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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.
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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.
We study the global wellposedness of the Euler-Monge-Amp`ere (EMA) system. We obtain a sharp, explicit critical threshold in the space of initial configurations which guarantees the global regularity of EMA system with radially symmetric initial data. The result is obtained using two independent approaches -- one using spectral dynamics of Liu & Tadmor [Comm. Math. Physics 228(3):435-466, 2002] and another based on the geometric approach of Brenier & Loeper [Geom. Funct. Analysis 14(6):1182--1218, 2004]. The results are extended to 2D radial EMA with swirl.
We investigate the inviscid compressible flow (Euler) equations constrained by an isentropic equation of state (EOS), whose functional form in pressure is an arbitrary function of density alone. Under the aforementioned condition, we interrogate using symmetry methods the scale-invariance of the homentropic inviscid Euler equations. We find that under general conditions, we can reduce the inviscid Euler equations into a system of two coupled ordinary differential equations. To exemplify the utility of these results, we formulate two example scale-invariant, self-similar solutions. The first example includes a shock-free expanding bubble scenario, featuring a modified Tait EOS. The second example features the classical Noh problem, coupled to an arbitrary isentropic EOS. In this case, in order to satisfy the conditions set forth in the classical Noh problem, we find that the solution for the flow is given by a transcendental algebraic equation in the shocked density.
In the first part of this paper we establish a uniqueness result for continuity equations with velocity field whose derivative can be represented by a singular integral operator of an $L^1$ function, extending the Lagrangian theory in cite{BouchutCrippa13}. The proof is based on a combination of a stability estimate via optimal transport techniques developed in cite{Seis16a} and some tools from harmonic analysis introduced in cite{BouchutCrippa13}. In the second part of the paper, we address a question that arose in cite{FilhoMazzucatoNussenzveig06}, namely whether 2D Euler solutions obtained via vanishing viscosity are renormalized (in the sense of DiPerna and Lions) when the initial data has low integrability. We show that this is the case even when the initial vorticity is only in~$L^1$, extending the proof for the $L^p$ case in cite{CrippaSpirito15}.
We present a simple model of alignment of a large number of rigid bodies (modeled by rotation matrices) subject to internal rotational noise. The numerical simulations exhibit a phenomenon of first order phase transition with respect the alignment intensity, with abrupt transition at two thresholds. Below the first threshold, the system is disordered in large time: the rotation matrices are uniformly distributed. Above the second threshold, the long time behaviour of the system is to concentrate around a given rotation matrix. When the intensity is between the two thresholds, both situations may occur. We then study the mean-field limit of this model, as the number of particles tends to infinity, which takes the form of a nonlinear Fokker--Planck equation. We describe the complete classification of the steady states of this equation, which fits with numerical experiments. This classification was obtained in a previous work by Degond, Diez, Merino-Aceituno and the author, thanks to the link between this model and a four-dimensional generalization of the Doi--Onsager equation for suspensions of rodlike polymers interacting through Maier--Saupe potential. This previous study concerned a similar equation of BGK type for which the steady-states were the same. We take advantage of the stability results obtained in this framework, and are able to prove the exponential stability of two families of steady-states: the disordered uniform distribution when the intensity of alignment is less than the second threshold, and a family of non-isotropic steady states (one for each possible rotation matrix, concentrated around it), when the intensity is greater than the first threshold. We also show that the other families of steady-states are unstable, in agreement with the numerical observations.
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