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

Lorenzo Mencherini

Bollettino dell'Unione Matematica Italiana (2006)

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

Abstract

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Given the Cauchy Problem t u ( x , t ) + A ( t ) x u ( x , t ) = 0    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 ) = ( d ( t ) a ( t ) b ( t ) - d ( t ) )    t [ 0 , T ] into the complex matrix A ( t ) = ( c ( t ) a ( t ) a ( t ) - c ( t ) ) = ( i a - b 2 a + b 2 + i d a + b 2 - i d - i a - b 2 ) and showed that the given Cauchy Problem is well posed in C in a neighborhood ofzero if and only if (see also [MS]) the following condition h | a | 2 C t 2 | D | 2 is satisfied, where D = a ˙ c - c ˙ a e h = - d e t A = | a | 2 - | c | 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 -

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