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Register a new userGeometric Structure of Mass Concentration Sets for Pressureless Euler Alignment Systems
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We study the limiting dynamics of the Euler Alignment system with a smooth, heavy-tailed interaction kernel $phi$ and unidirectional velocity $mathbf{u} = (u, 0, ldots, 0)$. We demonstrate a striking correspondence between the entropy function $e_0 = partial_1 u_0 + phi*rho_0$ and the limiting concentration set, i.e., the support of the singular part of the limiting density measure. In a typical scenario, a flock experiences aggregation toward a union of $C^1$ hypersurfaces: the image of the zero set of $e_0$ under the limiting flow map. This correspondence also allows us to make statements about the fine properties associated to the limiting dynamics, including a sharp upper bound on the dimension of the concentration set, depending only on the smoothness of $e_0$. In order to facilitate and contextualize our analysis of the limiting density measure, we also include an expository discussion of the wellposedness, flocking, and stability of the Euler Alignment system, most of which is new.
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We study a non-local hydrodynamic system with control. First we characterize the control dynamics as a sub-optimal approximation to the optimal control problem constrained to the evolution of the pressureless Euler alignment system. We then discuss the critical thresholds that leading to global regularity or finite-time blow-up of strong solutions in one and two dimensions. Finally we propose a finite volume scheme for numerical solutions of the controlled system. Several numerical simulations are shown to validate the theoretical and computational results of the paper.
Considering the isentropic Euler equations of compressible fluid dynamics with geometric effects included, we establish the existence of entropy solutions for a large class of initial data. We cover fluid flows in a nozzle or in spherical symmetry when the origin r=0 is included. These partial differential equations are hyperbolic, but fail to be strictly hyperbolic when the fluid mass density vanishes and vacuum is reached. Furthermore, when geometric effects are taken into account, the sup-norm of solutions can not be controlled since there exist no invariant regions. To overcome these difficulties and to establish an existence theory for solutions with arbitrarily large amplitude, we search for solutions with finite mass and total energy. Our strategy of proof takes advantage of the particular structure of the Euler equations, and leads to a versatile framework covering general compressible fluid problems. We establish first higher-integrability estimates for the mass density and the total energy. Next, we use arguments from the theory of compensated compactness and Young measures, extended here to sequences of solutions with finite mass and total energy. The third ingredient of the proof is a characterization of the unbounded support of entropy admissible Young measures. This requires the study of singular products involving measures and principal values.
We study partial analyticity of solutions to elliptic systems and analyticity of level sets of solutions to nonlinear elliptic systems. We consider several applications, including analyticity of flow lines for bounded stationary solutions to the 2-d Euler equation, and analyticity of water waves with and without surface tension.
In this paper, we investigate the well-posedness theory of compressible jet flows for two dimensional steady Euler system with non-zero vorticity. One of the key observations is that the stream function formulation for two dimensional compressible steady Euler system with non-zero vorticity enjoys a variational structure, so that the jet problem can be reformulated as a domain variation problem. This allows us to adapt the framework developed by Alt, Caffarelli and Friedman for the one-phase free boundary problems to obtain the existence and uniqueness of smooth solutions to the subsonic jet problem with non-zero vorticity. We also show that there is a critical mass flux, such that as long as the incoming mass flux does not exceed the critical value, the well-posedness theory holds true.
In this note, we extend the results on eigenfunction concentration in billiards as proved by the third author in cite{M1}. There, the methods developed in Burq-Zworski cite{BZ3} to study eigenfunctions for billiards which have rectangular components were applied. Here we take an arbitrary polygonal billiard $B$ and show that eigenfunction mass cannot concentrate away from the vertices; in other words, given any neighbourhood $U$ of the vertices, there is a lower bound $$ int_U |u|^2 geq c int_B |u|^2 $$ for some $c = c(U) > 0$ and any eigenfunction $u$.
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