Optimal control for a two-sidedly degenerate aggregation equation

In this paper, we are concerned with the study of the mathematical analysis for an optimal control of a nonlocal degenerate aggregation model. This model describes the aggregation of organisms such as pedestrian movements, chemotaxis, animal swarming. We establish the wellposedness (existence and uniqueness) for the weak solution of the direct problem by means of an auxiliary nondegenerate aggregation equation, the Faedo–Galerkin method (for the existence result) and duality method (for the uniqueness). Moreover, for the adjoint problem, we prove the existence result of minimizers and first-order necessary conditions. The main novelty of this work concerns the presence of a control to our nonlocal degenerate aggregation model. Our results are complemented with some numerical simulations

The regularity of the boundary of a multidimensional aggregation patch

Let $d \geq 2$ and let $N(y)$ be the fundamental solution of the Laplace equation in $R^d$ We consider the aggregation equation $$\frac{\partial \rho}{\partial t} + \operatorname{div}(\rho v) =0, v = -\nabla N * \rho$$ with initial data $\rho(x,0) = \chi_{D_0}$, where $\chi_{D_0}$ is the indicator function of a bounded domain $D_0 \subset R^d.$ We now fix $0 < \gamma < 1$ and take $D_0$ to be a bounded $C^{1+\gamma}$ domain (a domain with smooth boundary of class $C^{1+\gamma}$). Then we have Theorem: If $D_0$ is a $C^{1 + \gamma}$ domain, then the initial value problem above has a solution given by $$\rho(x,t) = \frac{1}{1 -t} \chi_{D_t}(x), \quad x \in R^d, \quad 0 \le t < 1$$ where $D_t$ is a $C^{1 + \gamma}$ domain for all $0 \leq t < 1$

The Filippov characteristic flow for the aggregation equation with mildly singular potentials

Existence and uniqueness of global in time measure solution for the multidimensional aggregation equation is analyzed. Such a system can be written as a continuity equation with a velocity field computed through a self-consistent interaction potential. In Carrillo et al. (Duke Math J (2011)), a well-posedness theory based on the geometric approach of gradient flows in measure metric spaces has been developed for mildly singular potentials at the origin under the basic assumption of being lambda-convex. We propose here an alternative method using classical tools from PDEs. We show the existence of a characteristic flow based on Filippov's theory of discontinuous dynamical systems such that the weak measure solution is the pushforward measure with this flow. Uniqueness is obtained thanks to a contraction argument in transport distances using the lambda-convexity of the potential. Moreover, we show the equivalence of this solution with the gradient flow solution. Finally, we show the convergence of a numerical scheme for general measure solutions in this framework allowing for the simulation of solutions for initial smooth densities after their first blow-up time in Lp-norms.Comment: 33 page

A repertoire of repulsive Keller–Segel models with logarithmic sensitivity: Derivation, traveling waves, and quasi-stationary dynamics

In this paper, we show how the chemotactic model {partial derivative(t)rho = d(1) Delta(x)rho - del(x) . (rho del(x)c) partial derivative(t)c = d(2) Delta(x)c + F(rho, c, del(x)rho, del(x)c, Delta x rho) introduced in Alejo and Lopez (2021), which accounts for a chemical production-degradation operator of Hamilton-Jacobi type involving first- and second-order derivatives of the logarithm of the cell concentration, namely, F = mu + tau c - sigma rho + A Delta(x)rho/rho + B vertical bar del(x)rho vertical bar(2)/rho(2) + C vertical bar del(x)c vertical bar(2), with mu, tau, sigma, A, B, C is an element of R, can be formally reduced to a repulsive Keller-Segel model with logarithmic sensitivity { partial derivative(t)rho = D-1 Delta(x)rho + chi del(x) . (rho del(x) log(c)), chi, lambda, beta > 0, partial derivative(t)c = D-2 Delta(x)c + lambda rho c - beta c whenever the chemotactic parameters are appropriately chosen and the cell concentration keeps strictly positive. In this way, some explicit solutions (namely, traveling waves and stationary cell density profiles) of the former system can be transferred to a number of variants of the the latter by means of an adequate change of variables.Spanish Government RTI2018-098850-B-I00 Junta de AndaluciaEuropean Commission PY18-RT-2422 B-FQM-580-UGRUniversidad de Granada/CBU

Convergence of a linearly transformed particle method for aggregation equations

We study a linearly transformed particle method for the aggregation equation with smooth or singular interaction forces. For the smooth interaction forces, we provide convergence estimates in $L^1$ and $L^\infty$ norms depending on the regularity of the initial data. Moreover, we give convergence estimates in bounded Lipschitz distance for measure valued solutions. For singular interaction forces, we establish the convergence of the error between the approximated and exact flows up to the existence time of the solutions in $L^1 \cap L^p$ norm

Swarm dynamics and equilibria for a nonlocal aggregation model

We consider the aggregation equation ρt − ∇ · (ρ∇K ∗ ρ) = 0 in Rn, where the interaction potential K models short-range repulsion and long-range attraction. We study a family of interaction potentials for which the equilibria are of finite density and compact support. We show global well-posedness of solutions and investigate analytically and numerically the equilibria and their global stability. In particular, we consider a potential for which the corresponding equilibrium solutions are of uniform density inside a ball of Rn and zero outside. For such a potential, various explicit calculations can be carried out in detail. In one dimension we fully solve the temporal dynamics, and in two or higher dimensions we show the global stability of this steady state within the class of radially symmetric solutions. Finally, we solve the following restricted inverse problem: given a radially symmetric density ρ ̄ that is zero outside some ball of radius R and is polynomial inside the ball, construct an interaction potential K for which ρ̄ is the steady state solution of the corresponding aggregation equation. Throughout the paper, numerical simulations are used to motivate and validate the analytical results