Wellposedness and regularity of solutions of an aggregation equation (Q971969)

Consider the aggregation equation in \(\mathbb{R}^d\) with fractional dissipation in the following form \[ u_t+\nabla\cdot (u\nabla K*u)=-\nu \Lambda^{\gamma}u, \] where \(d\geq 2\), \(\nu\geq 0\), \(0<\gamma\leq 2\), and the kernel \(K(x)=e^{-|x|}\). The fractional Laplacian \(\Lambda^{\gamma}\) is defined via the Fourier transform: \(\widehat{\Lambda^{\gamma}f}(\xi)=|\xi|^{\gamma}\hat{f}(\xi).\) Distinguishing the diffusion exponent \(\gamma\) into three different cases: supercritical (\(\gamma<1\)), critical \((\gamma=1)\) and subcritical \((\gamma>1)\), and assuming appropriate initial conditions, the authors prove certain results on the local or global existence, uniqueness, and regularity of solutions to the equation.