted \(L^2\) spaces. We moreover prove that the vorticity weakly converges back to radial symmetry as \(t \rightarrow \infty \) , a phenomenon known as vortex axisymmetrization in the physics literature, and characterize the dynamics in higher Sobolev spaces. Furthermore, we prove that the \(\theta \) -dependent angular Fourier modes in the vorticity are ejected from the origin as \(t \rightarrow \infty \) , resulting in faster inviscid damping rates than those possible with passive scalar evolution. This non-local effect is called vorticity depletion. Our work appears to be the first to find vorticity depletion relevant for the dynamics of vortices. PubDate: 2019-02-12

Abstract: Abstract In this paper, we prove the linear damping for the 2-D Euler equations around a class of shear flows under the assumption that the linearized operator has no embedding eigenvalues. For the symmetric flows, we obtain the same decay estimate of the velocity as the monotone shear flows. Moreover, we confirm a new dynamical phenomena found by Bouchet and Morita: the depletion of the vorticity at the stationary streamlines, which along with the vorticity mixing leads to the damping for the base flows with stationary streamlines. PubDate: 2019-01-25

Abstract: Abstract We prove boundedness and polynomial decay statements for solutions of the spin \(\pm \,2\) Teukolsky equation on a Kerr exterior background with parameters satisfying \( a \ll M\) . The bounds are obtained by introducing generalisations of the higher order quantities P and \({\underline{P}}\) used in our previous work on the linear stability of Schwarzschild. The existence of these quantities in the Schwarzschild case is related to the transformation theory of Chandrasekhar. In a followup paper, we shall extend this result to the general sub-extremal range of parameters \( a <M\) . As in the Schwarzschild case, these bounds provide the first step in proving the full linear stability of the Kerr metric to gravitational perturbations. PubDate: 2019-01-08

Abstract: Abstract In this paper, we prove the existence of smooth solutions near 0 of the degenerate hyperbolic Monge-Ampère equation \(\det (D^2 u) = K(x)h(x,u,Du)\) , where \(K(0)=0\) , \(K\le 0\) , \(h(0,0,0)>0\) . We also assume that, the zero set of small perturbation of \(D_n K\) has a simple structure. For the proof, we first transform the linearized equation into a simple form by a suitable change of variables. Then we proceed to derive a priori estimates for the linearized equation, which is degenerately hyperbolic. Finally we use Nash-Moser iteration to prove the existence of local solutions. PubDate: 2019-01-02

Abstract: Abstract We prove a priori estimates for the compressible Euler equations modeling the motion of a liquid with moving physical vacuum boundary with unbounded initial domain. The liquid is under influence of gravity but without surface tension. Our fluid is not assumed to be irrotational. But the physical sign condition needs to be assumed on the free boundary. We generalize the method used in Lindblad and Luo (Commun Pure Appl Math, 2008) to prove the energy estimates in an unbounded domain up to arbitrary order. In addition to that, the a priori energy estimates are in fact uniform in the sound speed \(\kappa \) . As a consequence, we obtain the convergence of solutions of compressible Euler equations with a free boundary to solutions of the incompressible equations, generalizing the result of Lindblad and Luo (2008) to when you have an unbounded domain. On the other hand, we prove that there are initial data satisfying the compatibility condition in some weighted Sobolev spaces, and this will propagate within a short time interval, which is essential for proving long time existence for slightly compressible irrotational water waves. PubDate: 2018-12-11

Abstract: Abstract We consider the time-dependent 2D Ginzburg–Landau equation in the whole plane with terms modeling impurities and applied currents. The Ginzburg–Landau vortices are then subjected to three forces: their mutual repulsive Coulomb-like interaction, the applied current pushing them in a fixed direction, and the pinning force attracting them towards the impurities. The competition between the three is expected to lead to complicated glassy effects. We rigorously study the limit in which the number of vortices \(N_\varepsilon \) blows up as the inverse Ginzburg–Landau parameter \(\varepsilon \) goes to 0, and we derive via a modulated energy method fluid-like mean-field evolution equations. These results hold for parabolic, conservative, and mixed-flow dynamics in appropriate regimes of \(N_\varepsilon \uparrow \infty \) . Finally, we briefly discuss some natural homogenization questions raised by this study. PubDate: 2018-12-10

Abstract: Abstract We construct a large class of examples of non-uniqueness for the linear transport equation and the transport-diffusion equation with divergence-free vector fields in Sobolev spaces \(W^{1, p}\) . PubDate: 2018-12-07

Abstract: Abstract We consider Wave Maps into the sphere and give a new proof of small data global well-posedness and scattering in the critical Besov space, in any space dimension \(n \geqslant 2\) . We use an adapted version of the atomic space \(U^2\) as the single building block for the iteration space. Our approach to the so-called division problem is modular as it systematically uses two ingredients: atomic bilinear (adjoint) Fourier restriction estimates and an algebra property of the iteration space, both of which can be adapted to other phase functions. PubDate: 2018-12-04

Abstract: Abstract We establish the dual notions of scaling and saturation from geometric control theory in an infinite-dimensional setting. This generalization is applied to the low-mode 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. PubDate: 2018-10-09

Abstract: We present a new vector field approach to almost-sharp decay for the wave equation on spherically symmetric, stationary and asymptotically flat spacetimes. Specifically, we derive a new hierarchy of higher-order weighted energy estimates by employing appropriate commutator vector fields. In cases where an integrated local energy decay estimate holds, like in the case of sub-extremal Reissner–Nordström black holes, this hierarchy leads to almost-sharp global energy and pointwise time-decay estimates with decay rates that go beyond those obtained by the traditional vector field method. Our estimates play a fundamental role in our companion paper where precise late-time asymptotics are obtained for linear scalar fields on such backgrounds. PubDate: 2018-09-28