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Abstract: Abstract A fundamental question in fluid dynamics concerns the formation of discontinuous shock waves from smooth initial data. We prove that from smooth initial data, smooth solutions to the 2d Euler equations in azimuthal symmetry form a first singularity, the so-called \(C^{\frac{1}{3}} \) pre-shock. The solution in the vicinity of this pre-shock is shown to have a fractional series expansion with coefficients computed from the data. Using this precise description of the pre-shock, we prove that a discontinuous shock instantaneously develops after the pre-shock. This regular shock solution is shown to be unique in a class of entropy solutions with azimuthal symmetry and regularity determined by the pre-shock expansion. Simultaneous to the development of the shock front, two other characteristic surfaces of cusp-type singularities emerge from the pre-shock. These surfaces have been termed weak discontinuities by Landau & Lifschitz [12, Chapter IX, §96], who conjectured some type of singular behavior of derivatives along such surfaces. We prove that along the slowest surface, all fluid variables except the entropy have \(C^{1, {\frac{1}{2}} }\) one-sided cusps from the shock side, and that the normal velocity is decreasing in the direction of its motion; we thus term this surface a weak rarefaction wave. Along the surface moving with the fluid velocity, density and entropy form \(C^{1, {\frac{1}{2}} }\) one-sided cusps while the pressure and normal velocity remain \(C^2\) ; as such, we term this surface a weak contact discontinuity. PubDate: 2022-11-19

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Abstract: Abstract In this work, we derive the globally precise late-time asymptotics for the spin- \({\mathfrak {s}}\) fields on a Schwarzschild background, including the scalar field \(({\mathfrak {s}}=0)\) , the Maxwell field \(({\mathfrak {s}}=\pm 1)\) and the linearized gravity \(({\mathfrak {s}}=\pm 2)\) . The conjectured Price’s law in the physics literature which predicts the sharp rates of decay of the spin \(s=\pm {\mathfrak {s}}\) components towards the future null infinity as well as in a compact region is shown. Further, we confirm the heuristic claim by Barack and Ori that the spin \(+1, +2\) components have an extra power of decay at the event horizon than the conjectured Price’s law. The asymptotics are derived via a unified, detailed analysis of the Teukolsky master equation that is satisfied by all these components. PubDate: 2022-11-15

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Abstract: Abstract Inspired by the numerical evidence of a potential 3D Euler singularity [54, 55], we prove finite time singularity from smooth initial data for the HL model introduced by Hou-Luo in [54, 55] for the 3D Euler equations with boundary. Our finite time blowup solution for the HL model and the singular solution considered in [54, 55] share some essential features, including similar blowup exponents, symmetry properties of the solution, and the sign of the solution. We use a dynamical rescaling formulation and the strategy proposed in our recent work in [11] to establish the nonlinear stability of an approximate self-similar profile. The nonlinear stability enables us to prove that the solution of the HL model with smooth initial data and finite energy will develop a focusing asymptotically self-similar singularity in finite time. Moreover the self-similar profile is unique within a small energy ball and the \(C^\gamma \) norm of the density \(\theta \) with \(\gamma \approx 1/3\) is uniformly bounded up to the singularity time. PubDate: 2022-11-13

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Abstract: Abstract Let \({{\mathcal {H}}}\) denote the future outgoing null hypersurface emanating from a spacelike 2-sphere S in a vacuum spacetime \(({{\mathcal {M}}},\textbf{g})\) . In this paper we study the so-called canonical foliation on \({{\mathcal {H}}}\) introduced in [13, 22] and show that the corresponding geometry is controlled locally only in terms of the initial geometry on S and the \(L^2\) curvature flux through \({{\mathcal {H}}}\) . In particular, we show that the ingoing and outgoing null expansions \({\textrm{tr}}\chi \) and \({\textrm{tr}}{{{\underline{\chi }}}}\) are both locally uniformly bounded. The proof of our estimates relies on a generalisation of the methods of [15–17] and [1, 2, 26, 32] where the geodesic foliation on null hypersurfaces \({{\mathcal {H}}}\) is studied. The results of this paper, while of independent interest, are essential for the proof of the spacelike-characteristic bounded \(L^2\) curvature theorem [12]. PubDate: 2022-10-20

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Abstract: Abstract In this paper we study the spacelike-characteristic Cauchy problem for the Einstein vacuum equations. Given initial data on a maximal spacelike hypersurface \(\Sigma \simeq \overline{B_1} \subset {{\mathbb {R}}}^3\) and the outgoing null hypersurface \({{\mathcal {H}}}\) emanating from \({\partial }\Sigma \) , we prove a priori estimates for the resulting future development in terms of low-regularity bounds on the initial data at the level of curvature in \(L^2\) . The proof uses the bounded \(L^2\) curvature theorem [22], the extension procedure for the constraint equations [12], Cheeger-Gromov theory in low regularity [13], the canonical foliation on null hypersurfaces in low regularity [15] and global elliptic estimates for spacelike maximal hypersurfaces. PubDate: 2022-10-20

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Abstract: Abstract We show novel types of uniqueness and rigidity results for Schrödinger equations in either the nonlinear case or in the presence of a complex-valued potential. As our main result we obtain that the trivial solution \(u=0\) is the only solution for which the assumptions \(u(t=0)\vert _{D}=0, u(t=T)\vert _{D}=0\) hold, where \(D\subset \mathbb {R}^d\) are certain subsets of codimension one. In particular, D is discrete for dimension \(d=1\) . Our main theorem can be seen as a nonlinear analogue of discrete Fourier uniqueness pairs such as the celebrated Radchenko–Viazovska formula in [21], and the uniqueness result of the second author and M. Sousa for powers of integers [22]. As an additional application, we deduce rigidity results for solutions to some semilinear elliptic equations from their zeros. PubDate: 2022-09-14 DOI: 10.1007/s40818-022-00138-1

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Abstract: Abstract This paper is concerned with the relativistic Boltzmann equation without angular cutoff. We establish the global-in-time existence, uniqueness and asymptotic stability for solutions nearby the relativistic Maxwellian. We work in the case of a spatially periodic box. We assume the generic hard-interaction and soft-interaction conditions on the collision kernel that were derived by Dudyński and Ekiel-Je \(\dot{\text {z}}\) ewska (Comm. Math. Phys. 115(4):607–629, 1985) in [32], and our assumptions include the case of Israel particles (J. Math. Phys. 4:1163–1181, 1963) in [56]. In this physical situation, the angular function in the collision kernel is not locally integrable, and the collision operator behaves like a fractional diffusion operator. The coercivity estimates that are needed rely crucially on the sharp asymptotics for the frequency multiplier that has not been previously established. We further derive the relativistic analogue of the Carleman dual representation for the Boltzmann collision operator. This resolves the open question of perturbative global existence and uniqueness without the Grad’s angular cut-off assumption. PubDate: 2022-08-17 DOI: 10.1007/s40818-022-00137-2

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Abstract: Abstract In this paper, we initiate the study of the global stability of nonlinear wave equations with initial data that are not required to be localized around a single point. More precisely, we allow small initial data localized around any finite collection of points which can be arbitrarily far from one another. Existing techniques do not directly apply to this setting because they require norms with radial weights away from some center to be small. The smallness we require on the data is measured in a norm which does not depend on the scale of the configuration of the data. Our method of proof relies on a close analysis of the geometry of the interaction between waves originating from different sources. We prove estimates on the bilinear forms encoding the interaction, which allow us to show improved bounds for the energy of the solution. We finally apply a variant of the vector field method involving modified Klainerman–Sobolev estimates to prove global stability. As a corollary of our proof, we are able to show global existence for a class of data whose \(H^1\) norm is arbitrarily large. PubDate: 2022-08-08 DOI: 10.1007/s40818-022-00136-3

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Abstract: Abstract This the first in a series of papers whose ultimate goal is to establish the full nonlinear stability of the Kerr family for \( a \ll m\) . The paper builds on the strategy laid out in [6] in the context of the nonlinear stability of Schwarzschild for axially symmetric polarized perturbations. In fact the central idea of [6] was the introduction and construction of generally covariant modulated (GCM) spheres on which specific geometric quantities take Schwarzschildian values. This was made possible by taking into account the full general covariance of the Einstein vacuum equations. The goal of this, and its companion paper [7], is to get rid of the symmetry restriction in the construction of GCM spheres in [6] and thus remove an essential obstruction in extending the result to a full stability proof of the Kerr family. PubDate: 2022-08-02 DOI: 10.1007/s40818-022-00131-8

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Abstract: Abstract This is a follow-up of our paper (Klainerman and Szeftel in Construction of GCM spheres in perturbations of Kerr, Accepted for publication in Annals of PDE) on the construction of general covariant modulated (GCM) spheres in perturbations of Kerr, which we expect to play a central role in establishing their nonlinear stability. We reformulate the main results of that paper using a canonical definition of \(\ell =1\) modes on a 2-sphere embedded in a \(1+3\) vacuum manifold. This is based on a new, effective, version of the classical uniformization theorem which allows us to define such modes and prove their stability for spheres with comparable metrics. The reformulation allows us to prove a second, intrinsic, existence theorem for GCM spheres, expressed purely in terms of geometric quantities defined on it. A natural definition of angular momentum for such GCM spheres is also introduced, which we expect to play a key role in determining the final angular momentum for general perturbations of Kerr. PubDate: 2022-08-02 DOI: 10.1007/s40818-022-00132-7

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Abstract: Abstract The question about behavior of gaps between zeros of polynomials under differentiation is classical and goes back to Marcel Riesz. Recently, Stefan Steinerberger [42] formally derived a nonlocal nonlinear partial differential equation which models dynamics of roots of polynomials under differentiation. In this paper, we connect rigorously solutions of Steinerberger’s PDE and evolution of roots under differentiation for a class of trigonometric polynomials. Namely, we prove that the distribution of the zeros of the derivatives of a polynomial and the corresponding solutions of the PDE remain close for all times. The global in time control follows from the analysis of the propagation of errors equation, which turns out to be a nonlinear fractional heat equation with the main term similar to the modulated discretized fractional Laplacian \((-\Delta )^{1/2}\) . PubDate: 2022-07-25 DOI: 10.1007/s40818-022-00135-4

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Abstract: Abstract In this article, we prove various illposedness results for the Cauchy problem for the incompressible Hall- and electron-magnetohydrodynamic (MHD) equations without resistivity. These PDEs are fluid descriptions of plasmas, where the effect of collisions is neglected (no resistivity), while the motion of the electrons relative to the ions (Hall current term) is taken into account. The Hall current term endows the magnetic field equation with a quasilinear dispersive character, which is key to our mechanism for illposedness. Perhaps the most striking conclusion of this article is that the Cauchy problems for the Hall-MHD (either viscous or inviscid) and the electron-MHD equations, under one translational symmetry, are ill-posed near the trivial solution in any sufficiently high regularity Sobolev space \(H^{s}\) and even in any Gevrey spaces. This result holds despite obvious wellposedness of the linearized equations near the trivial solution, as well as conservation of the nonlinear energy, by which the \(L^{2}\) norm (energy) of the solution stays constant in time. The core illposedness (or instability) mechanism is degeneration of certain high frequency wave packet solutions to the linearization around a class of linearly degenerate stationary solutions of these equations, which are essentially dispersive equations with degenerate principal symbols. The method developed in this work is sharp and robust, in that we also prove nonlinear \(H^{s}\) -illposedness (for s arbitrarily high) in the presence of fractional dissipation of any order less than 1, matching the previously known wellposedness results. The results in this article are complemented by a companion work, where we provide geometric conditions on the initial magnetic field that ensure wellposedness(!) of the Cauchy problems for the incompressible Hall and electron-MHD equations. In particular, in stark contrast to the results here, it is shown in the companion work that the nonlinear Cauchy problems are well-posed near any nonzero constant magnetic field. PubDate: 2022-07-21 DOI: 10.1007/s40818-022-00134-5

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Abstract: Abstract We consider the cubic and quintic nonlinear Schrödinger equations (NLS) under the \({\mathbb {R}}^{d}\) and \({\mathbb {T}}^{d}\) energy-supercritical setting. Via a newly developed unified scheme, we prove the unconditional uniqueness for solutions to NLS at critical regularity for all dimensions. Thus, together with [19, 20], the unconditional uniqueness problems for \(H^{1}\) -critical and \(H^{1}\) -supercritical cubic and quintic NLS are completely and uniformly resolved at critical regularity for these domains. One application of our theorem is to prove that defocusing blowup solutions of the type in [59] are the only possible \(C([0,T);{\dot{H}}^{s_{c}})\) solutions if exist in these domains. PubDate: 2022-06-18 DOI: 10.1007/s40818-022-00130-9

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Abstract: Abstract 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. PubDate: 2022-06-11 DOI: 10.1007/s40818-022-00127-4

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Abstract: Abstract 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, \(r\phi _\ell \sim t^{-1}\) as \(t\rightarrow -\infty \) , 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 \(\partial _v(r\phi _\ell )\) 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 \(\partial _v(r\phi _\ell )\) 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 \(\ell \) : \(r\phi _\ell _{{\mathscr {I}}^+}\sim u^{-2}\) as \(u\rightarrow \infty \) . PubDate: 2022-06-07 DOI: 10.1007/s40818-022-00129-2

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Abstract: Abstract 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. PubDate: 2022-05-27 DOI: 10.1007/s40818-022-00126-5

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Abstract: Abstract 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. PubDate: 2022-05-12 DOI: 10.1007/s40818-022-00123-8

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Abstract: Abstract 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. PubDate: 2022-05-04 DOI: 10.1007/s40818-022-00125-6

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Abstract: Abstract 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. PubDate: 2022-04-07 DOI: 10.1007/s40818-022-00119-4

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Abstract: Abstract 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. PubDate: 2022-04-07 DOI: 10.1007/s40818-022-00121-w