相关论文: Global regularity for 1D Eulerian dynamics with si…
We study global regularity of nonlinear systems of partial differential equations depending on the symmetric part of the gradient with Dirichlet boundary conditions. These systems arise from variational problems in plasticity with power…
This paper deals with the existence and uniqueness of solutions to kinetic equations describing alignment of self-propelled particles. The particularity of these models is that the velocity variable is not on the euclidean space but…
We overview recent developments in the study of alignment hydrodynamics, driven by a general class of symmetric communication kernels. A main question of interest is to characterize the emergent behavior of such systems, which we quantify…
In this article, we consider the Ericksen-Leslie's hyperbolic system for compressible liquid crystal model in three spatial dimensions. Global regularity for small and smooth initial data near equilibrium is proved for the case that the…
We introduce a novel regularization framework for the two-dimensional incompressible Euler equation that exactly preserves the transport structure of multi-phase vorticity fields. The key step is a reformulation of multi-phase vortex patch…
The dynamics of patterns in large two-dimensional domains remains a challenge in non-equilibrium phenomena. Often it is addressed through mild extensions of one-dimensional equations. We show that full 2D generalizations of the latter can…
We review the theoretical development in the study of critical thresholds for hyperbolic balance laws. The emphasis is on two classes of systems: Euler-Poisson-alignment (EPA) systems and hyperbolic relaxation systems. We start with an…
We study a class of partial differential equations (PDEs) in the family of the so-called Euler-Poincar\'e differential systems, with the aim of developing a foundation for numerical algorithms of their solutions. This requires particular…
Despite the strong focus of regularization on ill-posed problems, the general construction of such methods has not been fully explored. Moreover, many previous studies cannot be clearly adapted to handle more complex scenarios, albeit the…
The goal of this note is to study limiting behavior of a self-organized continuous flock evolving according to the 1D hydrodynamic Euler Alignment model. We provide a series of quantitative estimates that show how far the density of the…
This paper is concerned with the global-in-time convergence from bipolar Euler-Poisson system (BEP) to unipolar one (UEP) through the infinity-ion-mass limit by letting the ratio of the mass of ion $m_i$ over that of electron $m_e$ goes to…
Stable concurrent learning and control of dynamical systems is the subject of adaptive control. Despite being an established field with many practical applications and a rich theory, much of the development in adaptive control for nonlinear…
We study the formation of singularities for the Euler-Alignment system with influence function $\psi=\frac{k_\alpha}{|x|^\alpha}$ in 1D. As in [20] the problem is reduced to the analysis of a nonlocal 1D equation. We show the existence of…
Consider a collection of particles interacting through an attractive-repulsive potential given as a difference of power laws and normalized so that its unique minimum occurs at unit separation. For a range of exponents corresponding to mild…
Lipid bilayer membranes are commonly modeled as area-preserving fluid surfaces that resist bending. There appear to be two schools of thought in the literature concerning the actual area constraint. In some works the total or global area…
Dimensional regularization of Euclidean momentum space integrals is a highly successful technique in renormalization of quantum field theories. While it yields a straightforward algorithmic method, with which to evaluate diagrams beyond…
We provide the Alexandroff-Bakelman-Pucci estimate and global $C^{1, \alpha}$-regularity for a class of singular/degenerate fully nonlinear elliptic equations. We also derive the existence of a viscosity solution to the Dirichlet problem…
Biological networks of interacting agents exhibit similar topological properties for a wide range of scales, from cellular to ecological levels, suggesting the existence of a common evolutionary origin. A general evolutionary mechanism…
Active matter deals with systems whose particles consume energy at the individual level in order to move. To unravel features such as the emergence of collective structures several models have been suggested, such as the on-lattice model of…
Given any solution $u$ of the Euler equations which is assumed to have some regularity in space - in terms of Besov norms, natural in this context - we show by interpolation methods that it enjoys a corresponding regularity in time and that…