Piero D'Ancona, Vittoria Pierfelice
We prove an optimal dispersive $L^{\infty}$ decay estimate for a three dimensional wave equation perturbed with a large non smooth potential belonging to a particular Kato class. The proof is based on a spectral representation of the solution and suitable resolvent estimates for the perturbed operator.
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Consider the two-dimensional inverse elastic wave scattering by an infinite rough surface with a Dirichlet boundary condition. A non-interative sampling technique is proposed for detecting the rough surface by taking elastic wave measurements on a bounded line segment above the surface, based on reconstructing a modified near-field equation associated with a special surface, which generalized our pervious work for the Helmholtz equation (SIAM J. IMAGING. SCI. 10(3)(2017), 1579-1602) to the Navier equation. Several numerical examples are carried out to illustrate the effectiveness of the invers…
We show that in the presence of a rough external potential, a 3d cubic defocusing NLS is globally well-posed in H^s for s>5/6. The proof is based on the approach of Colliander-Keel-Staffilani-Takaoka-Tao, called the I-method, but in order to deal with a rough potential, we modify harmonic analysis tools by spectral theory.
We study a Follow-the-Leader (FtL) ODE model for traffic flow with rough road condition, and analyze stationary traveling wave profiles where the solutions of the FtL model trace along, near the jump in the road condition. We derive a discontinuous delay differential equation (DDDE) for these profiles. For various cases, we obtain results on existence, uniqueness and local stability of the profiles. The results here offer an alternative approximation, possibly more realistic than the classical vanishing viscosity approach, to the conservation law with discontinuous flux for traffic flow.
We prove global well-posedness for the $3D$ radial defocusing cubic wave equation with data in $H^{s} \times H^{s-1}$, $1>s>{7/10}$.
We obtain certain time decay and regularity estimates for 3D Schroedinger equation with a potential in the Kato class by using Besov spaces associated with Schroedinger operators.
Wave packet decomposition for Schrodinger evolution with rough potential and generic value of parameter
本文发展波包分解方法分析带粗糙势的 Schrödinger 演化，并据此在参数取一般值时改进波传播界。
This article deals with the weak and strong unique continuation principle for fractional Schrödinger equations with scaling-critical and rough potentials via Carleman estimates. Our methods allow to apply the results to variable coefficient versions of fractional Schrödinger equations.
In this note, we consider wave packet parametrices for Schrodinger-like evolution equations. Under an integrability condition along the flow, we prove that the flow is then globally well-defined and bilipschitz. Under an additional smallness assumption, we prove that the kernel of the phase space operator decays rapidly away from the graph of the Hamilton flow. This can be used to prove that smoothness is preserved for finite times by the flow of the equation.
We prove local and global existence from large, rough initial data for a wave map between 1+1 dimensional Minkowski space and an analytic manifold. Included here is global existence for large data in the scale-invariant norm $\dot L^{1,1}$, and in the Sobolev spaces $H^s$ for $s > 3/4$. This builds on previous work in 1+1 dimensions of Pohlmeyer, Gu, Ginibre-Velo and Shatah.
In this paper we establish square-function estimates on the double and single layer potentials with rough inputs for divergence form elliptic operators, of arbitrary even order 2m, with variable t-independent coefficients in the upper half-space.
We study the Cauchy problem for the half Ginzburg-Landau-Kuramoto (hGLK) equation with the second order elliptic operator having rough coefficients and potential type perturbation. The blow-up of solutions for hGLK equation with non-positive nonlinearity is shown by an ODE argument. The key tools in the proof are appropriate commutator estimates and the essential self-adjointness of the symmetric uniformly elliptic operator with rough metric and potential type perturbation.
In this paper, we obtain sharp Strichartz estimates for solutions of the wave equation $\square_\ggϕ=0$ where $\gg$ is a rough Lorentzian metric on a 4 dimensional space-time $\MM$. This is the last step of the proof of the bounded $L^2$ curvature conjecture proposed in [3], and solved by S. Klainerman, I. Rodnianski and the author in [8], which also relies on the sequence of papers [16][17][18][19]. Obtaining such estimates is at the core of the low regularity well-posedness theory for quasilinear wave equations. The difficulty is intimately connected to the regularity of the Eikonal equation…
We consider two scalar conservation laws with non-local flux functions, describing traffic flow on roads with rough conditions. In the first model, the velocity of the car depends on an averaged downstream density, while in the second model one considers an averaged downstream velocity. The road condition is piecewise constant with a jump at $x=0$. We study stationary traveling wave profiles cross $x=0$, for all possible cases. We show that, depending on the case, there could exit infinitely many profiles, a unique profile, or no profiles at all. Furthermore, some of the profiles are time asym…
We prove L^1 --> L^\infty estimates for the linear Schroedinger equation in three dimensions. The potential is assumed to belong to certain L^p spaces, but no pointwise decay estimates and no additional regularity is required.
We provide a reconstruction scheme for complex-valued potentials in $H^s (\mathbb{R}^2)$ for $s > 0$. The procedure extends the method of Bukhgeim relying on quadratic exponential solutions. We also see how the new reconstruction formulas proposed improve the convergence on a set of examples.
Wave packet decompositions and sharp bilinear estimates for rough Hamiltonian flows
本文针对满足非退化性和横截性假设的粗糙色散演化，利用波包分解和FBI变换构造参数化，证明了推广锥面和抛物面尖锐傅里叶扩展估计的双线性L^p估计。
Local smoothing for rough wave equations
该研究针对带$C^r$系数的粗糙波动方程，利用双线性限制估计证明局部光滑化估计，在二维和三维的$C^{1,1}$系数情形恢复了光滑系数下的最优界，并推导出非平凡的$L^p$-$L^p$局部光滑化估计。
We study the scattering theory for the Schrödinger and wave equations with rough potentials in a scale of homogeneous Sobolev spaces. The first half of the paper concerns with an inverse-square potential in both of subcritical and critical constant cases, which is a particular model of scaling-critical singular perturbations. In the subcritical case, the existence of the wave and inverse wave operators defined on a range of homogeneous Sobolev spaces is obtained. In particular, we have the scattering to a free solution in the homogeneous energy space for both of the Schrödinger and wave equati…
Global UCP For Parabolic Fractional $p$-Laplace Equation With Very Rough Potentials
研究抛物分数$p$-拉普拉斯方程在非常粗糙势函数下的全局唯一持续原理，提出新方法避免扩展技术和Carleman估计，解决局部问题仍存开放性。
In this paper, we obtain the existence and uniqueness of the strong solution to one spatial dimension stochastic wave equation $\frac{\partial^2 u(t,x)}{\partial t^2}=\frac{\partial^2 u(t,x)}{\partial x^2}+σ(t,x,u(t,x))\dot{W}(t,x)$ assuming $σ(t,x,0)=0$, where $\dot W$ is a mean zero Gaussian noise which is white in time and fractional in space with Hurst parameter $H\in(1/4, 1/2)$.
In this paper, we consider the direct and inverse problem of scattering of time-harmonic waves by an unbounded rough interface with a buried impenetrable obstacle. We first study the well-posedness of the direct problem with a local source by the variational method; the well-posedness result is then extended to scattering problems induced by point source waves (PSWs) and hyper-singular point source waves (HSPSWs). For PSW or HSPSW incident waves, the induced total field admits a uniformly bounded estimate in any compact subset far from the source position. Moreover, we show that the scattered …
This is the last of a sequence of four papers \cite{param1}, \cite{param2}, \cite{param3}, \cite{param4} dedicated to the construction and the control of a parametrix to the homogeneous wave equation $\square_{\bf g} ϕ=0$, where ${\bf g}$ is a rough metric satisfying the Einstein vacuum equations. Controlling such a parametrix as well as its error term when one only assumes $L^2$ bounds on the curvature tensor ${\bf R}$ of ${\bf g}$ is a major step of the proof of the bounded $L^2$ curvature conjecture proposed in \cite{Kl:2000}, and solved by S. Klainerman, I. Rodnianski and the author in \ci…
This is the second of a sequence of four papers \cite{param1}, \cite{param2}, \cite{param3}, \cite{param4} dedicated to the construction and the control of a parametrix to the homogeneous wave equation $\square_{\bf g} ϕ=0$, where ${\bf g}$ is a rough metric satisfying the Einstein vacuum equations. Controlling such a parametrix as well as its error term when one only assumes $L^2$ bounds on the curvature tensor ${\bf R}$ of ${\bf g}$ is a major step of the proof of the bounded $L^2$ curvature conjecture proposed in \cite{Kl:2000}, and solved by S. Klainerman, I. Rodnianski and the author in \…
This is the first of a sequence of four papers \cite{param1}, \cite{param2}, \cite{param3}, \cite{param4} dedicated to the construction and the control of a parametrix to the homogeneous wave equation $\square_{\bf g} ϕ=0$, where ${\bf g}$ is a rough metric satisfying the Einstein vacuum equations. Controlling such a parametrix as well as its error term when one only assumes $L^2$ bounds on the curvature tensor ${\bf R}$ of ${\bf g}$ is a major step of the proof of the bounded $L^2$ curvature conjecture proposed in \cite{Kl:2000}, and solved jointly with S. Klainerman and I. Rodnianski in \cit…
This is the third of a sequence of four papers \cite{param1}, \cite{param2}, \cite{param3}, \cite{param4} dedicated to the construction and the control of a parametrix to the homogeneous wave equation $\square_{\bf g} ϕ=0$, where ${\bf g}$ is a rough metric satisfying the Einstein vacuum equations. Controlling such a parametrix as well as its error term when one only assumes $L^2$ bounds on the curvature tensor ${\bf R}$ of ${\bf g}$ is a major step of the proof of the bounded $L^2$ curvature conjecture proposed in \cite{Kl:2000}, and solved by S. Klainerman, I. Rodnianski and the author in \c…
A solution of the heat equation with a distribution-valued potential is constructed by regularization. When the potential is the generalized derivative of a Hölder continuous function, regularity of the resulting solution is in line with the standard parabolic theory.
For the damped wave equation on a compact manifold with {\em continuous} dampings, the geometric control condition is necessary and sufficient for {uniform} stabilisation. In this article, on the two dimensional torus, in the special case where $a(x) = \sum\_{j=1}^N a\_j 1\_{x\in R\_j}$ ($R\_j$ are polygons), we give a very simple necessary and sufficient geometric condition for uniform stabilisation. We also propose a natural generalization of the geometric control condition which makes sense for $L^\infty$ dampings. We show that this condition is always necessary for uniform stabilisation (f…
In underwater acoustic waveguides a pressure field can be decomposed over three kinds of modes: the propagating modes, the radiating modes and the evanescent modes. In this paper, we analyze the effects produced by a randomly perturbed free surface and an uneven bottom topography on the coupling mechanism between these three kinds of modes. Using an asymptotic analysis based on a separation of scales technique we derive the asymptotic form of the distribution of the forward mode amplitudes. We show that the surface and bottom fluctuations affect the propagating-mode amplitudes mainly in the sa…
We study 4 problems in the area of scattering of time harmonic acoustic or electromagnetic waves by unbounded rough surfaces/inhomogeneous layers. Specifically we study: i) a boundary value problem (BVP) for the Helmholtz equation, in both 2 and 3 dimensions, modelling scattering of time harmonic waves due to a source that lies within a finite distance of the boundary and which decays along the boundary, by a layer of spatially varying refractive index above an unbounded rough surface on which the field vanishes; ii) a BVP for the Helmholtz equation with an impedance boundary condition, in 2 a…
In this paper we study the semiclassical limit of the Schrödinger equation. Under mild regularity assumptions on the potential $U$ which include Born-Oppenheimer potential energy surfaces in molecular dynamics, we establish asymptotic validity of classical dynamics globally in space and time for "almost all" initial data, with respect to an appropriate reference measure on the space of initial data. In order to achieve this goal we prove existence, uniqueness and stability results for the flow in the space of measures induced by the continuity equation.
We consider the semiclassical limit for the Heisenberg-von Neumann equation with a potential which consists of the sum of a repulsive Coulomb potential, plus a Lipschitz potential whose gradient belongs to $BV$; this assumption on the potential guarantees the well posedness of the Liouville equation in the space of bounded integrable solutions. We find sufficient conditions on the initial data to ensure that the quantum dynamics converges to the classical one. More precisely, we consider the Husimi functions of the solution of the Heisenberg-von Neumann equation, and under suitable assumptions…
We prove the existence of weak solutions in the space of energy for a class of non-linear Schroedinger equations in the presence of a external rough magnetic potential. Under our assumptions it is not possible to study the problem by means of usual arguments like resolvent techniques or Fourier integral operators, for example. We use a parabolic regularization and we solve the approximating Cauchy problem. This is achieved by obtaining suitable smoothing estimates for the dissipative evolution. The total mass and energy bounds allow to extend the solution globally in time. We then infer suffic…
A mathematical method for through-wall imaging via wave phenomena in the time domain is introduced. The method makes use of a single reflected wave over a finite time interval and gives us a criterion whether a penetrable obstacle exists or not in a general rough background medium. Moreover, if the obstacle exists, the lower and upper estimates of the distance between the obstacle and the center point of the support of the initial data are given. As an evidence of the potential of the method two applications are also given.
Consider the elastic scattering of a plane or point incident wave by an unbounded and rigid rough surface. The angular spectrum representation (ASR) for the time-harmonic Navier equation is derived in three dimensions. The ASR is utilized as a radiation condition to the elastic rough surface scattering problem. The uniqueness is proved through a Rellich-type identity for surfaces given by uniformly Lipschitz functions. In the case of flat surfaces with a local perturbation, we deduce an equivalent variational formulation in a truncated bounded domain and show the existence results for general …
This is a study of the Euler equations for free surface water waves in the case of varying bathymetry, considering the problem in the shallow water scaling regime. In the case of rapidly varying periodic bottom boundaries this is a problem of homogenization theory. In this setting we derive a new model system of equations, consisting of the classical shallow water equations coupled with nonlocal evolution equations for a periodic corrector term. We also exhibit a new resonance phenomenon between surface waves and a periodic bottom. This resonance, which gives rise to secular growth of surface …
This paper proves endpoint Strichartz estimates for the linear Schroedinger equation in $R^3$, with a time-dependent potential that keeps a constant profile and is subject to a rough motion, which need not be differentiable and may be large in norm. The potential is also subjected to a time-dependent rescaling, with a non-differentiable dilation parameter. We use the Strichartz estimates to prove the non-dispersion of bound states, when the path is small in norm, as well as boundedness of energy. We also include a sample nonlinear application of the linear results.
We establish dispersive and Strichartz estimates for solutions to the linear time-dependent Schrödinger equations with potential in three dimensions. Our main focus is on the small rough time-dependent potentials. Examples of such potentials are of the form $V(t,x)=T(t) V_0(x)$, where $T$ is quasiperiodic in time and $V_0$ is essentially an $L^{3/2}$ function of the spatial variables. We also prove the dispersive estimates for small time-independent potentials which belong to the interestion of the Rollnik and global Kato classes. Finally, we settle the question posed by Journe, Soffer, Sogge …
Observability of wave and plate equations with rough coefficients and interfaces: a multiplier approach
本文采用乘子方法，通过正则化过程推导主部含粗糙系数及界面的波动方程与板方程的可观测性，避免分析奇异集合附近解的正则性，建立了含界面情形的新可观测性结果。
We study a nonlocal particle model describing traffic flow on rough roads. In the model, each driver adjusts the speed of the car according to the condition over an interval in the front, leading to a system of nonlocal ODEs which we refer to as the FtLs (follow-the-leaders) model. Assuming that the road condition is discontinuous at the origin, we seek stationary wave profiles (see Definition 1.1) for the system of ODEs across this discontinuity. We derive a non-local delay differential equation with discontinuous coefficient, satisfied by the profiles, together with conditions on the asympto…
In this paper, we consider inverse time-harmonic acoustic and electromagnetic scattering from locally perturbed rough surfaces in three dimensions. The scattering interface is supposed to be the graph of a Lipschitz continuous function with compact support. It is proved that an acoustically sound-soft or sound-hard surface can be uniquely determined by the far-field pattern of infinite number of incident plane waves with distinct directions. Moreover, a single point source or plane wave can be used to uniquely determine a scattering surface of polyhedral type. These uniqueness results apply to…
Global Kato smoothing and Strichartz estimates for higher-order Schrödinger operators with rough decay potentials
本文研究带一般有界衰减势的高阶椭圆算子对应的薛定谔型方程，建立了其解的全局Kato光滑性估计与全套Strichartz估计，还引入了带附加导数项的新型Kenig-Ruiz-Sogge型一致Sobolev估计，结果可应用于非线性高阶薛定谔方程的研究。
In this paper, we study the cubic defocusing nonlinear wave equation on the three dimensional hyperbolic space. We use the Fourier truncation method to show that the equation is globally well-posed and scatters if the initial data lies in $H^s(\mathbb{H}^3)$, $s>\frac{182}{201}\approx 0.905$.
We reconstruct compactly supported potentials with only half a derivative in $L^2$ from the scattering amplitude at a fixed energy. For this we draw a connection between the recently introduced method of Bukhgeim, which uniquely determined the potential from the Dirichlet-to-Neumann map, and a question of Carleson regarding the convergence to initial data of solutions to time-dependent Schrödinger equations. We also provide examples of compactly supported potentials, with $s$ derivatives in $L^2$ for any $s<1/2$, which cannot be recovered by these means. Thus the recovery method has a differen…
We study the Cauchy problem for the Gross-Pitaevskii equation with a nonlocal interaction potential of Hartree type in three space dimensions. If the potential is even and positive definite or a positive function and its Fourier transform decays sufficiently rapidly the problem is shown to be globally well-posed for large rough data which not necessarily have finite energy and also in a situation where the energy functional is not positive definite. The proof uses a suitable modification of the I-method.
This is the first in a series Of papers in which we initiate the study Of very rough solutions to the initial value problem for the Einstein Vacuum equations expressed relative to wave coordinates. By very rough we mean solutions which cannot be constructed by the classical techniques Of energy estimates and Sobolev inequalities. Following our previous work on quasilinear wave equations we develop new analytic methods based on Strichartz type inequalities which results in a gain of half a derivative relative to the classical result. Thus our result requires only $H^{2+ε}$ regularity for the da…
For the Schrödinger equation, $ (i \partial_t + Δ) u = 0 $ on a torus, an arbitrary non-empty open set $ Ω$ provides control and observability of the solution: $ \| u |_{t = 0} \|_{L^2 (\T^2)} \leq K_T \| u \|_{L^2 ([0,T] \times Ω)} $. We show that the same result remains true for $ (i \partial_t + Δ- V) u = 0 $ where $ V \in L^2 (\T^2) $, and $ \T^2 $ is a (rational or irrational) torus. That extends the results of \cite{AM}, and \cite{BZ4} where the observability was proved for $ V \in C (\T^2) $ and conjectured for $ V \in L^\infty (\T^2) $. The higher dimensional generalization remains ope…
In this paper, we consider very rough solutions to Cauchy problem for the Einstein vacuum equations in CMC spacial harmonic gauge, and obtain the local well-posedness result in $H^s, s>2$. The novelty of our approach lies in that, without resorting to the standard paradifferential regularization over the rough, Einstein metric $\bg$, we manage to implement the commuting vector field approach to prove Strichartz estimate for geometric wave equation $\Box_\bg ϕ=0$ directly.
The cubic nonlinear Schrödinger equation with rough potential
研究具有粗糙势能的立方非线性薛定谔方程，提出PDE适定性新最优结果及高效数值方法，通过收敛性分析与数值实验验证了理论正则性并证明新方案在粗糙势能下精度更高。
$L^{p}$ and $\mathcal{H}^{p}_{FIO}$ regularity for wave equations with rough coefficients
该研究针对空间具$C^{1,1}$正则性的时不变系数波动方程，证明其标准非齐次初值问题在傅里叶积分算子相关的Sobolev空间中适定，二维、三维下覆盖全$p$范围，并得到最优固定时刻$L^p$正则性，推广了光滑系数情形的已有结果。
Local near-field scattering data enables unique reconstruction of rough electric potentials
本文研究逆点源散射问题，引入散射问题近场数据的Runge逼近结果结合内部正则性论证，得到局部近场数据下一类粗糙电势逆问题的全新唯一性结果，改进了已有研究且适用电势范围更广。
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