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.

Abstract (VO)

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.

Riassunto

Dato il Problema di Cauchy \[\partial_t u(x,t)+ A(t)\partial_x u(x,t)=0 \qquad u(0,x)=u_0(x)\] Nishitani [N], dopo aver effettuato, mediante una matrice di cambiamento di base costante, la trasformazione della matrice \[A(t)=\left(\begin{array}{cc} d(t) a(t) \\b(t) -d(t) \\ \end{array}\right) \qquad t\in [0, T]\] reale, analitica e iperbolica, nella matrice complessa \[A^\sharp(t)=\left(\begin{array}{cc}c^\sharp(t) a^\sharp(t) \\a^\sharp(t) -c^\sharp(t) \\\end{array} \right)= \left(\begin{array}{cc} i \frac{a-b}{2} \frac{a+b}{2}+id \\ \frac{a+b}{2}-id -i \frac{a-b}{2} \\ \end{array}\right)\] ha dimostrato che il Problema di Cauchy considerato è ben posto in \( C^\infty\) in un intorno di zero se e solo se vale la condizione \[h|a^\sharp |^2 \geq C t^2 |D^\sharp|^2\] dove \[D^\sharp= \dot{a}^\sharp c^\sharp- \dot{c}^\sharp a^\sharp\ \text{e} h=-detA=|a^\sharp |^2- |c^\sharp |^2.\] In questo breve lavoro invece diamo una semplicissima condizione equivalente a quella di Nishitani (e quindi necessaria e sufficiente per la buona positura), in cui compaiono solamente gli elementi di \(A(t)\) e non le loro derivate.

Bibliographie

[MS] L. Mencherini - S. Spagnolo, Well Posedness of 2x2 Systems with $C^\infty$-Coefficients, in «Hyperbolic Problems and Related Topics, Cortona 2002», F. Colombini and T. Nishitani Ed.s, International Press, Somerville, USA, 235-242. | MR

[N]T. Nishitani, Hyperbolicity of two by two systems with two independent variables, Comm. Part. Diff. Equat. 23 (1998), 1061-1110. | fulltext EuDML | DOI | MR | Zbl