Suppose we consider the heat equation in 2D: $u_t - \Delta u=0$ with initial value $u_0$. Then how to estimate the fundamental solution $u=e^{t\Delta}u_{0}$? More precisely, for $0<s<1$, if $u_{0} \in H^{s}(\mathbb R^2)$, then why is $u \in L^{1}(0,T; H^{2+\tilde s})$ for $0<\tilde s<s$?
I tried to use Young's inequality for convolution but it doesn't tell any information about Sobolev space of $u$.