Abstract In this paper, we study some properties of multi-solitons for the non-linear Schrödinger equations in Rd with general non-linearities. Multi-solitons have already been constructed in H1(Rd) in papers by Merle (1990), Martel and Merle (2006), and Côte, Martel and Merle (2011). We show here that multi-solitons are smooth, depending on the regularity of the non-linearity. We obtain also a result

Abstract We consider the half-wave equation with mass critical in two dimensions {iut=Du−|u|u, u(t0,x)=u0(x), First, we prove the existence of a family of traveling solitary waves. We then show the existence of finite-time blowup solutions with ground state mass ‖u0‖2=‖Q‖2, where Q is the ground state solution of equation DQ+Q=Q2.

Abstract In this paper we establish periodic homogenization for Hamilton-Jacobi-Bellman (HJB) equations, associated to nonlocal operators of integro-differential type. We consider the case when the fractional diffusion has the same order as the drift term, and is weakly elliptic. The outcome of the paper is two-fold. One one hand, we provide Lipschitz regularity results for weakly elliptic nonlocal

Abstract We prove that the half-wave maps problem on R4+1 with target S2 is globally well-posed for smooth initial data which are small in the critical l1 based Besov space. This is a formal analogue of the result proved by Tataru for wave maps.

Abstract We consider the regular Lagrangian flow X associated to a bounded divergence-free vector field b with bounded variation. We prove a Lusin-Lipschitz regularity result for X and we show that the Lipschitz constant grows at most linearly in time. As a consequence we deduce that both geometric and analytical mixing have a lower bound of order 1/t as t→∞.

Abstract We study the asymptotic dynamics for solutions to a system of nonlinear Schrödinger equations with cubic interactions, arising in nonlinear optics. We provide sharp threshold criteria leading to global well-posedness and scattering of solutions, as well as formation of singularities in finite time for (anisotropic) symmetric initial data. The free asymptotic results are proved by means of

Abstract We prove a long time existence result for the solutions of a two-dimensional Boussinesq system modeling the propagation of long, weakly nonlinear water waves. This system is exceptional in the sense that it is the only linearly well-posed system in the (abcd) family of Boussinesq systems whose eigenvalues of the linearized system have nontrivial zeroes. This new difficulty is solved by the

Abstract We obtain generalizations of classical versions of the Weyl formula involving Schrödinger operators HV=−Δg+V(x) on compact boundaryless Riemannian manifolds with critically singular potentials V. In particular, we extend the classical results of Avakumović (1956 Avakumović, V. G. (1956). Über die Eigenfunktionen auf geschlossenen Riemannschen Mannigfaltigkeiten. Math. Z. 65(1):327–344. DOI:

Abstract We study thermal insulating of a bounded body Ω⊂Rn. Under a prescribed heat source f≥0, we consider a model of heat transfer between Ω and the environment determined by convection; this corresponds, before insulation, to Robin boundary conditions. The body is then surrounded by a layer of insulating material of thickness of size ε>0, and whose conductivity is also proportional to ε. This corresponds

Abstract This article is devoted to the study of the Cauchy problem for the Muskat equation. We consider initial data belonging to the critical Sobolev space of functions with three-half derivative in L2, up to a fractional logarithmic correction. As a corollary, we obtain the first local and global well-posedness results for initial free surfaces which are not Lipschitz.

Abstract We study the existence of segregated solutions to a class of reaction-diffusion systems with strong interactions, arising in many physical applications. These special solutions are obtained as weak limits of minimizers of a family of perturbed functionals. We prove some a priori estimates through a minimization procedure which is nonstandard in the parabolic theory: our approach is purely

Abstract We study pointwise and Lp gradient estimates of the heat kernels of both the scalar Laplacian, as well as the Hodge Laplacian on k-forms, on manifolds that may have some amount of negative Ricci curvature, provided it is not too negative (in an integral sense) at infinity. Such heat kernel estimates have already been obtained by the author, together with Coulhon and Sikora, provided certain

Abstract We consider the cubic-quintic nonlinear Schrödinger equation in two space dimensions. For this model, X. Cheng established scattering for H1 data with mass strictly below that of the ground state for the cubic NLS. Subsequently, R. Carles and C. Sparber utilized the pseudoconformal energy estimate to obtain scattering at the sharp threshold for data belonging to a weighted Sobolev space. In

Abstract We prove that on a smooth bounded set, the positive least energy solution of the Lane-Emden equation with sublinear power is isolated. As a corollary, we obtain that the first q−eigenvalue of the Dirichlet-Laplacian is not an accumulation point of the q−spectrum, on a smooth bounded set. Our results extend to a suitable class of Lipschitz domains, as well.

Abstract We prove that certain Sobolev-type norms, slightly stronger than those given by energy conservation, stay bounded uniformly in time and N. This allows one to extend the local existence results of the second and third author globally in time. The proof is based on interaction Morawetz-type estimates and Strichartz estimates (including some new end-point results) for the equation {1i∂t−Δx−Δy+1NVN(x−y)}Λ(t

Abstract The Vlasov-Poisson system with massless electrons (VPME) is widely used in plasma physics to model the evolution of ions in a plasma. It differs from the Vlasov-Poisson system (VP) for electrons in that the Poisson coupling has an exponential nonlinearity that creates several mathematical difficulties. In particular, while global well-posedness in 3 D is well understood in the electron case

Abstract This paper is concerned with the nodal set of weak solutions to a broken quasilinear partial differential equation, div(a+∇u+−a−∇u−)=div f, where a+ and a− are uniformly elliptic, Dini continuous coefficient matrices, subject to a strong correlation that a+ and a− are a multiple of some scalar function to each other. Under such a structural condition, we develop an iteration argument to achieve

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 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 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 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 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 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 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 g0 on R4 with monotone warping coefficients and whose restriction to any hypersphere is a Berger metric. If g0 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 gTNUT in the

Abstract We consider the Zakharov-Kutznesov (ZK) equation posed in Rd, with d = 2 and 3. Both equations are globally well-posed in L2(Rd). In this article, we prove local energy decay of global solutions: if u(t) is a solution to ZK with data in L2(Rd), then lim inft→∞∫Ωd(t)u2(x,t)dx=0,for suitable regions of space Ωd(t)⊆Rd around the origin, growing unbounded in time, not containing the soliton region

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 Ẇ−1,p, where max(0,12−1n−ε)<1p<12. Our arguments are inspired by an argument of Shen and build on known well posedness results in the case p = 2. We use the same techniques to establish

Abstract Consider the higher order parabolic operator ∂t+(−Δx)m and the higher order Schrödinger operator i−1∂t+(−Δx)m in X={(t,x)∈R1+n; |t|0, then the solutions vanish in X. Such results are global if n > 1, and we also prove some relevant local results.

Abstract We consider the mixed problem on the exterior of the unit ball in Rn,n≥2, for a defocusing Schrödinger equation with a power nonlinearity |u|p−1u, 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 inflate, and the

Abstract In this paper, we study the differential inclusion associated with the minimal surface system for two-dimensional graphs in R2+n. We prove regularity of W1,2 solutions and a compactness result for approximate solutions of this differential inclusion in W1,p. Moreover, we make a perturbation argument to infer that for every R > 0, there exists α(R)>0 such that R-Lipschitz stationary points

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=λϕ in Ω for rapidly varying potentials V in bounded domains Ω⊂Rd. 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 showed that 1/u arises

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 study two overdetermined elliptic boundary value problems on exterior domains (the complement of a ball and the complement of a solid cylinder in R3 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 The chemotaxis-Stokes system {nt+u·∇n=Δn−∇·(n∇c),ct+u·∇c=Δc−nc,ut=Δu+∇P+n∇ϕ,  ∇·u=0, (⋆) is considered in a bounded domain Ω⊂R3 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 u. In such situations, namely, a quasi-Lyapunov structure

Abstract We characterize the denseness of the singular set of the distance function from a C1-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 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 ω0=αezδx=y=0, where δx=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 Oseen vortex

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 Consider an isotropic elastic medium Ω⊂R3 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

This paper is concerned with the intrinsic geometric structures of conductive transmission eigenfunctions. The geometric properties of interior transmission eigenfunctions were first studied in [9]. It is shown in two scenarios that the interior transmission eigenfunction must be locally vanishing near a corner of the domain with an interior angle less than $\pi$. We significantly extend and generalize

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 towards 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 Euler

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

In this paper, we study the stability of the conical Kahler-Ricci flows on Fano manifolds. That is, if there exists a conical Kahler-Einstein metric with cone angle $2\pi\beta$ along the divisor, then for any $\beta'$ sufficiently close to $\beta$, the corresponding conical Kahler-Ricci flow converges to a conical Kahler-Einstein metric with cone angle $2\pi\beta'$ along the divisor. Here, we only

This paper 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 $\delta^2$ for $0<\delta<\infty$. Our study also covers the case of soft inclusions ($\delta=0$) as well as the case of stiff inclusions ($\delta=\infty$)

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 $f=\sum_{k=1}^m a_k \phi_{\lambda_{j_k}}$, where $-\Delta_g\phi_{\lambda}=\lambda\phi_{\lambda}$. Denoting by $Z_f$ the zero-set of $f$, we show that for any $x\in M$, $dist(x,Z_f)\leq C(m)\lambda_{j_1}^{-1/2}$

We give resolvent estimates near zero energy on Riemannian scattering, i.e. asymptotically conic, spaces, and their generalizations, using a uniform microlocal Fredholm analysis framework.

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 spherical

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 Calderon problem by using an integral identity. We also determine both the magnetic potential and the nonlinearity in the fractional semilinear magnetic Calderon problem by using a first order linearization.

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 in a unified

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 needed,