Michael Christ
Generalized solutions of the Cauchy problem for the one-dimensional periodic nonlinear Schrödinger equation, with certain nonlinearities, are not unique. For any $s<0$ there exist nonzero generalized solutions varying continuously in the Sobolev space $H^s$, with identically vanishing initial data.
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We prove that weak solutions of the inviscid SQG equations are not unique, thereby answering Open Problem 11 in the survey arXiv:1111.2700 by De Lellis and Székelyhidi Jr. Moreover, we also show that weak solutions of the dissipative SQG equation are not unique, even if the fractional dissipation is stronger than the square root of the Laplacian.
We prove nonuniqueness of weak solutions to multi-dimensional generalisation of the Aw-Rascle model of vehicular traffic. Our generalisation includes the velocity offset in a form of gradient of density function, which results in a dissipation effect, similar to viscous dissipation in the compressible viscous fluid models. We show that despite this dissipation, the extension of the method of convex integration can be applied to generate infinitely many weak solutions connecting arbitrary initial and final states. We also show that for certain choice of data, ill posedness holds in the class of…
Consider the unforced incompressible homogeneous Navier-Stokes equations on the $d$-torus $\mathbb{T}^d$ where $d\geq 4$ is the space dimension. It is shown that there exist nontrivial steady-state weak solutions $u\in L^{2}(\mathbb{T}^d)$. The result implies the nonuniqueness of finite energy weak solutions for the Navier-Stokes equations in dimensions $d \geq 4$. And it also suggests that the uniqueness of forced stationary problem is likely to fail however smooth the given force is.
We give an alternative proof of the nonuniqueness of weak solutions to the surface quasigeostrophic equation (SQG) first shown in [Buckmaster-Shkoller-Vicol, '16]. Our approach proceeds directly at the level of the scalar field. Furthermore, we prove that every smooth scalar field with compact support that conserves the integral can be realized as a weak limit of solutions to SQG.
For suitable initial and boundary data, we construct infinitely many weak solutions to the nematic liquid crystal flows in dimension three. These solutions are in the axisymmetric class with bounded energy and backward bubbling at a large time.
We consider the linear transport equations driven by an incompressible flow in dimensions $d\geq 3$. For divergence-free vector fields $u \in L^1_t W^{1,q}$, the celebrated DiPerna-Lions theory of the renormalized solutions established the uniqueness of the weak solution in the class $L^\infty_t L^p$ when $\frac{1}{p} + \frac{1}{q} \leq 1$. For such vector fields, we show that in the regime $\frac{1}{p} + \frac{1}{q} > 1$, weak solutions are not unique in the class $ L^1_t L^p$. One crucial ingredient in the proof is the use of both temporal intermittency and oscillation in the convex integrat…
We provide some counterexamples concerning the uniqueness and regularity of weak solutions to the initial-boundary value problem for gradient flows of certain strongly polyconvex functionals by showing that such a problem can possess a trivial classical solution as well as infinitely many weak solutions that are nowhere smooth. Such polyconvex functions have been constructed in the previous work, and the nonuniqueness and nonregularity will be achieved by reformulating the gradient flow as a space-time partial differential relation and then using the convex integration method to construct cert…
Nonuniqueness of generalised weak solutions to the primitive and Prandtl equations
本文通过凸积分方案构造了二维和三维流体静力欧拉方程、三维粘性原始方程及二维Prandtl方程的非唯一广义弱解，证明了大类初值下解的存在性与非唯一性，并构造了无穷多个不守恒能量的解。
Sharp nonuniqueness for the Navier-Stokes equations
本文在周期情形下证明了不可压缩Navier-Stokes方程弱解在 $L^{p}_t L^\infty$（$p<2$）类中的尖锐非唯一性，并构造了奇异时间集Hausdorff维数任意小的非Leray-Hopf弱解，同时给出反常耗散实例。
Nonuniqueness for high-dimensional ideal MHD equations via differential inclusion
本文通过微分包含框架，证明了高维理想磁流体方程存在无限多个弱解，揭示了解的非唯一性及能量守恒的失效。
In this short note we partially extend the recent nonuniqueness results on admissible weak solutions to the Riemann problem for the 2D compressible isentropic Euler equations. We prove nonuniqueness of admissible weak solutions that start from the Riemann initial data allowing a contact discontinuity to emerge.
For initial datum of finite kinetic energy, Leray has proven in 1934 that there exists at least one global in time finite energy weak solution of the 3D Navier-Stokes equations. In this paper we prove that weak solutions of the 3D Navier-Stokes equations are not unique in the class of weak solutions with finite kinetic energy. Moreover, we prove that Holder continuous dissipative weak solutions of the 3D Euler equations may be obtained as a strong vanishing viscosity limit of a sequence of finite energy weak solutions of the 3D Navier-Stokes equations.
$L^2$-critical nonuniqueness for the 2D Navier-Stokes equations
针对环面二维不可压缩 Navier–Stokes 方程，本文构造 $p<2$ 时 $C_tL^p$ 类中的非唯一弱解，表明唯一性在接近 $L^2$ 临界尺度处失效。
Nonuniqueness analysis on the Navier-Stokes equation in $C_{t}L^{q}$ space
本文采用带 $L^{q}$ 归一化间歇射流的凸积分，证明 Navier-Stokes 方程在接近临界指数 $3$ 的 $C_{t}L^{q}$ 空间中从零初值可产生无穷多非平凡弱解，并提升其分数阶正则性。
Sharp Nonuniqueness in the Transport Equation with Sobolev Velocity Field
本文通过构造反例证明DiPerna--Lions理论中输运方程弱解唯一性条件在Sobolev速度场下的改进范围是尖锐的，并提出一种兼具凸积分与自相似特征的新颖流动机制。
Sharp spectral asymptotics for 2- and 3-dimensional Schroedinger operators with strong magnetic field are derived under rather weak smoothness conditions. In comparison with version 1 of 2005 new results are added and minor errors corrected.
The Schroedinger equation with the nonlinearity concentrated at a single point proves to be an interesting and important model for the analysis of long-time behavior of solutions, such as the asymptotic stability of solitary waves and properties of weak global attractors. In this note, we prove global well-posedness of this system in the energy space $H\sp 1$.
We study the well-posedness of the Cauchy problem for a fractional porous medium equation with a varying density. We establish existence of weak energy solutions; uniqueness and nonuniqueness is studied as well, according with the behavior of the density at infinity.
In this expository work, we present Vishik's theorem on non-unique weak solutions to the two-dimensional Euler equations on the whole space, \[ \partial_t ω+ u \cdot \nabla ω= f \, , \quad u = \frac{1}{2π} \frac{x^\perp}{|x|^2} \ast ω\, , \] with initial vorticity $ω_0 \in L^1 \cap L^p$ and $f \in L^1_t (L^1 \cap L^p)_x$, $p < \infty$. His theorem demonstrates, in particular, the sharpness of the Yudovich class. An important intermediate step is the rigorous construction of an unstable vortex, which is of independent physical and mathematical interest. We follow the strategy of Vishik but allo…
Consider a regular $d$-dimensional metric tree $Γ$ with root $o$. Define the Schroedinger operator $-Δ- V$, where $V$ is a non-negative, symmetric potential, on $Γ$, with Neumann boundary conditions at $o$. Provided that $V$ decays like $x^{-γ}$ at infinity, where $1 < γ\leq d \leq 2, γ\neq 2$, we will determine the weak coupling behavior of the bottom of the spectrum of $-Δ- V$. In other words, we will describe the asymptotical behavior of $\inf σ(-Δ- αV)$ as $α\to 0+$
We study the Cauchy problem for a system of two coupled nonlinear focusing Schroedinger equations arising in nonlinear optics. We discuss when the solutions are global in time or blow-up in finite time. Some results, in dependence of the data of the problem, are proved; in particular we give a bound, depending on the coupling parameter, for the blow-up threshold.
We prove that, for any real $λ$, the system $-Δu +λu = u^3-βuv^2$, $ -Δv+λv =v^3-βvu^2$, $ u,v\in H^1_0(Ω),$ where $Ω$ is a bounded smooth domain of $R^3$, admits a bounded family of positive solutions $(u_β, v_β)$ as $β\to +\infty$. An upper bound on the number of nodal sets of the weak limits of difference $u_β-v_β$ is also provided. Moreover, for any sufficiently large fixed value of $β>0$ the system admits infinitely many positive solutions.
A slightly modified variant of the cubic periodic one-dimensional nonlinear Schroedinger equation is shown to admit weak solutions for all initial data in certain function spaces wider than L^2. These solutions depend uniformly continuously on the initial data, in the norms considered. The solutions are constructed as sums of infinite series of multilinear operators applied to initial data; no fixed point argument or energy inequality are used. In a companion paper we have shown that weak solutions in these same function spaces are however not unique.
Motivated by some models arising in quantum plasma dynamics, in this paper we study the Maxwell-Schrödinger system with a power-type nonlinearity. We show the local well-posedness in $H^2(\mathbb{R}^3)\times H^{3/2}(\mathbb{R}^3)$ and the global existence of finite energy weak solutions, these results are then applied to the analysis of finite energy weak solutions for Quantum Magnetohydrodynamic systems.
Resonances which result from perturbation of embedded eigenvalues are studied by time dependent methods. A general theory is developed, with new and weaker conditions, allowing for perturbations of threshold eigenvalues and relaxed Fermi Golden rule. The exponential decay rate of resonances is addressed; its uniqueness in the time dependent picture is shown is certain cases. The relation to the existence of meromorphic continuation of the properly weighted Green's function to time dependent resonance is further elucidated, by giving an equivalent time dependent asymptotic expansion of the solu…
We derive the dispersion decay for solutions of the 1D discrete Schroedinger and wave equations. Based on previous works, we weaken the conditions on potentials.
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…
The large time asymptotics of nonlinear multichannel Schroedinger equations
该研究针对带一般空间局域相互作用的非线性多通道薛定谔方程，通过相空间分析与新的外区域先验传播估计，证明径向对称且H¹一致有界的解强渐近于自由波加弱局域化解，可统一分析线性含时势与非线性相互作用。
Sharp non-uniqueness for the Navier-Stokes equations in R^3
该研究证明R^3中不可压缩Navier-Stokes方程弱解在L^p([0,T];L^∞(R^3))（1≤p<2）中具有强尖锐非唯一性，推广了环面情形的结果，核心是新迭代平衡雷诺应力误差紧支集与解非紧支集。
In this article we study limits of Wigner distributions (the so-called semiclassical measures) corresponding to sequences of solutions to the semiclassical Schroedinger equation at times scales $α_{h}$ tending to infinity as the semiclassical parameter $h$ tends to zero (when $α_{h}=1/h$ this is equivalent to consider solutions to the non-semiclassical Schreodinger equation). Some general results are presented, among which a weak version of Egorov's theorem that holds in this setting. A complete characterization is given for the Euclidean space and Zoll manifolds (that is, manifolds with perio…
We investigate the asymptotic behavior of solutions to semi-classical Schroedinger equations with nonlinearities of Hartree type. For a weakly nonlinear scaling, we show the validity of an asymptotic superposition principle for slowly modulated highly oscillatory pulses. The result is based on a high-frequency averaging effect due to the nonlocal nature of the Hartree potential, which inhibits the creation of new resonant waves. In the proof we make use of the framework of Wiener algebras.
Compactly Supported Finite-Energy Stationary Solutions of the Three-Dimensional Navier-Stokes Equations
本文构造了范数可任意小的三维不可压缩纳维-斯托克斯方程非零紧支定常分布解，通过引入对数Mikado剖面解决了单尺度Mikado机制在有限能量端点的代数增益消失问题，完成了定常凸积分方案的闭合。
New exact solutions are obtained for several nonlinear physical equations, namely the Navier-Stokes and Euler systems, an isentropic compressible fluid system and a vector nonlinear Schroedinger equation. The solution methods make use of the symmetry group of the system in situations when the standard Lie method of symmetry reduction is not applicable.
Adhesion and volume filling in one-dimensional population dynamics under no-flux boundary condition
本文研究无通量边界下含黏附与体积填充的一维种群模型，针对前向-后向-前向型对流-扩散方程，用凸积分证明五类初值情形中除一种外均存在无穷多个全局弱解，揭示问题不适定性。
We study the spectrum of random ergodic Schroedinger-type operators in the weak disorder regime. We give upper and lower bounds on how much the spectrum expands at its bottom for very general perturbations. The background operator is assumed to be a periodic elliptic differential operator on R^d, not necessarily of second order.
In many situations, the notion of function is not sufficient and it needs to be extended. A classical way to do this is to introduce the notion of weak solution; another approach is to use generalized functions. Ultrafunctions are a particular class of generalized functions that has been previously introduced and used to define generalized solutions of stationary problems in [4,7,9,11,12]. In this paper we generalize this notion in order to study also evolution problems. In particular, we introduce the notion of Generalized Ultrafunction Solution (GUS) for a large family of PDE's, and we confr…
Hamiltonian compactness and dissipation for the generalized SQG equation in the inviscid limit
研究无黏极限下广义SQG方程的耗散机制，通过强紧致性防止哈密顿量异常耗散，揭示更一般的机制，证明在临界积分条件下全局解的存在性。
Flexibility for the Three-Dimensional Navier-Stokes Equations via Moving Hill Vortices
该研究基于凸积分格式，利用移动山涡旋构造三维环面上不可压缩纳维-斯托克斯方程的弱解，证明了特定指数下的非唯一性，为方程解的灵活性提供了新结果。
Flexibility for the SQG Equation with an $L^{4/3+}$ Active Scalar
该研究针对二维环面上的无粘表面准地转方程，利用凸积分格式在$p=4/3+10^{-5$的主动标量下，证明了弱解的灵活性与非唯一性，拓展了浓度临界指数相关结论。
Forward Self-Similar Solutions to the 2D Hypodissipative Navier-Stokes Equations
研究分数阶扩散参数α∈(1/2,1)的二维次耗散Navier-Stokes方程的前向自相似解，证明对于任意大的(1-2α)-齐次初始数据存在弱解，当α∈(2/3,1)时解光滑，并提供数值证据表明相关时变黏性Navier-Stokes方程解的非唯一性。
We consider a nonlinear Schroedinger equation in two spatial dimensions subject to a periodic honeycomb lattice potential. Using a multi-scale expansion together with rigorous error estimates, we derive an effective model of nonlinear Dirac type. The latter describes the propagation of slowly modulated, weakly nonlinear waves spectrally localized near a Dirac point.
We consider two-dimensional Schroedinger equations with honeycomb potentials and slow time-periodic forcing of the form: $$iψ_t (t,x) = H^\varepsilon(t)ψ=\left(H^0+2i\varepsilon A (\varepsilon t) \cdot \nabla \right)ψ,\quad H^0=-Δ+V (x) .$$ The unforced Hamiltonian, $H^0$, is known to generically have Dirac (conical) points in its band spectrum. The evolution under $H^\varepsilon(t)$ of {\it band limited Dirac wave-packets} (spectrally localized near the Dirac point) is well-approximated on large time scales ($t\lesssim \varepsilon^{-2+}$) by an effective time-periodic Dirac equation with a ga…
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