Abstract This paper concerns the modified fractional Korteweg-de Vries (modified fKdV) and fractional cubic nonlinear Schrödinger (fNLS) equations, with the dispersions |D|α∂x and |D|α+1, respectively. We prove the global existence of small solutions for both the Cauchy problems to the modified fKdV and fNLS equations, with a modified scattering which has a logarithmic phase correction. Our results

Abstract On finite metric graphs the set of all realizations of the Laplace operator in the edgewise defined L2-spaces are studied. These are defined by coupling boundary conditions at the vertices most of which define non-self-adjoint operators. In [Hussein, Krejčiřík, Siegl, Trans. Amer. Math. Soc., 367(4):2921–2957, 2015] a notion of regularity of boundary conditions by means of the Cayley transform

Abstract Let (Xt)t≥0 be the overdamped Langevin process on Rd, i.e. the solution of the stochastic differential equation dXt=−∇f(Xt) dt+h dBt. Let Ω⊂Rd be a bounded domain. In this work, when X0=x∈Ω, we derive new sharp asymptotic equivalents (with optimal error terms) in the limit h→0 of the mean exit time from Ω of the process (Xt)t≥0 (which is the solution of (−h2Δ+∇f·∇)w=1 in Ω and w=0 on ∂Ω),

Abstract We consider the semilinear problem Δu=λ+(− log u+)1{u>0}−λ−(− log u−)1{u<0} in B1, where B1 is the unit ball in Rn and assume λ+,λ−>0. Using a monotonicity formula argument, we prove an optimal regularity result for solutions: ∇u is a log-Lipschitz function. This problem introduces two main difficulties. The first is the lack of invariance in the scaling and blow-up of the problem. The other

Abstract We consider the scattering problem governed by the Helmholtz equation with inhomogeneity in both “conductivity” in the divergence form and “potential” in the lower order term. The support of the inhomogeneity is assumed to contain a convex corner. We prove that, due to the presence of such corner under appropriate assumptions on the potential and conductivity in the vicinity of the corner

Abstract We study the self-similar behavior of the exchange-driven growth model, which describes a process in which pairs of clusters, consisting of an integer number of monomers, interact through the exchange of a single monomer. The rate of exchange is given by an interaction kernel K(k, l) which depends on the sizes k and l of the two interacting clusters and is assumed to be of product form (k

Abstract For some deterministic nonlinear PDEs on the torus whose solutions may blow up in finite time, we show that, under suitable conditions on the nonlinear term, the blow-up is delayed by multiplicative noise of transport type in a certain scaling limit. The main result is applied to the 3D Keller–Segel, 3D Fisher–KPP, and 2D Kuramoto–Sivashinsky equations, yielding long-time existence for large

Abstract The Ray–Singer analytic torsion is the classical zeta-trace of a sum of logarithm operators of p-form Laplacians on the de Rham complex. In this article, we examine residue-analytic torsion, defined using the residue trace in place of the zeta function regularised trace.

Abstract This paper is concerned with the existence of normalized solutions of the nonlinear Schrödinger equation −Δu+V(x)u+λu=|u|p−2u  in RN in the mass supercritical and Sobolev subcritical case 2+4N

Abstract We prove local well-posedness for the L2 critical generalized Zakharov-Kuznetsov equation in Hs, s∈(3/4,1). We also prove that the equation is “almost well-posedness” for initial data u0∈Hs, s∈[1,2), in the sense that the solution belongs to a certain intersection C([0,T]:Hs(R3))∩XTs and is unique within that class, where we can ensure continuity of the data-to-solution map in an only slightly

Abstract We prove that the Debye screening length emerges in an infinitely extended plasma with uniform background described by the nonlinear Vlasov-Poisson equation, interacting with a point charge. While screened stationary states as well as the time-dependent problem are well-studied for the linearized equation, there are few rigorous results on screening in the nonlinear setting. As such, the results

Abstract We consider convex solutions of nonlinear elliptic equations which satisfy the structure condition of Bian and Guan. We prove a weak Harnack inequality for the eigenvalues of the Hessian of these solutions. This can be viewed as a quantitative version of the constant rank theorem.

Abstract In a recent joint paper with Hwang and Mitrea we have introduced new solvability methods for strongly elliptic second order systems in divergence form on a domain above a Lipschitz graph, satisfying Lp-boundary data for p near 2. The main novel aspect of our result is that it applies to operators with coefficients of limited regularity and applies to operators satisfying a natural Carleson

Abstract We prove a Lipschitz extension lemma in which the extension procedure simultaneously preserves the Lipschitz continuity for two nonequivalent distances. The two distances under consideration are the Euclidean distance and, roughly speaking, the geodesic distance along integral curves of a (possibly multi-valued) flow of a continuous vector field. The Lipschitz constant for the geodesic distance

Abstract We study the Ricci flow starting at an SU(2) cohomogeneity-1 metric g 0 on R 4 with monotone warping coefficients and whose restriction to any hypersphere is a Berger metric. If g 0 has bounded Hopf-fiber, curvature controlled by the size of the orbits and opens faster than a paraboloid in the directions orthogonal to the Hopf-fiber, then the flow converges to the Taub-NUT metric g TNUT in

Abstract We consider the Zakharov-Kutznesov (ZK) equation posed in R d , with d = 2 and 3. Both equations are globally well-posed in L 2 ( R d ) . In this article, we prove local energy decay of global solutions: if u(t) is a solution to ZK with data in L 2 ( R d ) , then lim  inf t → ∞ ∫ Ω d ( t ) u 2 ( x , t ) d x = 0 , for suitable regions of space Ω d ( t ) ⊆ R d around the origin, growing unbounded

Abstract The system leading to phase segregation in two-component Bose-Einstein condensates can be generalized to hyperfine spin states with a Rabi term coupling. This leads to domain wall solutions having a monotone structure for a non-cooperative system. We use the moving plane method to prove monotonicity and one-dimensionality of the phase transition solutions. This relies on totally new estimates

Abstract We solve the Neumann problem in the half space R + n + 1 , for higher order elliptic differential equations with variable self-adjoint t-independent coefficients, and with boundary data in the negative smoothness space W ̇ − 1 , p , where max ( 0 , 1 2 − 1 n − ε ) < 1 p < 1 2 . Our arguments are inspired by an argument of Shen and build on known well posedness results in the case p = 2. We

Abstract Consider the higher order parabolic operator ∂ t + ( − Δ x ) m and the higher order Schrödinger operator i − 1 ∂ t + ( − Δ x ) m in X = { ( t , x ) ∈ R 1 + n ;   | t | < A , | x n | < B } , where m and n are any positive integers. Under certain lower order and regularity assumptions, we prove that if solutions to the linear problems vanish when x n > 0 , then the solutions vanish in X. Such

Abstract We consider the mixed problem on the exterior of the unit ball in R n , n ≥ 2 , for a defocusing Schrödinger equation with a power nonlinearity | u | p − 1 u , with zero boundary data. Assuming that the initial data are non-radial, sufficiently small perturbations of large radial initial data, we prove that for all powers p > n + 6 the solution exists for all times, its Sobolev norms do not

Abstract For each pair ε=(ε1,ε2) of positive parameters, we define a perforated domain Ωε by making a small hole of size ε1ε2 in an open regular subset Ω of ℝn ( n≥3). The hole is situated at distance ε1 from the outer boundary ∂Ω of the domain. Thus, when ε→(0,0) both the size of the hole and its distance from ∂Ω tend to zero, but the size shrinks faster than the distance. Next, we consider a Dirichlet

Abstract We consider a class of equations in divergence form with a singular/degenerate weight −div(|y|aA(x,y)∇u)=|y|af(x,y) or div(|y|aF(x,y)). Under suitable regularity assumptions for the matrix A and f (resp. F) we prove Hölder continuity of solutions which are even in y∈R, and possibly of their derivatives up to order two or more (Schauder estimates). In addition, we show stability of the C0,α

Abstract We analyse the asymptotic behaviour of the eigenvalues and eigenvectors of a Steklov problem in a dumbbell domain consisting of two Lipschitz sets connected by a thin tube with vanishing width. All the eigenvalues are collapsing to zero, the speed being driven by some power of the width which multiplies the eigenvalues of a one dimensional problem. In two dimensions of the space, the behaviour

Abstract In this paper, we study the differential inclusion associated with the minimal surface system for two-dimensional graphs in R 2 + n . We prove regularity of W 1 , 2 solutions and a compactness result for approximate solutions of this differential inclusion in W 1 , p . Moreover, we make a perturbation argument to infer that for every R > 0, there exists α ( R ) > 0 such that R-Lipschitz stationary

Abstract We introduce a notion of uniform convergence for local and nonlocal curvatures. Then, we propose an abstract method to prove the convergence of the corresponding geometric flows, within the level set formulation. We apply such a general theory to characterize the limits of s-fractional mean curvature flows as s → 0 + and s → 1 − . In analogy with the s-fractional mean curvature flows, we introduce

Abstract We study localization properties of low-lying eigenfunctions ( − Δ + V ) ϕ = λ ϕ   in   Ω for rapidly varying potentials V in bounded domains Ω ⊂ R d . Filoche & Mayboroda introduced the landscape function ( − Δ + V ) u = 1 and showed that the function u has remarkable properties: localized eigenfunctions prefer to localize in the local maxima of u. Arnold, David, Filoche, Jerison & Mayboroda

Abstract The chemotaxis-Stokes system { n t + u · ∇ n = Δ n − ∇ · ( n ∇ c ) , c t + u · ∇ c = Δ c − n c , u t = Δ u + ∇ P + n ∇ ϕ ,     ∇ · u = 0 , ( ⋆ ) is considered in a bounded domain Ω ⊂ R 3 with smooth boundary. The corresponding solution theory is quite well-developed in the case when ( ⋆ ) is accompanied by homogeneous boundary conditions of no-flux type for n and c, and of Dirichlet type for

Abstract We characterize the denseness of the singular set of the distance function from a C 1 -hypersurface in terms of an inner ball condition and we address the problem of the existence of viscosity solutions of the Eikonal equation whose singular set (i.e. set of non-differentiability points) is not no-where dense.

Abstract In this article, we consider a class of prescribed Weingarten curvature equations. Under some sufficient conditions, we obtain an existence result by the standard degree theory based on the a priori estimates for the solutions to the prescribed Weingarten curvature equations.

Abstract We study two overdetermined elliptic boundary value problems on exterior domains (the complement of a ball and the complement of a solid cylinder in R 3 respectively). The Neumann condition is non-constant and involves the mean curvature of the boundary. We show there exists a family of bifurcation branches of domains which are small deformations of the complement of a ball and of the complement

Abstract We construct dynamical many-black-hole spacetimes with well-controlled asymptotic behavior as solutions of the Einstein vacuum equation with positive cosmological constant. We accomplish this by gluing Schwarzschild–de Sitter or Kerr–de Sitter black hole metrics into neighborhoods of points on the future conformal boundary of de Sitter space, under certain balance conditions on the black hole

Abstract In this paper, we consider the uniqueness of solutions to the 3d Navier-Stokes equations with initial vorticity given by ω 0 = α e z δ x = y = 0 , where δ x = y = 0 is the one dimensional Hausdorff measure of an infinite, vertical line and α ∈ R is an arbitrary circulation. This initial data corresponds to an idealized, infinite vortex filament. One smooth, mild solution is given by the self-similar

Abstract Consider an isotropic elastic medium Ω ⊂ R 3 whose Lamé parameters are piecewise smooth. In the elastic wave initial value inverse problem, we are given the solution operator for the elastic wave equation, but only outside Ω and only for initial data supported outside Ω. Using the recently introduced scattering control series in the acoustic case, we prove that piecewise smooth Lamé parameters

Abstract This paper is concerned with the intrinsic geometric structures of conductive transmission eigenfunctions. The geometric properties of interior transmission eigenfunctions were first studied in Blåsten, E., Liu, H. (2017). On vanishing near corners of transmission eigenfunctions. J. Funct. Anal. 273(11):3616–3632. It is shown in two scenarios that the interior transmission eigenfunction must

Abstract We analyze the Cauchy problem for symmetric hyperbolic equations with a time singularity of Fuchsian type and establish a global existence theory along with decay estimates for evolutions toward the singular time under a small initial data assumption. We, then, apply this theory to semilinear wave equations near spatial infinity on Minkowski and Schwarzschild spacetimes, and to the relativistic

Abstract It is shown that negative Einstein metrics are attractors of the Einstein-Klein-Gordon system. As an essential part of the proof we upgrade a technique that uses the continuity equation complementary to L2-estimates to control massive matter fields. In contrast to earlier applications of this idea we require a correction to the energy density to obtain sufficiently strong pointwise bounds

Abstract Under general assumptions on the velocity field, it is possible to construct a flow that is forward untangled. Once such a flow has been selected, the associated transport problem is well-posed.

Abstract We study the homogenization of nonlinear, first-order equations with highly oscillatory mixing spatio-temporal dependence. It is shown in a variety of settings that the homogenized equations are stochastic Hamilton-Jacobi equations with deterministic, spatially homogenous Hamiltonians driven by white noise in time. The paper also contains proofs of some general regularity and path stability

Abstract In this paper, we study stability of the conical Kähler-Ricci flows on Fano manifolds. That is, if there exists a conical Kähler-Einstein metric with cone angle 2 π β along the divisor, then for any β ′ sufficiently close to β, the corresponding conical Kähler-Ricci flow converges to a conical Kähler-Einstein metric with cone angle 2 π β ′ along the divisor. Here, we only use the condition

Abstract This article is concerned with the large-scale regularity in the homogenization of elliptic systems of elasticity with periodic high-contrast coefficients. We obtain the large-scale Lipschitz estimate that is uniform with respect to the contrast ratio δ 2 for 0 < δ < ∞ . Our study also covers the case of soft inclusions (δ = 0) as well as the case of stiff inclusions ( δ = ∞ ). The large-scale

Abstract We consider linear combinations of eigenfunctions of the Laplace-Beltrami operator on a compact Riemannian manifold (M, g) and investigate a density property of their zero sets. More precisely, let f = ∑ k = 1 m a k ϕ λ j k , where − Δ g ϕ λ = λ ϕ λ . Denoting by Zf the zero-set of f, we show that for any x ∈ M , dist ( x , Z f ) ≤ C ( m ) λ j 1 − 1 / 2 . The proof is based on a new integral

Abstract We use a Lagrangian regularity perspective to discuss resolvent estimates near zero energy on Riemannian scattering, i.e. asymptotically conic, spaces, and their generalizations. In addition to the Lagrangian perspective we introduce and use a resolved pseudodifferential algebra to deal with zero energy degeneracies in a robust manner.

Abstract We use a Lagrangian perspective to show the limiting absorption principle on Riemannian scattering, i.e. asymptotically conic, spaces, and their generalizations. More precisely we show that, for non-zero spectral parameter, the ‘on spectrum’, as well as the ‘off-spectrum’, spectral family is Fredholm in function spaces which encode the Lagrangian regularity of generalizations of ‘outgoing

Abstract We determine both the magnetic potential and the electric potential from the exterior partial measurements of the Dirichlet-to-Neumann map in the fractional linear magnetic Calderón problem by using an integral identity. We also determine both the magnetic potential and the nonlinearity in the fractional semi-linear magnetic Calderón problem by using a first order linearization.

Abstract In this paper we study existence, nonexistence and classification of radial positive solutions of some weighted fully nonlinear equations involving Pucci extremal operators. Our results are entirely based on the analysis of the dynamics induced by an autonomous quadratic system which is obtained after a suitable transformation. This method allows to treat both regular and singular solutions

Abstract We prove the concavity of classical solutions to a wide class of degenerate elliptic differential equations on strictly convex domains of the unit sphere. The proof employs a suitable two-point maximum principle, a technique which originates in works of Korevaar, Kawohl and Kennington for equations on Euclidean domains. We emphasize that no differentiability of the differential operator is

Abstract In this paper, we discuss the general existence theory of Dirac-harmonic maps from closed surfaces via the heat flow for α-Dirac-harmonic maps and blow-up analysis. More precisely, given any initial map along which the Dirac operator has nontrivial minimal kernel, we first prove the short time existence of the heat flow for α-Dirac-harmonic maps. The obstacle to the global existence is the

Abstract We provide a rigorous derivation of Einstein’s formula for the effective viscosity of dilute suspensions of n rigid balls, n ≫ 1 , set in a volume of size 1. So far, most justifications were carried under a strong assumption on the minimal distance between the balls: d min ≥ c n − 1 3 , c > 0. We relax this assumption into a set of two much weaker conditions: one expresses essentially that

Abstract We consider the nonlinear Hartree and Vlasov equations around a translation-invariant (homogeneous) stationary state in infinite volume, for a short range interaction potential. For both models, we consider time-dependent solutions which have a finite relative energy with respect to the reference translation-invariant state. We prove the convergence of the Hartree solutions to the Vlasov ones

Abstract We study the large-time behavior of bounded from below solutions of parabolic viscous Hamilton-Jacobi Equations in the whole space RN in the case of superquadratic Hamiltonians. Existence and uniqueness of such solutions are shown in a very general framework, namely when the source term and the initial data are only bounded from below with an arbitrary growth at infinity. Our main result is

Abstract Considering the defocusing nonlinear Schrödinger equation (NLSE) in generic (bounded or unbounded) open sets U⊆Rn for n = 1, 2, and 3, we prove the regularity of weak, non-vanishing solutions at infinity or at the boundary of U. Our approach is based on suitably defined extension operators, along with a priori estimates for regular functions, under certain assumptions on the smoothness of

Abstract We consider the mixed Dirichlet-conormal problem on irregular domains in Rd. Two types of regularity results will be discussed: the W1,p regularity and a non-tangential maximal function estimate. The domain is assumed to be Reifenberg-flat, and the interfacial boundary is either Reifenberg-flat of co-dimension 2 or is locally sufficiently close to a Lipschitz function of m variables, where

Abstract We study the first Dirichlet eigenfunction of the Laplacian in a n-dimensional convex domain. For domains of a fixed inner radius, estimates of Chiti [1 Chiti, G. (1982). A reverse Hölder inequality for the eigenfunctions of linear second order elliptic operators. Z. angew. Math. Phys. 33(1):143–148.[Crossref], [Web of Science ®] , [Google Scholar],2 Chiti, G. (1982). An isoperimetric inequality

Abstract We study the Schrödinger equation driven by a weak Brownian forcing, and derive Gaussian fluctuations in the form of a time-inhomogeneous Ornstein-Uhlenbeck process. As a result, when evaluated at a fixed frequency, the intensity of the incoherent wave is of exponential distribution.

Abstract We prove the applicability of the Weighted Energy-Dissipation (WED) variational principle to nonlinear parabolic stochastic partial differential equations in abstract form. The WED principle consists in the minimization of a parameter-dependent convex functional on entire trajectories. Its unique minimizers correspond to elliptic-in-time regularizations of the stochastic differential problem

Abstract We obtain new semiclassical estimates for pseudodifferential operators with low regular symbols. Such symbols appear naturally in a Cauchy Problem related to recent weak solutions to the unstable Muskat problem constructed via convex integration. In particular, our new estimates reveal the tight relation between the speed of opening of the mixing zone and the regularity of the interphase.

Abstract We investigate the existence and uniqueness issues of the 3D incompressible Hall-magnetohydrodynamic system supplemented with initial velocity u0 and magnetic field B0 in critical regularity spaces. In the case where u0, B0 and the current J0:=∇×B0 belong to the homogeneous Besov space Ḃp,13p−1, 1≤p<∞, and are small enough, we establish a global result and the conservation of higher regularity

Abstract We consider the barotropic Navier–Stokes system describing the motion of a viscous compressible fluid interacting with the outer world through general in/out flux boundary conditions. We consider a hard-sphere type pressure EOS and show that all trajectories eventually enter a bounded absorbing set. In particular, the associated ω − limit sets are compact and support a stationary statistical

Abstract In this manuscript, we consider a non-local porous medium equation with non-local diffusion effects given by a fractional heat operator { ∂ t u = div ( u ∇ p ) , ∂ t p = − ( − Δ ) s p + u β , in two space dimensions for β > 1 , 1 β < s < 1 . Global in time existence of weak solutions is shown by employing a time semi-discretization of the equations, an energy inequality and the Div-Curl lemma

In subdomains of R d , we consider uniformly elliptic equations H ( v ( x ) , D v ( x ) , D 2 v ( x ) , x ) = 0 with the growth of H with respect to | D v | controlled by the product of a function from Ld and | D v | . The dependence of H on x is assumed to be of BMO type. Among other things we prove that there exists d 0 ∈ ( d / 2 , d ) such that for any p ∈ ( d 0 , d ) the equation with prescribed