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

• Volume: 9-B, Issue: 1, page 215-220
• ISSN: 0392-4041

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## Abstract

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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.

## How to cite

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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 9-B.1 (2006): 215-220. <http://eudml.org/doc/289618>.

@article{Mencherini2006,
abstract = {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. },
author = {Mencherini, Lorenzo},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {215-220},
publisher = {Unione Matematica Italiana},
title = { A simple necessary and sufficient condition for well-posedness of $2 \times 2$ differential systems with time-dependent coefficients},
url = {http://eudml.org/doc/289618},
volume = {9-B},
year = {2006},
}

TY - JOUR
AU - Mencherini, Lorenzo
TI - A simple necessary and sufficient condition for well-posedness of $2 \times 2$ differential systems with time-dependent coefficients
JO - Bollettino dell'Unione Matematica Italiana
DA - 2006/2//
PB - Unione Matematica Italiana
VL - 9-B
IS - 1
SP - 215
EP - 220
AB - 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.
LA - eng
UR - http://eudml.org/doc/289618
ER -

## References

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1. 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. MR2056853
2. NISHITANI, T., Hyperbolicity of two by two systems with two independent variables, Comm. Part. Diff. Equat.23 (1998), 1061-1110. Zbl0913.35079MR1632796DOI10.1080/03605309808821378

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