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

Mencherini, Lorenzo

A simple necessary and sufficient condition for well-posedness of

2\times 2

differential systems with time-dependent coefficients

Mencherini, Lorenzo

Bollettino dell'Unione Matematica Italiana, Tome 9-A (2006), p. 215-220 / Harvested from Biblioteca Digitale Italiana di Matematica

Access to full text
Full (PDF)
Full (DjVu)

Abstract
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 $h|a^{\sharp}|^{2}\geq Ct^{2}|D^{\sharp}|^{2}$ 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.

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 $h|a^{\sharp}|^{2}\geq Ct^{2}|D^{\sharp}|^{2}$ 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.

Publié le : 2006-02-01

MR 2204908 | Zbl 1150.35065

@article{BUMI_2006_8_9B_1_215_0,
     author = {Lorenzo Mencherini},
     title = { A simple necessary and sufficient condition for well-posedness of $2 \times 2$ differential systems with time-dependent coefficients},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {9-A},
     year = {2006},
     pages = {215-220},
     zbl = {1150.35065},
     mrnumber = {2204908},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2006_8_9B_1_215_0}
}

Mencherini, Lorenzo.  A simple necessary and sufficient condition for well-posedness of $2 \times 2$ differential systems with time-dependent coefficients. Bollettino dell'Unione Matematica Italiana, Tome 9-A (2006) pp. 215-220. http://gdmltest.u-ga.fr/item/BUMI_2006_8_9B_1_215_0/

Bibliographie

[MS] Mencherini, L. - Spagnolo, S., 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 2056853

[N] Nishitani, T., Hyperbolicity of two by two systems with two independent variables, Comm. Part. Diff. Equat.23 (1998), 1061-1110. | MR 1632796 | Zbl 0913.35079