Showing papers in "Communications on Pure and Applied Mathematics in 2015"

K‐Stability and Kähler‐Einstein Metrics

Abstract: We prove that if a Fano manifold M is K-stable, then it admits a Kahler-Einstein metric. It affirms a longstanding conjecture for Fano manifolds. © 2015 Wiley Periodicals, Inc.

Global Ill-Posedness of the Isentropic System of Gas Dynamics

Abstract: We consider the isentropic compressible Euler system in 2 space dimensions with pressure law p () = (2) and we show the existence of classical Riemann data, i.e. pure jump discontinuities across a line, for which there are infinitely many admissible bounded weak solutions (bounded away from the void). We also show that some of these Riemann data are generated by a 1-dimensional compression wave: our theorem leads therefore to Lipschitz initial data for which there are infinitely many global bounded admissible weak solutions. (c) 2015 Wiley Periodicals, Inc.

Local-in-Time Existence and Uniqueness of Solutions to the Prandtl Equations by Energy Methods

Abstract: We prove local existence and uniqueness for the two-dimensional Prandtl system in weighted Sobolev spaces under Oleinik's monotonicity assumption. In particular we do not use the Crocco transform or any change of variables. Our proof is based on a new nonlinear energy estimate for the Prandtl system. This new energy estimate is based on a cancellation property that is valid under the monotonicity assumption. To construct the solution, we use a regularization of the system that preserves this nonlinear structure. This new nonlinear structure may give some insight into the convergence properties from the Navier-Stokes system to the Euler system when the viscosity goes to 0. © 2015 Wiley Periodicals, Inc.

Threshold dynamics for networks with arbitrary surface tensions

Abstract: We present and study a new algorithm for simulating the N-phase mean curvature motion for an arbitrary set of (isotropic) surface tensions. The departure point is the threshold dynamics algorithm of Merriman, Bence, and Osher for the two-phase case. A new energetic interpretation of this algorithm allows us to extend it in a natural way to the case of N phases, for arbitrary surface tensions and arbitrary (isotropic) mobilities. For a large class of surface tensions, the algorithm is shown to be consistent in the sense that at every time step, it decreases an energy functional that is an approximation (in the sense of Gamma convergence) of the interfacial energy. A broad range of numerical tests shows good convergence properties. An important application is the motion of grain boundaries in polycrystalline materials: It is also established that the above-mentioned large class of surface tensions contains the Read-Shockley surface tensions, both in the two-dimensional and three-dimensional settings.© 2015 Wiley Periodicals, Inc.

Bogoliubov Spectrum of Interacting Bose Gases

Abstract: We study the large-N limit of a system of N bosons interacting with a potential of intensity 1/N. When the ground state energy is to the first order given by Hartree's theory, we study the next order, predicted by Bogoliubov's theory. We show the convergence of the lower eigenvalues and eigenfunctions towards that of the Bogoliubov Hamiltonian (up to a convenient unitary transform). We also prove the convergence of the free energy when the system is sufficiently trapped. Our results are valid in an abstract setting, our main assumptions being that the Hartree ground state is unique and non-degenerate, and that there is complete Bose-Einstein condensation on this state. Using our method we then treat two applications: atoms with ''bosonic'' electrons on one hand, and trapped 2D and 3D Coulomb gases on the other hand.

Well-posedness of Compressible Euler Equations in a Physical Vacuum

Abstract: An important problem in gas and fluid dynamics is to understand the behavior of vacuum states, namely the behavior of the system in the presence of a vacuum. In particular, physical vacuum, in which the boundary moves with a nontrivial finite normal acceleration, naturally arises in the study of the motion of gaseous stars or shallow water. Despite its importance, there are only a few mathematical results available near a vacuum. The main difficulty lies in the fact that the physical systems become degenerate along the vacuum boundary. In this paper, we establish the local-in-time well-posedness of three-dimensional compressible Euler equations for polytropic gases with a physical vacuum by considering the problem as a free boundary problem. © 2015 Wiley Periodicals, Inc.

Quantitative Derivation of the Gross-Pitaevskii Equation

Abstract: Starting from first-principle many-body quantum dynamics, we show that the dynamics of Bose-Einstein condensates can be approximated by the time-dependent nonlinear Gross-Pitaevskii equation, giving a bound on the rate of the convergence. Initial data are constructed on the bosonic Fock space applying an appropriate Bogoliubov transformation on a coherent state with expected number of particles $\mathit{N}$. The Bogoliubov transformation plays a crucial role; it produces the correct microscopic correlations among the particles. Our analysis shows that, on the level of the one-particle reduced density, the form of the initial data is preserved by the many-body evolution, up to a small error that vanishes as $\mathit{N}^{-1/2}$ in the limit of large $\mathit{N}$.

Generalizations and Properties of the Principal Eigenvalue of Elliptic Operators in Unbounded Domains

Abstract: Using three different notions of the generalized principal eigenvalue of linear second-order elliptic operators in unbounded domains, we derive necessary and sufficient conditions for the validity of the maximum principle, as well as for the existence of positive eigenfunctions for the Dirichlet problem. Relations between these principal eigenvalues, their simplicity, and several other properties are further discussed. © 2015 Wiley Periodicals, Inc.

Mean Field Equation of Liouville Type with Singular Data: Topological Degree

Abstract: We consider the following mean field equation: Δgv+ρ(h*ev∫Mh*ev−1)=4π∑j=1Nαj(δqj−1) on M, where M is a compact Riemann surface with volume 1, h* is a positive C1 function on M, and ρ and αj are constants satisfying αj > −1. In this paper, we derive the topological-degree-counting formula for noncritical values of ρ. We also give several applications of this formula, including existence of the curvature + 1 metric with conic singularities, doubly periodic solutions of electroweak theory, and a special case of self-gravitating strings. © 2015 Wiley Periodicals, Inc.

Global Existence for Capillary Water Waves

Abstract: Consider the capillary water waves equations, set in the whole space with infinite depth, and consider small data (i.e., sufficiently close to zero velocity, and constant height of the water). We prove global existence and scattering. The proof combines in a novel way the energy method with a cascade of energy estimates, the space-time resonance method and commuting vector fields. © 2015 Wiley Periodicals, Inc.

Unconditional Uniqueness for the Cubic Gross-Pitaevskii Hierarchy via Quantum de Finetti

Abstract: We present a new, simpler proof of the unconditional uniqueness of solutions to the cubic Gross-Pitaevskii hierarchy in . One of the main tools in our analysis is the quantum de Finetti theorem. Our uniqueness result is equivalent to the one established in the celebrated works of Erdős, Schlein, and Yau. © 2015 Wiley Periodicals, Inc.

Global Regularity for Some Oldroyd-B Type Models

Abstract: We investigate some critical models for visco-elastic flows of Oldroyd-B type in dimension 2. We use a transformation that exploits the Oldroyd-B structure to prove an L∞ bound on the vorticity which allows us to prove global regularity for our systems.© 2014 Wiley Periodicals, Inc.

General β‐Jacobi Corners Process and the Gaussian Free Field

Abstract: We prove that the two-dimensional Gaussian free field describes the asymptotics of global fluctuations of a multilevel extension of the general β-Jacobi random matrix ensembles. Our approach is based on the connection of the Jacobi ensembles to a degeneration of the Macdonald processes that parallels the degeneration of the Macdonald polynomials to the Heckman-Opdam hypergeometric functions (of type A). We also discuss the β ∞ limit. © 2015 Wiley Periodicals, Inc.

Critical Sets of Elliptic Equations

Abstract: Given a solution u to a linear, homogeneous, second-order elliptic equation with Lipschitz coefficients, we introduce techniques for giving improved estimates of the critical set (u)u {x :|δu|(x) = 0}, as well as the first estimates on the effective critical set r(u), which roughly consists of points x such that the gradient of u is small somewhere on Br(x) compared to the nonconstancy of u. The results are new even for harmonic functions on ℝn. Given such a u, the standard first-order stratification {lk} of u separates points x based on the degrees of symmetry of the leading-order polynomial of u-u(x). In this paper we give a quantitative stratification of u, which separates points based on the number of almost symmetries of approximate leading-order polynomials of u at various scales. We prove effective estimates on the volume of the tubular neighborhood of each , which lead directly to (n-2 + ɛ)-Minkowski type estimates for the critical set of u. With some additional regularity assumptions on the coefficients of the equation, we refine the estimate to give new proofs of the uniform (n-2)-Hausdorff measure estimate on the critical set and singular sets of u.© 2014 Wiley Periodicals, Inc.

Global C2‐Estimates for Convex Solutions of Curvature Equations

Abstract: We establish a C2 a priori estimate for convex hypersurfaces whose principal curvatures κ=(κ1,…, κn) satisfy σk(κ(X))=f(X,ν(X)), the Weingarten curvature equation. We also obtain such an estimate for admissible 2-convex hypersurfaces in the case k=2. Our estimates resolve a longstanding problem in geometric fully nonlinear elliptic equations.© 2015 Wiley Periodicals, Inc.

The Small Deborah Number Limit of the Doi‐Onsager Equation to the Ericksen‐Leslie Equation

Abstract: We present a rigorous derivation of the Ericksen-Leslie equation starting from the Doi-Onsager equation. As in the fluid dynamic limit of the Boltzmann equation, we first make the Hilbert expansion for the solution of the Doi-Onsager equation. The existence of the Hilbert expansion is connected to an open question whether the energy of the Ericksen- Leslie equation is dissipated. We show that the energy is dissipated for the Ericksen-Leslie equation derived from the Doi-Onsager equation. The most difficult step is to prove a uniform bound for the remainder in the Hilbert expansion. This question is connected to the spectral stability of the linearized Doi-Onsager operator around a critical point. By introducing two important auxiliary operators, the detailed spectral information is obtained for the linearized operator around all critical points. However, these are not enough to justify the small Deborah number limit for the inhomogeneous Doi-Onsager equation, since the elastic stress in the velocity equation is also strongly singular. For this, we need to establish a precise lower bound for a bilinear form associated with the linearized operator. In the bilinear form, the interactions between the part inside the kernel and the part outside the kernel of the linearized operator are very complicated. We find a coordinate transform and introduce a five dimensional space called the Maier-Saupe space such that the interactions between two parts can been seen explicitly by a delicate argument of completing the square. However, the lower bound is very weak for the part inside the Maier-Saupe space. In order to apply them to the error estimates, we have to analyze the structure of the singular terms and introduce a suitable energy functional.

Local Propagation of Impulsive GravitationalWaves

Abstract: In this paper, we initiate the rigorous mathematical study of the problem of impulsive gravitational spacetime waves. We construct such spacetimes as solutions to the characteristic initial value problem of the Einstein vacuum equations with a data curvature delta singularity. We show that in the resulting spacetime, the delta singularity propagates along a characteristic hypersurface, while away from that hypersurface the spacetime remains smooth. Unlike the known explicit examples of impulsive gravitational spacetimes, this work in particular provides the first construction of an impulsive gravitational wave of compact extent and does not require any symmetry assumptions. The arguments in the present paper also extend to the problem of existence and uniqueness of solutions to a larger class of nonregular characteristic data. © 2015 Wiley Periodicals, Inc.

On the Soliton Resolution for Equivariant Wave Maps to the Sphere

Abstract: We consider finite energy corotational wave maps with target manifold S2. We prove that for a sequence of times, they decompose as a sum of decoupled harmonic maps in the light cone, and a smooth wave map (in the blowup case) or a linear scattering term (in the global case), up to an error which tends to 0 in the energy space. © 2015 Wiley Periodicals, Inc.

A General Existence Theorem for Embedded Minimal Surfaces with Free Boundary

Abstract: In this paper, we prove a general existence theorem for properly embedded minimal surfaces with free boundary in any compact Riemannian 3-manifold M with boundary ∂M. These minimal surfaces are either disjoint from ∂M or meet ∂M orthogonally. The main feature of our result is that there is no assumptions on the curvature of M or convexity of ∂M. We prove the boundary regularity of the minimal surfaces at their free boundaries. Furthermore, we define a topological invariant, the filling genus, for compact 3-manifolds with boundary and show that we can bound the genus of the minimal surface constructed above in terms of the filling genus of the ambient manifold M. Our proof employs a variant of the min-max construction used by Colding and De Lellis on closed embedded minimal surfaces, which were first developed by Almgren and Pitts.© 2014 Wiley Periodicals, Inc.

Coercivity of Combined Boundary Integral Equations in High-Frequency Scattering

Abstract: We prove that the standard second-kind integral equation formulation of the exterior Dirichlet problem for the Helmholtz equation is coercive (i.e., sign-definite) for all smooth convex domains when the wavenumber k is sufficiently large. (This integral equation involves the so-called combined potential, or combined field, operator.) This coercivity result yields k-explicit error estimates when the integral equation is solved using the Galerkin method, regardless of the particular approximation space used (and thus these error estimates apply to several hybrid numerical-asymptotic methods developed recently). Coercivity also gives k-explicit bounds on the number of GMRES iterations needed to achieve a prescribed accuracy when the integral equation is solved using the Galerkin method with standard piecewise-polynomial subspaces. The coercivity result is obtained by using identities for the Helmholtz equation originally introduced by Morawetz in her work on the local energy decay of solutions to the wave equation. © 2015 Wiley Periodicals, Inc.

Asymptotic Localization of Energy in Nondisordered Oscillator Chains

Abstract: We study two popular one-dimensional chains of classical anharmonic oscillators: the rotor chain and a version of the discrete nonlinear Schrodinger chain. We assume that the interaction between neighboring oscillators, controlled by the parameter ɛ > 0, is small. We rigorously establish that the thermal conductivity of the chains has a nonperturbative origin with respect to the coupling constant ɛ, and we provide strong evidence that it decays faster than any power law in ɛ as ɛ → 0. The weak coupling regime also translates into a high-temperature regime, suggesting that the conductivity vanishes faster than any power of the inverse temperature. To our knowledge, it is the first time that a clear connection has been established between KAM-like phenomena and thermal conductivity. © 2015 Wiley Periodicals, Inc.

Strong Asymptotics of the Orthogonal Polynomials with Respect to a Measure Supported on the Plane

Abstract: We consider the orthogonal polynomials { Pn(z) } with respect to the measure | z−a |2Nce−N| z |2dA(z) over the whole complex plane. We obtain the strong asymptotic of the orthogonal polynomials in the complex plane and the location of their zeros in a scaling limit where n grows to infinity with N. The asymptotics are described in terms of three (probability) measures associated with the problem. The first measure is the limit of the counting measure of zeros of the polynomials, which is captured by the g-function much in the spirit of ordinary orthogonal polynomials on the real line. The second measure is the equilibrium measure that minimizes a certain logarithmic potential energy, supported on a region K of the complex plane. The third measure is the harmonic measure of Kc with a pole at ∞. This appears as the limit of the probability measure given (up to the normalization constant) by the squared modulus of the nth orthogonal polynomial times the orthogonality measure, i.e., | Pn(z) |2| z−a |2Nce−N| z |2dA(z). The compact region K that is the support of the second measure undergoes a topological transition under the variation of the parameter t=n/N; in a double scaling limit near the critical point given by tc=a(a+2c), we observe the Hastings-McLeod solution to Painleve II in the asymptotics of the orthogonal polynomials. © 2014 Wiley Periodicals, Inc.

Global Uniqueness for an Inverse Stochastic Hyperbolic Problem with Three Unknowns

Abstract: This paper is addressed to an inverse stochastic hyperbolic problem with three unknowns, i.e., a random force intensity, an initial displacement, and an initial velocity. The global uniqueness for this inverse problem is proved by means of a new global Carleman estimate for the stochastic hyperbolic equation. It is found that both the formulation of stochastic inverse problems and the tools to solve them differ considerably from their deterministic counterpart. © 2015 Wiley Periodicals, Inc.

Characteristic Classes of Singular Toric Varieties

Abstract: In this paper we compute the motivic Chern classes and homology Hirzebruch characteristic classes of (possibly singular) toric varieties, which in the context of complete toric varieties fit nicely with a generalized Hirzebruch-Riemann-Roch theorem. As important special cases, we obtain new (or recover well-known) formulae for the Baum-Fulton-MacPherson Todd (or MacPherson's Chern) classes of toric varieties, as well as for the Thom-Milnor L-classes of simplicial projective toric varieties. We present two different perspectives for the computation of these characteristic classes of toric varieties. First, we take advantage of the torus-orbit decomposition and the motivic properties of the motivic Chern and respectively homology Hirzebruch classes to express the latter in terms of dualizing sheaves and respectively the (dual) Todd classes of closures of orbits. This method even applies to torus-invariant subspaces of a given toric variety. The obtained formula is then applied to weighted lattice-point counting in lattice polytopes and their subcomplexes, yielding generalized Pick-type formulae. Second, in the case of simplicial toric varieties, we compute our characteristic classes by using the Lefschetz-Riemann-Roch theorem of Edidin-Graham in the context of the geometric quotient description of such varieties. In this setting, we define mock Hirzebruch classes of simplicial toric varieties (which specialize to the mock Chern, mock Todd, and mock L-classes of such varieties) and investigate the difference between the (actual) homology Hirzebruch class and the mock Hirzebruch class. We show that this difference is localized on the singular locus, and we obtain a formula for it in which the contribution of each singular cone is identified explicitly. Finally, the two methods of computing characteristic classes are combined for proving several characteristic class formulae originally obtained by Cappell and Shaneson in the early 1990s.© 2015 Wiley Periodicals, Inc.

Large Deviations from a Stationary Measure for a Class of Dissipative PDEs with Random Kicks

Abstract: We study a class of dissipative PDEs perturbed by a bounded random kick force. It is assumed that the random force is nondegenerate, so that the Markov process obtained by the restriction of solutions to integer times has a unique stationary measure. The main result of the paper is a large deviations principle for occupation measures of the Markov process in question. The proof is based on Kifer's large-deviation criterion, a coupling argument for Markov processes, and an abstract result on large-time asymptotic for generalized Markov semigroups. © 2015 Wiley Periodicals, Inc.

Partial Regularity for Singular Solutions to the Monge-Ampère Equation

Abstract: We prove that solutions to the Monge-Ampere inequality in ℝn are strictly convex away from a singular set of Hausdorff (n-1)-dimensional measure zero. Furthermore, we show this is optimal by constructing solutions to det D2u = 1 with singular set of Hausdorff dimension as close as we like to n-1. As a consequence we obtain W2,1 regularity for the Monge-Ampere equation with bounded right-hand side and unique continuation for the Monge-Ampere equation with sufficiently regular right-hand side. © 2015 Wiley Periodicals, Inc.

Blister Patterns and Energy Minimization in Compressed Thin Films on Compliant Substrates

Abstract: This paper is motivated by the complex blister patterns sometimes seen in thin elastic films on thick, compliant substrates. These patterns are often induced by an elastic misfit that compresses the film. Blistering permits the film to expand locally, reducing the elastic energy of the system. It is therefore natural to ask: what is the minimum elastic energy achievable by blistering on a fixed area fraction of the substrate? This is a variational problem involving both the elastic deformation of the film and substrate and the geometry of the blistered region. It involves three small parameters: the nondimensionalized thickness of the film, the compliance ratio of the film/substrate pair, and the mismatch strain. In formulating the problem, we use a small-slope (Foppl–von Karman) approximation for the elastic energy of the film, and a local approximation for the elastic energy of the substrate. For a one-dimensional version of the problem, we obtain “matching” upper and lower bounds on the minimum energy, in the sense that both bounds have the same scaling behavior with respect to the small parameters. The upper bound is straightforward and familiar: it is achieved by periodic blistering on a specific length scale. The lower bound is more subtle, since it must be proved without any assumption on the geometry of the blistered region. For a two-dimensional version of the problem, our results are less complete. Our upper and lower bounds only “match” in their scaling with respect to the nondimensionalized thickness, not in the dependence on the compliance ratio and the mismatch strain. The lower bound is an easy consequence of our one-dimensional analysis. The upper bound considers a two-dimensional lattice of blisters and uses ideas from the literature on the folding or “crumpling” of a confined elastic sheet. Our main two-dimensional result is that in a certain parameter regime, the elastic energy of this lattice is significantly lower than that of a few large blisters. © 2015 Wiley Periodicals, Inc.

Nontopological Condensates for the Self-Dual Chern-Simons-Higgs Model

Abstract: For the abelian self-dual Chern-Simons-Higgs model we address existence issues of periodic vortex configurations – the so-called condensates– of non-topological type as k ! 0, where k > 0 is the Chern-Simons parameter. We provide a positive answer to the long-standing problem on the existence of non-topological condensates with magnetic field concentrated at some of the vortex points (as a sum of Dirac measures) as k ! 0, a question which is of definite physical interest.

On the space of Kahler potentials

Abstract: We consider the geodesic equation for the generalized Kahler potential with only mixed second derivatives bounded. We show that given two such generalized Kahler potentials, there is a unique geodesic segment such that for each point on the geodesic, the generalized Kahler potential has uniformly bounded mixed second derivatives (in manifold directions). This generalizes a fundamental theorem of Chen (2000) on the space of Kahler potentials.© 2014 Wiley Periodicals, Inc.

Quasi‐Periodic Almost‐Collision Orbits in the Spatial Three‐Body Problem

Abstract: In a system of particles, quasi-periodic almost-collision orbits are collisionless orbits along which two bodies become arbitrarily close to each other - the lower limit of their distance is zero but the upper limit is strictly positive - and which are quasi-periodic in a regularized system up to a change of time. The existence of such orbits was shown in the restricted planar circular three-body problem by A. Chenciner and J. Llibre, and later, in the planar three-body problem by J. Fejoz. In the spatial three-body problem, the existence of a set of positive measure of such orbits was predicted by C. Marchal. In this article, we present a proof of this fact.