The Cauchy problem for the system \(\partial_ tz - \partial_ x(\bar z^ y) = 0\)
The authors consider the Cauchy problem \[ \partial_ t z- \partial_ x \overline {z}^ \gamma=0, \;\;z=u+iv, \quad -\infty<x<\infty, \;\;t>0,\qquad z(x,0)= z_ 0(x), \quad -\infty<x<\infty, \tag{1} \] where \(\gamma\) is a constant, \(1<\gamma<2\). The main result is the existence of a global solution if \(z_ 0\in L^ 2\cap L^ \infty\), and \(z_ 0\) takes values only in a closed sector of the plane with vertex at the origin and angle \(\pi/ (\gamma+1)\). The proof using compensated compactness consists of establishing bounds in \(L^ 2\) (rather than the usual \(L^ \infty\) bounds) for the corresponding viscous system, and then using generalized Young measures. The paper uses and extends ideas from earlier work of DiPerna, Majda and Schonbek on generalized Young measures, and more recent work of Serre and Kan specifically aimed at systems of two equations.
scientific article; zbMATH DE number 635888