\(L^q\)-solvability for an equation of viscoelasticity in power type material
The authors study the existence, uniqueness, regularity, continuous dependence, unique continuation, a blow-up alternative for mild solutions, and global well-posedness of the nonlinear Volterra equation \[ u_t = \int_0^t dg_\alpha(s) \Delta u(t-s,x)- \nabla p +h - (u\cdot \nabla u),\quad \textrm{div}(u)=0, \] in \((0,\infty)\times \Omega\), where \(u=0\) on \((0,\infty)\times \partial \Omega\) and \(u(0,x)=u_0(x)\) in \(\Omega\). Here the kernel is taken to be \(g_\alpha (t)= t^{\alpha}/\Gamma(\alpha+1)\) with \(0\leq \alpha <1\) and a mild solution is a solution to the equation \[ u(t)= S_\alpha (tA)u_0 + \int_0^t S_\alpha((t-s)A)(F(u)(s)+Ph(s))\, ds, \] where \(P\) is the Leray projection on divergence free functions, \(F(u)= P(u\cdot \nabla)u\), \(A=P\Delta\) and \[S_\alpha(tA)= \frac 1{2\pi i} \int_{Ha}e^{\lambda t}\lambda^\alpha(\lambda^{\alpha+1}I+A)^{-1}\, d\lambda,\] where \(t>0\) and \(Ha\) is a suitable path. The existence results show that the mild solutions have more spatial regularity in terms of estimates on norms in fractional power spaces when \(\alpha\) is closer to \(0\), the case of the Navier-Stokes equations. The linear estimates needed are stated in an abstract setting for sectorial operators which makes it possible to restate the results for some other equations as well.
scientific article; zbMATH DE number 7349777
scientific article