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 consider a class of equations in divergence form with a singular/degenerate weight − div ( | y | a A ( x , y ) ∇ u ) = | y | a f ( x , y )   or div ( | y | a F ( 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)

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 For each pair ε = ( ε 1 , ε 2 ) of positive parameters, we define a perforated domain Ω ε by making a small hole of size ε 1 ε 2 in an open regular subset Ω of ℝ n ( n ≥ 3 ). The hole is situated at distance ε 1 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

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 study the large-time behavior of bounded from below solutions of parabolic viscous Hamilton-Jacobi Equations in the whole space R N 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

Abstract Considering the defocusing nonlinear Schrödinger equation (NLSE) in generic (bounded or unbounded) open sets U ⊆ R n 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

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 consider the mixed Dirichlet-conormal problem on irregular domains in R d . Two types of regularity results will be discussed: the W 1 , 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,

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

Abstract 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

We introduce a kinetic formulation for scalar conservation laws with nonlocal and nonlinear diffusion terms. We deal with merely L1 initial data, general self-adjoint pure jump Lévy operators, and locally Lipschitz nonlinearities of porous medium kind possibly strongly degenerate. The cornerstone of the formulation and the uniqueness proof is an adequate explicit representation of the dissipation measure

We consider the radial focusing energy critical nonlinear wave equation in three spatial dimensions. We establish the stability of the ODE-blowup under random perturbations below the energy space. The argument relies on probabilistic Strichartz estimates in similarity coordinates.

Abstract We consider the radial focusing energy critical nonlinear wave equation in three spatial dimensions. We establish the stability of the ODE-blowup under random perturbations below the energy space. The argument relies on probabilistic Strichartz estimates in similarity coordinates.

Abstract We consider the 2 D incompressible Navier-Stokes equations on T × R , with initial vorticity that is δ close in H x log L y 2 to −1(the vorticity of the Couette flow ( y , 0 ) ). We prove that if δ ≪ ν 1 / 2 , where ν denotes the viscosity, then the solution of the Navier-Stokes equation approaches some shear flow which is also close to Couette flow for time t ≫ ν − 1 / 3 by a mixing-enhanced

Abstract We study the defocusing inhomogeneous mass-critical nonlinear Schrödinger equation on R2 i∂tun+Δun=g(nx)|un|2un for initial data in L2(R2). We obtain sufficient conditions on g to ensure existence and uniqueness of global solutions for n sufficiently large, as well as homogenization.

Abstract We answer affirmatively a question posed by Aviles in 1983, concerning the construction of singular solutions of semilinear equations without using phase-plane analysis. Fully exploiting the semilinearity and the stability of the linearized operator in any dimension, our techniques involve a careful gluing in weighted L∞ spaces that handles multiple occurrences of criticality, without the

Abstract We present the sharp characterization of the behavior at the isolated singularities of positive solutions of some equations on singular manifolds with conical metrics. It is seen that the equations on singular manifolds with conical metrics are equivalent to weighted elliptic equations in B \ { 0 } , where B ⊂ R N is the unit ball. The weights can be singular at x = 0. We present the sharp

Abstract Eigenspaces of the quantum isotropic Harmonic Oscillator Ĥℏ:=−ℏ22Δ+12||x||2 on Rd have extremally high multiplicites and the eigenspace projections Πℏ,EN(ℏ) have special asymptotic properties. This article gives a detailed study of their Wigner distributions Wℏ,EN(ℏ)(x,ξ). Heuristically, if EN(ℏ)=E,Wℏ,EN(ℏ)(x,ξ) is the “quantization” of the energy surface ΣE, and should be like the delta-function

Abstract In this article, we present two mechanisms for deducing logarithmic quantitative unique continuation bounds for certain classes of integral operators. In our first method, expanding the corresponding integral kernels, we exploit the logarithmic stability of the moment problem. In our second method we rely on the presence of branch-cut singularities for certain Fourier multipliers. As an application

Abstract A system of wave equations on a Lorentzian manifold, the coefficients of which depend on time relates to the Einstein equation in general relativity. We consider inverse source problem for the system in this paper. Having established the Carleman estimate with a second large parameter for the Laplace–Beltrami operator on a Lorentzian manifold under assumptions independent of a choice of local

Abstract We prove gradient bounds for solutions of a class of boundary value problems related to the capillary problem, using the maximum principle. Our results extend those of the author and those of Ma and Xu. In addition to the capillary boundary condition, we can prove results for the general boundary condition (1+|Du|2)q−1Du·γ=ψ(x) for any choice of the parameter q≥0. (The range 0

Abstract We study the long-time behavior of solutions to an ODE- Schrödinger type system that models the interaction of a particle with a Bose gas. We show that the particle has sonic or subsonic ballistic trajectory asymptotically, and that the wave function describing the Bose gas converges to a soliton in L∞.

Abstract In this paper, we consider the Laplcian operator on theta line bundle over the quasi-torus, which is called the Bochner Laplacian. This operator has a canonical realization as a magnetic Laplacian acting on complex valued functions satisfying a functional equation. We study the spectral properties of such Laplacian and we show that its spectrum is reduced to eigenvalues πm; m=0,1,…. Then,

Abstract We consider scattering theory of the Laplace Beltrami operator on differential forms on a Riemannian manifold that is Euclidean at infinity. The manifold may have several boundary components caused by obstacles at which relative boundary conditions are imposed. Scattering takes place because of the presence of these obstacles and possible non-trivial topology and geometry. Unlike in the case

Abstract Let Ω⊂Cm be a bounded pseudoconvex domain with smooth boundary. For each k∈N, we give a sufficient condition to estimate the ∂¯-Neumann operator in the Sobolev space Wk(Ω). The key feature of our results is a precise formula for k in terms of the geometry of the boundary of Ω. It also shows the bound in the precise formula for Sobolev estimates of order k is comparably reciprocal of the Sobolev

Abstract This paper concerns the random source problems for the time-harmonic acoustic and elastic wave equations in two and three dimensions. The goal is to determine the compactly supported external force from the radiated wave field measured in a domain away from the source region. The source is assumed to be a microlocally isotropic generalized Gaussian random function such that its covariance

Abstract The L2-gradient flow of the elastic energy of networks leads to a Willmore type evolution law with non-trivial nonlinear boundary conditions. We show local in time existence and uniqueness for this elastic flow of networks in a Sobolev space setting under natural boundary conditions. In addition, we show a regularisation property and geometric existence and uniqueness. The main result is a

This paper addresses several problems associated to local energy solutions (in the sense of Lemarié-Rieusset) to the Navier-Stokes equations with initial data which is sufficiently small at large or small scales as measured using truncated Morrey-type quantities, namely: (1) global existence for a class of data including the critical L2-based Morrey space; (2) initial and eventual regularity of local

We establish magnetic improvements upon the classical Hardy inequality for two specific choices of singular magnetic fields. First, we consider the Aharonov-Bohm field in all dimensions and establish a sharp Hardy-type inequality that takes into account both the dimensional as well as the magnetic flux contributions. Second, in the three-dimensional Euclidean space, we derive a non-trivial magnetic

This work presents a new constructive uniqueness proof for Calderón’s inverse problem of electrical impedance tomography, subject to local Cauchy data, for a large class of piecewise constant conductivities that we call piecewise constant layered conductivities (PCLC). The resulting reconstruction method only relies on the physically intuitive monotonicity principles of the local Neumann-to-Dirichlet

In this paper, we study the defocusing nonlinear Schrödinger equation with a locally distributed damping on a smooth bounded domain as well as on the whole space and on an exterior domain. We first construct approximate solutions using the theory of monotone operators. We show that approximate solutions decay exponentially fast in the L2-sense by using the multiplier technique and a unique continuation

We study the quasi-mode of Stokes system posed on a smooth bounded domain Ω with Dirichlet boundary condition. We prove that the semi-classical defect measure associated with a sequence of solutions concentrates on the bicharacteristics of Laplacian as a matrix-valued Radon measure. Moreover, we show that the support of the measure is invariant under the Melrose-Sjöstrand flow.