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

Bollettino dell'Unione Matematica Italiana (2006)

- Volume: 9-B, Issue: 1, page 215-220
- ISSN: 0392-4033

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topMencherini, 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

top- MENCHERINI, L. - SPAGNOLO, S., Well Posedness of 2x2 Systems with ${C}^{\mathrm{\infty}}$-Coefficients, in «Hyperbolic Problems and Related Topics, Cortona 2002», F. Colombini and T. Nishitani Ed.s, International Press, Somerville, USA, 235-242. MR2056853
- 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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