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.

We investigate the existence and uniqueness issues of the 3D incompressible Hall-magnetohydrodynamic system supplemented with initial velocity u 0 and magnetic field B 0 in critical regularity spaces. In the case where u 0, B 0 and the current J 0 : = ∇ × B 0 belong to the homogeneous Besov space B ̇ p , 1 3 p − 1 , 1 ≤ p < ∞ , and are small enough, we establish a global result and the conservation

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 in a

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 solution

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

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.

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 dissipation

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⊂RN is the unit ball. The weights can be singular at x = 0. We present the sharp asymptotic behavior

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 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.

We establish both a local and a global well-posedness theories for the nonlinear Hartree–Fock equations and its reduced analog in the setting of the modulation spaces on ℝd. In addition, we prove similar results when a harmonic potential is added to the equations. In the process, we prove the boundedeness of certain multilinear operators on products of the modulation spaces which may be of independent

We prove existence of L2-weak solutions of a quasilinear wave equation with boundary conditions. This describes the isothermal evolution of a one dimensional non-linear elastic material, attached to a fixed point on one side and subject to a force (tension) applied to the other side. The L2-valued solutions appear naturally when studying the hydrodynamic limit from a microscopic dynamics of a chain

We prove existence of L2-weak solutions of a quasilinear wave equation with boundary conditions. This describes the isothermal evolution of a one dimensional non-linear elastic material, attached to a fixed point on one side and subject to a force (tension) applied to the other side. The L2-valued solutions appear naturally when studying the hydrodynamic limit from a microscopic dynamics of a chain

We consider almost minimizers to the one-phase energy functional and we prove their optimal Lipschitz regularity and partial regularity of their free boundary. These results were recently obtained by David and Toro, and David, Engelstein, and Toro. Our proofs provide a different method based on a non-infinitesimal notion of viscosity solutions that we introduced.

Upper and lower bounds on the heat kernel on complete Riemannian manifolds were obtained in a series of pioneering works due to Cheng-Li-Yau, Cheeger-Yau and Li-Yau. However, these estimates do not give a complete picture of the heat kernel for all times and all pairs of points — in particular, there is a considerable gap between available upper and lower bounds at large distances and/or large times

We present a class of pseudo-differential elliptic systems with anti-self-dual potentials on R satisfying compensation phenomena similar to the ones discovered by the second author for elliptic systems with anti-symmetric potentials. These compensation phenomena are based on new “multi-commutator” structures generalizing the 3-commutators recently introduced by the authors.

We study the heat flow for Yang-Mills connections on Rd×SO(d). It is well-known that in dimensions 5≤d≤9 this model admits homothetically shrinking solitons, i.e., self-similar blowup solutions, with an explicit example given by Weinkove. We prove the nonlinear asymptotic stability of the Weinkove solution under small equivariant perturbations and thus extend a result by the second author and Donninger

We develop a regularity theory for integro-differential equations with kernels deforming in space like sections of a convex solution of a Monge–Ampère equation. We prove an ABP estimate and a Harnack inequality, and derive Hölder and C1,α regularity results for solutions.

Abstract We prove unique continuation properties of solutions to a large class of nonlinear, non-local dispersive equations. The goal is to show that if u1, u2 are two suitable solutions of the equation defined in Rn×[0,T] such that for some non-empty open set Ω⊂Rn×[0,T],u1(x,t)=u2(x,t) for (x,t)∈Ω, then u1(x,t)=u2(x,t) for any (x,t)∈Rn×[0,T]. The proof is based on static arguments. More precisely

Models of tissue growth are now well established, in particular, in relation to their applications to cancer. They describe the dynamics of cells subject to motion resulting from a pressure gradient generated by the death and birth of cells, itself controlled primarily by pressure through contact inhibition. In the compressible regime, pressure results from the cell densities and when two different

We prove local in time, uniform in N, estimates for the solutions ϕ, Λ, and Γ of a coupled system of Hartree–Fock–Bogoliubov type with interaction potential vN(x−y)=N3βv(Nβ(x−y)), with β<1 and v a Schwartz function (satisfying additional technical requirements). The initial conditions are general functions in a Sobolev-type space, and the expected correlations in Λ develop dynamically in time. As shown

We consider the operator Ah=−h2Δ+iV in the semi-classical limit h→0, where V is a smooth real potential with no critical points. We obtain both the left margin of the spectrum, as well as resolvent estimates on the left side of this margin. We extend here previous results obtained for the Dirichlet realization of Ah by removing significant limitations that were formerly imposed on V. In addition, we

In this article, we focus on the uniqueness question for (expanding) solutions of the harmonic map flow coming out of smooth 0-homogeneous maps with values into a closed Riemannian manifold. We introduce a relative entropy for two purposes. On the one hand, we prove the existence of two expanding solutions associated to any suitable solution coming out of a 0-homogeneous map by a blow-up and a blow-down

On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that Atiyah–Singer Dirac operator /DB in L2 depends Riesz continuously on L∞ perturbations of local boundary conditions B. The Lipschitz bound for the map B→/DB(1+/DB2)−12 depends on Lipschitz smoothness and ellipticity of B and bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity

Uniqueness of positive solutions to viscous Hamilton–Jacobi–Bellman (HJB) equations of the form −Δu(x)+1γ|Du(x)|γ=f(x)−λ, with f a coercive function and λ a constant, in the subquadratic case, that is, γ∈(1,2), appears to be an open problem. Barles and Meireles [Comm. Partial Differential Equations 41 (2016)] show uniqueness in the case that f(x)≈|x|β and |Df(x)|≲|x|(β−1)+ for some β>0, essentially

We study two resonant Hamiltonian systems on the phase space L2(R→C): the quintic one-dimensional continuous resonant equation, and a cubic resonant system that has appeared in the literature as a modified scattering limit for an NLS equation with cigar shaped trap. We prove that these systems approximate the dynamics of the quintic and cubic one-dimensional NLS with harmonic trapping in the small

We study bulk/edge aspects of continuous honeycomb lattices in a magnetic field. We compute the bulk index of Bloch eigenbundles: it equals 2 or –2, with sign depending on nearby Dirac points and on the magnetic field. We then prove the existence of two topologically protected unidirectional waves propagating along line defects. This shows the bulk/edge correspondence for our class of Hamiltonians

In this work, we investigate the question of preventing the three-dimensional, incompressible Navier-Stokes equations from developing singularities, by controlling one component of the velocity field only, in space-time scale invariant norms. In particular we prove that it is not possible for one component of the velocity field to tend to 0 too fast near blow up. We also introduce a space “almost”

The Nonlinear Noisy Leaky Integrate and Fire (NNLIF) model is widely used to describe the dynamics of neural networks after a diffusive approximation of the mean-field limit of a stochastic differential equation system. When the total activity of the network has an instantaneous effect on the network, in the average-excitatory case, a blow-up phenomenon occurs. This article is devoted to the theoretical

We consider the localization of eigenfunctions for the operator L=−div A grad+V on a Lipschitz domain Ω and, more generally, on manifolds with and without boundary. In earlier work, two authors of the present paper demonstrated the remarkable ability of the landscape, defined as the solution to Lu = 1, to predict the location of the localized eigenfunctions. Here, we explain and justify a new framework

In a recent work, Bourgain gave a fine description of the expectation of solutions of discrete linear elliptic equations on Zd with random coefficients in a perturbative regime using tools from harmonic analysis. This result is surprising for it goes beyond the expected accuracy suggested by recent results in quantitative stochastic homogenization. In this short article we reformulate Bourgain’s result

In this article, we study some quantitative unique continuation properties of solutions to second-order elliptic equations with singular lower order terms. First, we quantify the strong unique continuation property by estimating the maximal vanishing order of solutions. That is, when u is a nontrivial solution to Δu+W·∇u+Vu=0 in some open, connected subset of Rn, where n≥3, we characterize the vanishing

The dissipative solutions can be seen as a convenient generalization of the concept of weak solution to the isentropic Euler system. They can be seen as expectations of the Young measures associated to a suitable measure-valued solution of the problem. We show that dissipative solutions coincide with weak solutions starting from the same initial data on condition that: (i) the weak solution enjoys

We study the relativistic Euler equations on the Minkowski spacetime background. We make assumptions on the equation of state and the initial data that are relativistic analogs of the well-known physical vacuum boundary condition, which has played an important role in prior work on the non-relativistic compressible Euler equations. Our main result is the derivation, relative to Lagrangian (also known

We consider a class of manifolds M obtained by taking the connected sum of a finite number of N-dimensional Riemannian manifolds of the form (Rni,δ)×(Mi,g), where Mi is a compact manifold, with the product metric. The case of greatest interest is when the Euclidean dimensions ni are not all equal. This means that the ends have different ‘asymptotic dimension’, and implies that the Riemannian manifold

In this article, we study the logarithmic Laplacian operator LΔ, which is a singular integral operator with symbol 2 log |ζ|. We show that this operator has the integral representation LΔu(x)=cN∫RNu(x)1B1(x)(y)−u(y)|x−y|Ndy+ρNu(x) with cN=π−N2Γ(N2) and ρN=2 log 2+ψ(N2)−γ, where Γ is the Gamma function, ψ=Γ′Γ is the Digamma function and γ=−Γ′(1) is the Euler Mascheroni constant. This operator arises

We prove existence and uniqueness of solution of a class of semi-linear wave equations with initial data prescribed on the light-cone with vertex the origin of a Minkowski space-time. The nonlinear term is assumed to satisfy a nullity condition which guarantees that the neighborhood of the initial cone on which we obtain our solution does not shrink to zero as one approaches infinity. This result is

We study the asymptotic behavior of the generalized Bergman kernel of the renormalized Bochner-Laplacian on high tensor powers of a positive line bundle on a symplectic manifold of bounded geometry. First, we establish the off-diagonal exponential estimate for the generalized Bergman kernel. As an application, we obtain the relation between the generalized Bergman kernel on a Galois covering of a compact

Following a framework initiated by Blanc, Le Bris and Lions, this article aims at obtaining quantitative homogenization results in a simple case of interface between two periodic media. By using Avellaneda and Lin’s techniques, we provide pointwise estimates for the gradient of the solution to the multiscale problem and for the associated Green function. Also we generalize the classical two-scale expansion

Let N be a smooth (n + l)-dimensional Riemannian manifold. We show that if V is an area-stationary union of three or more C1,μ n-dimensional submanifolds-with-boundary Mk⊂N with a common boundary Γ, then Γ is smooth and each Mk is smooth up to Γ (real-analytic in the case N is real-analytic). This extends a previous result of the author for codimension l = 1. We additionally show that if {Vt}t∈(−1

Motivated by recent papers describing the formation of biological transport networks we study a discrete model proposed by Hu and Cai consisting of an energy consumption function constrained by a linear system on a graph. For the spatially two-dimensional rectangular setting we prove the rigorous continuum limit of the constrained energy functional as the number of nodes of the underlying graph tends

Existence of mass-conserving self-similar solutions with a sufficiently small total mass is proved for a specific class of homogeneous coagulation and fragmentation coefficients. The proof combines a dynamical approach to construct such solutions for a regularized coagulation–fragmentation equation in scaling variables and a compactness method.