齐性空间上关于测度值Lagrange量的非线性方程非平凡解的存在性

Singularities and heteroclinic connections in complex-valued evolutionary equations with a quadratic nonlinearity

In this paper, we consider the dynamics of solutions to complex-valued evolutionary partial differential equations (PDEs) and show existence of heteroclinic orbits from nontrivial equilibria to zero via computer-assisted proofs. We also show that the existence of unbounded solutions along unstable manifolds at the equilibrium follows from the existence of heteroclinic orbits. Our computer-assisted proof consists of three separate techniques of rigorous numerics: an enclosure of a local unstable manifold at the equilibria, a rigorous integration of PDEs, and a constructive validation of a trapp…

A Directional Lipschitz Extension Lemma, with Applications to Uniqueness and Lagrangianity for the Continuity Equation

We prove a Lipschitz extension lemma in which the extension procedure simultaneously preserves the Lipschitz continuity for two non-equivalent distances. The two distances under consideration are the Euclidean distance and, roughly speaking, the geodesic distance along integral curves of a (possibly multi-valued) flow of a continuous vector field. The Lipschitz constant for the geodesic distance of the extension can be estimated in terms of the Lipschitz constant for the geodesic distance of the original function. This Lipschitz extension lemma allows us to remove the high integrability assump…

Weak Measure-Valued Solutions of a Nonlinear Hyperbolic Conservation Law

We revisit a well-established model for highly re-entrant semi-conductor manufacturing systems, and analyze it in the setting of states, in- and outfluxes being Borel measures. This is motivated by the lack of optimal solutions in the L1-setting for transitions from a smaller to a larger equilibrium with zero backlog. Key innovations involve dealing with discontinuous velocities in the presence of point masses, and a finite domain with in- and outfluxes. Taking a Lagrangian point of view, we establish existence and uniqueness of solutions, and formulate a notion of weak solution. We prove cont…

Chirality transitions in frustrated $S^{2}$-valued spin systems

We study the discrete-to-continuum limit of the helical XY $S^{2}$-spin system on the lattice $\mathbb{Z}^{2}$. We scale the interaction parameters in order to reduce the model to a spin chain in the vicinity of the Landau-Lifschitz point and we prove that at the same energy scaling under which the $S^{1}$-model presents scalar chirality transitions, the cost of every vectorial chirality transition is now zero. In addition we show that if the energy of the system is modified penalizing the distance of the $S^{2}$ field from a finite number of copies of $S^{1}$, it is still possible to prove th…

无旋轴对称Euler- $α$方程：解的存在性、唯一性和Radon测度

Axisymmetric Euler-$α$ Equations without Swirl: Existence, Uniqueness, and Radon Measure Valued Solutions

本文证明了无旋三维轴对称Euler- $α$方程（也称为Lagrangian-averaged Euler- $α$）整体弱解的存在性，只要初始未过滤速度$v_0$满足$\frac{\nabla \times v_0}{r}$是紧支撑的有限Randon测度.此外，在这种情况下，还建立了全局存在性和唯一性，提供$\frac{\nabla \times v_0}{r} \in L^p_c(\mathbb{R}^3)$与$p>{3/2}$。值得一提的是，迄今为止，对于理想不可压缩流的三维欧拉方程，还没有这样的结果。

A Lagrangian perspective on nonautonomous advection-diffusion processes in the low-diffusivity limit

We study mass preserving transport of passive tracers in the low-diffusivity limit using Lagrangian coordinates. Over finite-time intervals, the solution-operator of the nonautonomous diffusion equation is approximated by that of a time-averaged diffusion equation. We show that leading order asymptotics that hold for functions [Krol, 1991] extend to the dominant nontrivial singular value. This answers questions raised in [Karrasch & Keller, 2020]. The generator of the time-averaged diffusion/heat semigroup is a Laplace operator associated to a weighted manifold structure on the material manifo…

具有全实边界条件的巴拿赫空间值柯西-黎曼方程

Banach space valued Cauchy-Riemann equations with totally real boundary conditions

本文的主要目的是给出复巴拿赫空间中具有全实边界条件的柯西-黎曼方程的一个一般正则性结果。通常的椭圆$L^p$ -正则性结果在一个关键假设下成立：全实子流形必须在$L^p$ -空间或其闭子空间上建模。其次，我们描述了一类这样的全实子流形的例子，即黎曼曲面上的联络空间中的规范不变拉格朗日子流形。这些构成自然边界条件的反自对偶方程的4流形与边界空间-时间分裂，导致对一个弗洛尔同调的定义3流形的边界，这是第一步的程序由Salamon的证明阿蒂亚-弗洛尔猜想。这类边值问题的主部是一个具有全实边界条件的巴拿赫空间值Cauchy-Riemann方程的例子。