Global well-posedness for the one-phase Muskat problem

The free boundary problem for a two-dimensional fluid permeating a porous medium is studied. This is known as the one-phase Muskat problem and is mathematically equivalent to the vertical Hele-Shaw problem driven by gravity force. We prove that if the initial free boundary is the graph of a periodic Lipschitz function, then there exists a global-in-time Lipschitz solution in the strong $L_t^{\infty }{L}_x^2$ sense and it is the unique viscosity solution. The proof requires quantitative estimates for layer potentials and pointwise elliptic regularity in Lipschitz domains. This is the first construction of unique global strong solutions for the Muskat problem with initial data of arbitrary size.     

Fully Nonlinear Equations with Applications to Grad Equations in Plasma Physics

In this paper we generalize an equation studied by Mossino and Temam in [7], to the fully nonlinear case. This equation arises in plasma physics as an approximation to Grad equations, which were introduced by Harold Grad in [4], to model the behavior of plasma confined in a toroidal vessel called TOKAMAK. We prove existence of a -viscosity solution and regularity up to for any (we improve this regularity near the boundary). The difficulty of this problem lies in the right-hand side which involves the measure of the superlevel sets, making the problem nonlocal. © 2021 Wiley Periodicals LLC.     

Upper Bounds on Liouville First-Passage Percolation and Watabiki's Prediction

Given a planar continuum Gaussian free field h^𝒰 in a domain 𝒰 with Dirichlet boundary condition and any δ > 0, we let be a real-valued smooth Gaussian process where is the average of h^𝒰 along a circle of radius δ with center v. For γ > 0, we study the Liouville first-passage percolation (in scale δ), i.e., the shortest path distance in 𝒰 where the weight of each path P is given by . We show that the distance between two typical points is for all sufficiently small but fixed γ > 0 and some constant c^* > 0. In addition, we obtain similar upper bounds on the Liouville first-passage percolation for discrete Gaussian free fields, as well as the Liouville graph distance, which roughly speaking is the minimal number of euclidean balls with comparable Liouville quantum gravity measure whose union contains a continuous path between two endpoints. Our results contradict some reasonable interpretations of Watabiki's prediction (1993) on the random distance of Liouville quantum gravity at high temperatures.© 2019 Wiley Periodicals, Inc.     

Prescribing Morse Scalar Curvatures: Pinching and Morse Theory

We consider the problem of prescribing conformally the scalar curvature on compact manifolds of positive Yamabe class in dimension . We prove new existence results using Morse theory and some analysis on blowing-up solutions under suitable pinching conditions on the curvature function. We also provide new nonexistence results showing the sharpness of some of our assumptions, both in terms of the dimension and of the Morse structure of the prescribed function. © 2021 Wiley Periodicals, Inc.     

Traveling Wave Solutions to the Free Boundary Incompressible Navier-Stokes Equations

In this paper we study a finite-depth layer of viscous incompressible fluid in dimension $n\ge 2$, modeled by the Navier-Stokes equations. The fluid is assumed to be bounded below by a flat rigid surface and above by a free, moving interface. A uniform gravitational field acts perpendicularly to the flat surface, and we consider the cases with and without surface tension acting on the free interface. In addition to these gravity-capillary effects, we allow for a second force field in the bulk and an external stress tensor on the free interface, both of which are posited to be in traveling wave form, i.e., time-independent when viewed in a coordinate system moving at a constant velocity parallel to the rigid lower boundary. We prove that, with surface tension in dimension $n\ge 2$ and without surface tension in dimension $n=2$, for every nontrivial traveling velocity there exists a nonempty open set of force and stress data that give rise to traveling wave solutions. While the existence of inviscid traveling waves is well-known, to the best of our knowledge this is the first construction of viscous traveling wave solutions. Our proof involves a number of novel analytic ingredients, including: the study of an overdetermined Stokes problem and its underdetermined adjoint problem, a delicate asymptotic development of the symbol for a normal-stress to normal-Dirichlet map defined via the Stokes operator, a new scale of specialized anisotropic Sobolev spaces, and the study of a pseudodifferential operator that synthesizes the various operators acting on the free surface functions. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.     

Geodesics in the Space of Kähler Cone Metrics II: Uniqueness of Constant Scalar Curvature Kähler Cone Metrics

In this article, we give a complete construction of geodesics in the space of Kähler cone metrics (cone geodesics), and we address the problem on the uniqueness of constant scalar curvature Kähler (cscK) cone metrics when the cone angle β stays in the whole interval (0, 1] . The part requires new weighted function spaces and new analytic techniques. We determine the asymptotic behavior of both cone geodesics and cscK cone metrics, prove the reductivity of the automorphism group, and establish the linear theory for the Lichnerowicz operator, which immediately implies the openness of the path deforming the cone angles of cscK cone metrics. © 2019 Wiley Periodicals, Inc.     

Compressible Navier-Stokes equations with ripped density

We are concerned with the Cauchy problem for the two-dimensional compressible Navier-Stokes equations  supplemented with general H¹ initial velocity and bounded initial density not necessarily strictly positive: it may be the characteristic function of any set, for instance. In the perfect gas case, we establish global-in-time existence and uniqueness, provided the volume (bulk) viscosity coefficient is large enough. For more general pressure laws (like e.g., $P={\rho }^{\gamma }$ with $\gamma >1$), we still get global existence, but uniqueness remains an open question. As a by-product of our results, we give a rigorous justification of the convergence to the inhomogeneous incompressible Navier-Stokes equations when the bulk viscosity tends to infinity. In the three-dimensional case, similar results are proved for short time without restriction on the viscosity, and for large time if the initial velocity field is small enough.     

A De Giorgi–Type Conjecture for Minimal Solutions to a Nonlinear Stokes Equation

We study the one-dimensional symmetry of solutions to the nonlinear Stokes equation which are periodic in the d − 1 last variables (living on the torus 𝕋^d−1) and globally minimize the corresponding energy in Ω = ℝ × 𝕋^d−1, i.e., Namely, we find a class of nonlinear potentials W ≥ 0 such that any global minimizer u of E connecting two zeros of W as x₁ → ± ∞ is one-dimensional; i.e., u depends only on the x₁ -variable. In particular, this class includes in dimension d = 2 the nonlinearities with w being a harmonic function or a solution to the wave equation, while in dimension d ≥ 3 , this class contains a perturbation of the Ginzburg-Landau potential as well as potentials W having d + 1 wells with prescribed transition cost between the wells. For that, we develop a theory of calibrations relying on the notion of entropy (coming from scalar conservation laws). We also study the problem of the existence of global minimizers of E for general potentials W providing in particular compactness results for uniformly finite energy maps u in Ω connecting two wells of W as x₁ → ± ∞ . © 2019 Wiley Periodicals, Inc.     

From 1 to 6: A Finer Analysis of Perturbed Branching Brownian Motion

The logarithmic correction for the order of the maximum for two-speed branching Brownian motion changes discontinuously when approaching slopes , which corresponds to standard branching Brownian motion. In this article we study this transition more closely by choosing and . We show that the logarithmic correction for the order of the maximum now smoothly interpolates between the correction in the i.i.d. case , and when . This is due to the localization of extremal particles at the time of speed change, which depends on α and differs from the one in standard branching Brownian motion. We also establish in all cases the asymptotic law of the maximum and characterize the extremal process, which turns out to coincide essentially with that of standard branching Brownian motion. © 2020 the Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC     

Min-Oo Conjecture for Fully Nonlinear Conformally Invariant Equations

In this paper we show rigidity results for supersolutions to fully nonlinear, elliptic, conformally invariant equations on subdomains of the standard n -sphere under suitable conditions along the boundary. We emphasize that our results do not assume concavity on the fully nonlinear equations we will work with. This proves rigidity for compact, connected, locally conformally flat manifolds (M, g) with boundary such that the eigenvalues of the Schouten tensor satisfy a fully nonlinear elliptic inequality and whose boundary is isometric to a geodesic sphere ∂D(r) , where D(r) denotes a geodesic ball of radius r ∈ (0, π/2] in , and totally umbilical with mean curvature bounded below by the mean curvature of this geodesic sphere. Under the above conditions, (M, g) must be isometric to the closed geodesic ball . As a side product, in dimension 2 our methods provide a new proof to Toponogov's theorem about rigidity of compact surfaces carrying a shortest simple geodesic. Roughly speaking, Toponogov's theorem is equivalent to a rigidity theorem for spherical caps in the hyperbolic three-space ℍ³ . In fact, we extend it to obtain rigidity for supersolutions to certain Monge-Ampère equations. © 2019 Wiley Periodicals, Inc.     

Free Boundary Regularity in the Parabolic Fractional Obstacle Problem

The parabolic obstacle problem for the fractional Laplacian naturally arises in American option models when the asset prices are driven by pure‐jump Lévy processes. In this paper we study the regularity of the free boundary. Our main result establishes that, when , the free boundary is a C^1,α graph in x and t near any regular free boundary point . Furthermore, we also prove that solutions u are C^{1 + s} in x and t near such points, with a precise expansion of the form (1) with , and . © 2018 Wiley Periodicals, Inc.     

Upper Tail Large Deviations of Regular Subgraph Counts in Erdős-Rényi Graphs in the Full Localized Regime

For a -regular connected graph H the problem of determining the upper tail large deviation for the number of copies of H in , an Erdős-Rényi graph on n vertices with edge probability p, has generated significant interest. For and , where is the number of vertices in H, the upper tail large deviation event is believed to occur due to the presence of localized structures. In this regime the large deviation event that the number of copies of H in exceeds its expectation by a constant factor is predicted to hold at a speed , and the rate function is conjectured to be given by the solution of a mean-field variational problem. After a series of developments in recent years, covering progressively broader ranges of p, the upper tail large deviations for cliques of fixed size were proved by Harel, Mousset, and Samotij in the entire localized regime. This paper establishes the conjecture for all connected regular graphs in the whole localized regime. © 2021 Wiley Periodicals LLC.     

Unique Continuation at the Boundary for Harmonic Functions in C1 Domains and Lipschitz Domains with Small Constant

Let be a domain, or more generally, a Lipschitz domain with small local Lipschitz constant. In this paper it is shown that if u is a function harmonic in and continuous in , which vanishes in a relatively open subset ; moreover, the normal derivative vanishes in a subset of with positive surface measure; then u is identically zero. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.     

Direct Epiperimetric Inequalities for the Thin Obstacle Problem and Applications

We introduce a new logarithmic epiperimetric inequality for the 2m -Weiss energy in any dimension, and we recover with a simple direct approach the usual epiperimetric inequality for the 3/2-Weiss energy. In particular, even in the latter case, unlike the classical statements, we do not assume any a priori closeness to a special class of homogeneous functions. In dimension 2 , we also prove for the first time the classical epiperimetric inequality for the (2m − 1/2)-Weiss energy, thus covering all the admissible energies. As a first application, we classify the global λ -homogeneous minimizers of the thin obstacle problem, with , showing as a consequence that the frequencies 3/2 and 2m are isolated and thus improving on the previously known results. Moreover, we give an example of a new family of (2m − 1/2) -homogeneous minimizers in dimension higher than 2 . Second, we give a short and self-contained proof of the regularity of the free boundary of the thin obstacle problem, previously obtained by Athanasopoulos, Caffarelli, and Salsa (2008) for regular points and Garofalo and Petrosyan (2009) for singular points. In particular, we improve the C¹ regularity of the singular set with frequency 2m by an explicit logarithmic modulus of continuity. © 2019 Wiley Periodicals, Inc.     

Global Regularity of 2D Density Patches for Viscous Inhomogeneous Incompressible Flow with General Density: Low Regularity Case

This paper presents some progress toward an open question proposed by P.-L. Lions [26] concerning the propagation of regularities of density patches for viscous inhomogeneous incompressible flow. We first establish the global-in-time well-posedness of the two-dimensional inhomogeneous incompressible Navier-Stokes system with initial density . Here is any pair of positive constants and Ω₀ is a bounded, simply connected domain. We then prove that for any positive time t, the density , with the domain Ω(t) preserving the -boundary regularity. © 2018 Wiley Periodicals, Inc.     

A Liouville Theorem for the Complex Monge-Ampère Equation on Product Manifolds

Let Y be a closed Calabi-Yau manifold. Let ω be the Kähler form of a Ricci-flat Kähler metric on . We prove that if ω is uniformly bounded above and below by constant multiples of , where is the standard flat Kähler form on and ω_Y is any Kähler form on Y, then ω is a product Kähler form up to a certain automorphism of . © 2018 Wiley Periodicals, Inc.     

Online Prediction with History-Dependent Experts: The General Case

We study the problem of prediction of binary sequences with expert advice in the online setting, which is a classic example of online machine learning. We interpret the binary sequence as the price history of a stock, and view the predictor as an investor, which converts the problem into a stock prediction problem. In this framework, an investor, who predicts the daily movements of a stock, and an adversarial market, who controls the stock, play against each other over N turns. The investor combines the predictions of $n\ge 2$ experts in order to make a decision about how much to invest at each turn, and aims to minimize their regret with respect to the best-performing expert at the end of the game. We consider the problem with history-dependent experts, in which each expert uses the previous d days of history of the market in making their predictions. We prove that the value function for this game, rescaled appropriately, converges as $N\to \mathrm{\infty }$ at a rate of $O\left(N^{-1/6}\right)$ to the viscosity solution of a nonlinear degenerate elliptic PDE, which can be understood as the Hamilton-Jacobi-Issacs equation for the two-person game. As a result, we are able to deduce asymptotically optimal strategies for the investor. Our results extend those established by the first author and R.V. Kohn [14] for $n=2$ experts and $d\le 4$ days of history. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.     

Inviscid Limit of Vorticity Distributions in the Yudovich Class

We prove that given initial data , forcing and any T > 0, the solutions u^ν of Navier-Stokes converge strongly in for any p ∈ [1, ∞) to the unique Yudovich weak solution u of the Euler equations. A consequence is that vorticity distribution functions converge to their inviscid counterparts. As a by-product of the proof, we establish continuity of the Euler solution map for Yudovich solutions in the L^p vorticity topology. The main tool in these proofs is a uniformly controlled loss of regularity property of the linear transport by Yudovich solutions. Our results provide a partial foundation for the Miller-Robert statistical equilibrium theory of vortices as it applies to slightly viscous fluids. © 2020 Wiley Periodicals LLC.     

On Growth of Sobolev Norms for Energy Critical NLS on Irrational Tori: Small Energy Case

We consider the energy-critical nonlinear Schrödinger equation on a generic irrational torus. Using the long-time Strichartz estimates proved in [8], we establish polynomial upper bounds for higher Sobolev norms for solutions with small energy. © 2018 Wiley Periodicals, Inc.     

Normalized Harmonic Map Heat Flow

For any closed Riemannian manifold N we propose the normalized harmonic map heat flow as a means to obtain nonconstant harmonic maps , m ≥ 3 . © 2019 Wiley Periodicals, Inc.