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 study the defocusing inhomogeneous mass-critical nonlinear Schrödinger equation on R 2 i ∂ t u n + Δ u n = g ( n x ) | u n | 2 u n for initial data in L 2 ( R 2 ) . We obtain sufficient conditions on g to ensure existence and uniqueness of global solutions for n sufficiently large, as well as homogenization.

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 need of derivative

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

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

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

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∞.

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

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, we give a

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 of functions

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 order

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 operator

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 long time

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 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