\(H^1\)-solutions for the Hele-Shaw equation (Q2046191)

The Cauchy problem \[ u_t = - (|u|^p u_x^3)_x, t>0, x \in\mathbb{R}; \quad u(0,x)=u_0(x), u_0 \in H^1(\mathbb{R}) \] is studied, which is a model for a thin layer of fluid in the Hele-Shaw cell. Some previous results of the authors (obtained for \(p<3\)) are improved in the case \(p< 8\). The distributional solution is introduced, whose existence in \(L^{\infty}(0,T; H^1(\mathbb{R}))\) is given in Theorem 1.1, for arbitrary \(T>0\). An approximate problem is considered, depending on a vanishing viscosity. A classical solution for this problem is obtained, along with some a priori estimates. The Theorem 1.1 is proved by by passing to the limit when the vanishing viscosity converges to zero. The main tool is the Aubin-Lions Lemma. The existence of local in the time solutions of the initial problem for \(p=8\) is given in the last part.