A method of energy estimates in \(L^\infty\) and its application to porous medium equations (Q5960513)

For the Cauchy problem for one-dimencional porous medium equation NEWLINE\[NEWLINE u_t=(u^l u_x)_x , \quad (x,t)\in \mathbb{R}\times [0,\infty), NEWLINE\]NEWLINE NEWLINE\[NEWLINE u(x,0)=u_0(x), \quad x\in \mathbb{R}, NEWLINE\]NEWLINE the \(C^\infty\)-smoothness of local solution is investigated. Main result is the next: NEWLINENEWLINENEWLINELet \(l\) is an even natural number and \(u_0\in \bigcap_{m=0}^\infty H^m(\mathbb{R})\). Then there exists \(T>0\) depending on \(\|u_0\|_{L^\infty}\) and \(\|u_{0_x}\|_{L^\infty}\) such that the problem has a unique solution \(u\in C^\infty (\mathbb{R}\times [0,T])\) such that NEWLINE\[NEWLINE \sup_{0\leq t\leq T}\|u(\cdot ,t)\|_{L^\infty}\leq \|u_0\|_{L^\infty}. NEWLINE\]NEWLINE The arguments rely on the ``\(L^\infty\)-energy method'' developed by authors and on the decomposition of perturbation into monotone part and a small part in Sobolev space of high.