A blob method for the aggregation equation (Q2796015)

The authors start from the aggregation equation \(\rho_t+\nabla \cdot (v\rho )=0\), with \(v=-\nabla K\ast \rho \), for some kernel \(K\), with the initial condition \(\rho (x,0)=\rho_0(x)\). Considering the trajectory of a particle NEWLINE\[NEWLINE\frac{d}{dt}X^{t}(\alpha )=-\nabla K\ast \rho (X^t(\alpha ),t),\quad X^0 (\alpha )=\alpha ,NEWLINE\]NEWLINE they rewrite the aggregation equation as NEWLINE\[NEWLINE\begin{cases} \frac{d}{dt}\rho (X^{t}(\alpha ),t)=(\Delta K\ast \rho (X^t(\alpha ),t))\rho (X^t (\alpha ),t), \\ \rho (X^{0}(\alpha ),0)=\rho _{0}(\alpha ).\end{cases}NEWLINE\]NEWLINE They introduce the approximate velocity field along the exact particle trajectories as NEWLINE\[NEWLINEv^{h}(x,t)=-\int_{\mathbb{R}^{d}}\nabla K_{\delta }(x-X^{t}(\alpha ))\rho _{0}^{\text{particle}}(\alpha )d\alpha =-\sum_{j}\nabla K_{\delta }(x-X_{j}(t))\rho _{0_{j}}h^{d}NEWLINE\]NEWLINE where \(K_{\delta }\) is the convolution product between the kernel \(K\) and \(\psi _{\delta }\) defined through \(\psi _{\delta }(x)=\delta ^{-d}\psi (x/\delta )\) \(\psi \) being a smooth mollifier or blob function. This allows defining approximate particle trajectories and the authors build the associated blob method. The main result of the paper proves the convergence of the approximate solution built through this blob method, under assumptions on the kernel and on the mollifier. In the last part of their paper, the authors present the results of numerical simulations choosing either regular or discontinuous initial data.