A simple necessary and sufficient condition for well-posedness of $2 \times 2$ differential systems with time-dependent coefficients

Mencherini, Lorenzo

Bollettino della Unione matematica italiana, Série 8, 9B (2006) no. 1, pp. 215-220

Résumé

Given the Cauchy Problem \[\partial_t u(x,t)+ A(t)\partial_x u(x,t)=0 \qquad u(0,x)=u_0(x)\] Nishitani [N], by making use of a change of basis by a constant matrix, transformed the real, analytic, hyperbolic matrix \[A(t)=\left(\begin{array}{cc} d(t) a(t) \\b(t) -d(t) \\ \end{array}\right) \qquad t\in [0, T]\] into the complex matrix \[A^\sharp(t)=\left(\begin{array}{cc}c^\sharp(t) a^\sharp(t) \\a^\sharp(t) -c^\sharp(t) \\\end{array} \right)= \left(\begin{array}{cc} i \frac{a-b}{2} \frac{a+b}{2}+id \\ \frac{a+b}{2}-id -i \frac{a-b}{2} \\ \end{array}\right)\]and showed that the given Cauchy Problem is well posed in \( C^\infty\) in a neighborhood ofzero if and only if (see also [MS]) the following condition \[h|a^\sharp |^2 \geq C t^2 |D^\sharp|^2\] is satisfied, where \[D^\sharp= \dot{a}^\sharp c^\sharp- \dot{c}^\sharp a^\sharp\ \text{e} h=-detA=|a^\sharp |^2- |c^\sharp |^2.\] In this short note, we give a very simple condition, which is equivalent to that of Nishitani (and then a necessary and sufficient for the Well-Posedness), but where only the elements of appear and not their derivatives.