Namespaces
Variants
Actions

Difference between revisions of "Abstract hyperbolic differential equation"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (fixing ))
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
 +
<!--This article has been texified automatically. Since there was no Nroff source code for this article,
 +
the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist
 +
was used.
 +
If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category.
 +
 +
Out of 84 formulas, 84 were replaced by TEX code.-->
 +
 +
{{TEX|semi-auto}}{{TEX|done}}
 
Consider the [[Cauchy problem|Cauchy problem]] for the symmetric hyperbolic system (cf. also [[Hyperbolic partial differential equation|Hyperbolic partial differential equation]])
 
Consider the [[Cauchy problem|Cauchy problem]] for the symmetric hyperbolic system (cf. also [[Hyperbolic partial differential equation|Hyperbolic partial differential equation]])
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a1200601.png" /></td> </tr></table>
+
$$
 
+
  \begin{cases}  \frac { \partial u } { \partial t } + \sum _ { j = 1 } ^ { m } a _ { j } ( x ) \frac { \partial u } { \partial x _ { j } } + c ( x ) u = f ( x , t ) , \\ { ( x , t ) \in \Omega \times [ 0 , T ] }, \\ { u ( x , 0 ) = u _ { 0 } ( x ) , \quad x \in \Omega, } \end{cases} 
 +
$$
 
with the boundary conditions
 
with the boundary conditions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a1200602.png" /></td> </tr></table>
+
\begin{equation*} u ( x , t ) \in P ( x ) , \quad ( x , t ) \in \partial \Omega \times [ 0 , T ]. \end{equation*}
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a1200603.png" /> is a bounded domain with smooth boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a1200604.png" /> (when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a1200605.png" />, no boundary conditions are necessary), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a1200606.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a1200607.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a1200608.png" /> are smooth functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a1200609.png" /> with as values real matrices in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006010.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006011.png" /> being symmetric. It is assumed that the boundary matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006013.png" />, is non-singular, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006014.png" /> is the unit outward normal vector to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006015.png" />. Also, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006016.png" /> denotes the maximal non-negative subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006017.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006018.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006020.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006021.png" /> is not a proper subset of any other subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006022.png" /> with this property. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006023.png" /> is the unknown function.
+
Here, $\Omega \subset \mathbf{R} ^ { m }$ is a bounded domain with smooth boundary $\partial \Omega$ (when $\Omega = \mathbf{R} ^ { m }$, no boundary conditions are necessary), and $a_{j}  ( x )$, $j = 1 , \ldots , m$, and $c ( x )$ are smooth functions on $\Omega$ with as values real matrices in $L ( \mathbf{R} ^ { p } )$, the $a_{j}  ( x )$ being symmetric. It is assumed that the boundary matrix $b ( x ) = \sum _ { j = 1 } ^ { m } n _ { j } ( x ) a _ { j } ( x )$, $x \in \partial \Omega$, is non-singular, where $n = ( n _{1} , \ldots , n _ { m } )$ is the unit outward normal vector to $\partial \Omega$. Also, $P ( x )$ denotes the maximal non-negative subspace of $\mathbf{R} ^ { p }$ with respect to $b ( x )$, i.e. $( b ( x ) u , u ) \geq 0$, $u \in P ( x )$, and $P ( x )$ is not a proper subset of any other subspace of $\mathbf{R} ^ { p }$ with this property. The function $u = ( u _ { 1 } , \ldots , u _ { p } )$ is the unknown function.
  
One can handle this problem as the Cauchy problem for an [[Evolution equation|evolution equation]] in a Banach space (cf. also [[Linear differential equation in a Banach space|Linear differential equation in a Banach space]]). Indeed, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006024.png" /> be the smallest closed extension in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006025.png" /> of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006026.png" /> defined by
+
One can handle this problem as the Cauchy problem for an [[Evolution equation|evolution equation]] in a Banach space (cf. also [[Linear differential equation in a Banach space|Linear differential equation in a Banach space]]). Indeed, let $A$ be the smallest closed extension in $X = [ L ^ { 2 } ( \Omega ) ] ^ { p }$ of the operator $\mathcal{A}$ defined by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006027.png" /></td> </tr></table>
+
\begin{equation*} {\cal A} u = \sum _ { j = 1 } ^ { m } a _ { j } ( x ) \frac { \partial u } { \partial x _ { j } } + c ( x ) u \end{equation*}
  
 
with domain
 
with domain
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006028.png" /></td> </tr></table>
+
\begin{equation*} D ( \mathcal{A} ) = \left\{ u \in [ H ^ { 1 } ( \Omega )] ^ { p } : u ( x ) \in P ( x ) \text { a.e. on } \partial \Omega \right\}. \end{equation*}
  
Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006029.png" /> is the negative generator of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006030.png" /> semi-group on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006031.png" /> (cf. [[#References|[a1]]], [[#References|[a2]]]; see also [[Semi-group of operators|Semi-group of operators]]). Hence, the Hille–Yoshida theorem proves the existence of a unique solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006032.png" /> to the Cauchy problem
+
Then $A$ is the negative generator of a $C _ { 0 }$ semi-group on $X$ (cf. [[#References|[a1]]], [[#References|[a2]]]; see also [[Semi-group of operators|Semi-group of operators]]). Hence, the Hille–Yoshida theorem proves the existence of a unique solution $u \in C ( [ 0 , T ] ; D ( \mathcal{A} ) ) \cap C ^ { 1 } ( [ 0 , T ] ; X )$ to the Cauchy problem
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006033.png" /></td> </tr></table>
+
\begin{equation*} \frac { d u } { d t } + A u = f ( t ) , t \in [ 0 , T ], \end{equation*}
  
 
which is given in the form
 
which is given in the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006034.png" /></td> </tr></table>
+
\begin{equation*} u ( t ) = e ^ { - t A } u _ { 0 } + \int _ { 0 } ^ { t } e ^ { - ( t - s ) A } f ( s ) d s, \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006035.png" /></td> </tr></table>
+
\begin{equation*} u ( 0 ) = u _ { 0 } \in D ( \mathcal{A} ) , f \in C ( [ 0 , T ] ; D ( A ) ). \end{equation*}
  
 
Next to this idea of an abstract formulation for hyperbolic systems, the study of the linear evolution equation
 
Next to this idea of an abstract formulation for hyperbolic systems, the study of the linear evolution equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006036.png" /></td> </tr></table>
+
\begin{equation*} \left\{ \begin{array} { l l } { \frac { d u } { d t } + A ( t ) u = f ( t ) , } &amp; { t \in [ 0 , T ], } \\ { u ( 0 ) = u _ { 0, } } \end{array} \right. \end{equation*}
  
was originated by T. Kato, and was developed by him and many others (cf. [[#References|[a3]]], Chap. 7). Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006037.png" /> denotes a given function with values in the space of closed linear operators acting in a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006038.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006040.png" /> are the initial data, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006041.png" /> is the unknown function with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006042.png" />.
+
was originated by T. Kato, and was developed by him and many others (cf. [[#References|[a3]]], Chap. 7). Here, $A ( t )$ denotes a given function with values in the space of closed linear operators acting in a Banach space $X$; $f ( t )$ and $u_{0}$ are the initial data, and $u = u ( t )$ is the unknown function with values in $X$.
  
 
Among others, Kato's theorem in [[#References|[a4]]] is fundamental: Suppose that
 
Among others, Kato's theorem in [[#References|[a4]]] is fundamental: Suppose that
  
I) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006043.png" /> is a stable family on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006044.png" />, in the sense that
+
I) $A ( t )$ is a stable family on $X$, in the sense that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006045.png" /></td> </tr></table>
+
\begin{equation*} \left\| ( \lambda + A ( t _ { k } ) ) ^ { - 1 } \ldots ( \lambda + A ( t _ { 1 } ) ) ^ { - 1 } \right\| _ { L ( X ) } \leq \frac { M } { ( \lambda - \beta ) ^ { k } } \end{equation*}
  
for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006046.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006047.png" /> with some fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006049.png" />.
+
for any $0 \leq t _ { 1 } \leq t _ { k } \leq T$ and any $\lambda &gt; \beta$ with some fixed $M$ and $\beta$.
  
II) There is a second Banach space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006050.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006051.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006052.png" /> is a continuous function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006053.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006054.png" />.
+
II) There is a second Banach space, $Y$, such that $Y \subset D ( A ( t ) )$, and $A ( t )$ is a continuous function of $t$ with values in $L ( X , Y )$.
  
III) There is an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006055.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006056.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006057.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006058.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006059.png" /> a strongly continuous function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006060.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006061.png" />. Then there is a unique solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006062.png" />, and it is given by
+
III) There is an isomorphism $S$ from $Y$ onto $X$ such that $S A ( t ) S ^ { - 1 } = A ( t ) + B ( t )$, with $B ( t )$ a strongly continuous function of $t$ with values in $L ( X )$. Then there is a unique solution $u \in C ( [ 0 , T ] ; Y ) \cap C ^ { 1 } ( [ 0 , T ] ; X )$, and it is given by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006063.png" /></td> </tr></table>
+
\begin{equation*} u ( t ) = U ( t , 0 ) u _ { 0 } + \int _ { 0 } ^ { t } U ( t , s ) f ( s ) d s, \end{equation*}
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006064.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006065.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006066.png" /> is a unique evolution operator. It is easily seen that III) implies, in particular, the stability of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006067.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006068.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006069.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006070.png" /> are Hilbert spaces, III) can be replaced by the simpler condition [[#References|[a5]]]:
+
$u _ { 0 } \in Y$, $f \in C ( [ 0 , T ] ; Y )$, where $U ( t , s )$ is a unique evolution operator. It is easily seen that III) implies, in particular, the stability of $A ( t )$ on $Y$. When $X$ and $Y$ are Hilbert spaces, III) can be replaced by the simpler condition [[#References|[a5]]]:
  
III') There exists a positive-definite [[Self-adjoint operator|self-adjoint operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006071.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006072.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006073.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006074.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006075.png" />, with some constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006076.png" />.
+
III') There exists a positive-definite [[Self-adjoint operator|self-adjoint operator]] $S$ on $X$ with $D ( S ) = Y$ such that $| \operatorname { Re } ( A ( t ) u , S ^ { 2 } u ) _ { X } | \leq \gamma \| S u \| _ { X } ^ { 2 }$ for any $u \in D ( S ^ { 2 } )$, with some constants $\gamma$.
  
 
The Cauchy problem for the quasi-linear differential equation
 
The Cauchy problem for the quasi-linear differential equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006077.png" /></td> </tr></table>
+
\begin{equation*} \left\{ \begin{array} { l } { \frac { d u } { d t } + A ( t , u ) u = f ( t , u ) , \quad t \in [ 0 , T ], } \\ { u ( 0 ) = u _ { 0 }, } \end{array} \right. \end{equation*}
  
has been studied by several mathematicians on the basis of results for linear problems, [[#References|[a7]]]. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006078.png" /> depends also on the unknown function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006079.png" />. In [[#References|[a6]]], [[#References|[a7]]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006080.png" />, defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006081.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006082.png" /> is a bounded open set, is assumed to satisfy conditions similar to I)–III) and a Lipschitz condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006083.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120060/a12006084.png" />. Under such conditions, the existence and uniqueness of a local solution, continuous dependence on the initial data and applications to quasi-linear hyperbolic systems have been given.
+
has been studied by several mathematicians on the basis of results for linear problems, [[#References|[a7]]]. Here, $A ( t , u )$ depends also on the unknown function $u$. In [[#References|[a6]]], [[#References|[a7]]], $A ( t , u )$, defined for $( t , u ) \in [ 0 , T ] \times W$, where $W \subset Y$ is a bounded open set, is assumed to satisfy conditions similar to I)–III) and a Lipschitz condition $\| A ( t , u ) - A ( t , u ^ { \prime } ) \| _ { L ( Y , X ) } \leq \mu \| u - u ^ { \prime } \| _ { X }$ with respect to $u$. Under such conditions, the existence and uniqueness of a local solution, continuous dependence on the initial data and applications to quasi-linear hyperbolic systems have been given.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Hille,  R.S. Phillips,  "Functional analysis and semi-groups" , Amer. Math. Soc.  (1957)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Yoshida,  "Functional analysis" , Springer  (1957)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Tanabe,  "Functional analytic methods for partial differential equations" , M. Dekker  (1997)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  T. Kato,  "Linear evolution equations of  "hyperbolic"  type"  ''J. Fac. Sci. Univ. Tokyo'' , '''17'''  (1970)  pp. 241–248</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  N. Okazawa,  "Remarks on linear evolution equations of hyperbolic type in Hilbert space"  ''Adv. Math. Sci. Appl.'' , '''8'''  (1998)  pp. 399–423</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  K. Kobayashi,  N. Sanekata,  "A method of iterations for quasi-linear evolution equations in nonreflexive Banach spaces"  ''Hiroshima Math. J.'' , '''19'''  (1989)  pp. 521–540</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  T. Kato,  "Abstract evolution equations, linear and quasilinear, revisited"  J. Komatsu (ed.) , ''Funct. Anal. and Rel. Topics. Proc. Conf. in Memory of K. Yoshida (RIMS, 1991)'' , ''Lecture Notes Math.'' , '''1540''' , Springer  (1991)  pp. 103–125</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  E. Hille,  R.S. Phillips,  "Functional analysis and semi-groups" , Amer. Math. Soc.  (1957)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  K. Yoshida,  "Functional analysis" , Springer  (1957)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  H. Tanabe,  "Functional analytic methods for partial differential equations" , M. Dekker  (1997)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  T. Kato,  "Linear evolution equations of  "hyperbolic"  type"  ''J. Fac. Sci. Univ. Tokyo'' , '''17'''  (1970)  pp. 241–248</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  N. Okazawa,  "Remarks on linear evolution equations of hyperbolic type in Hilbert space"  ''Adv. Math. Sci. Appl.'' , '''8'''  (1998)  pp. 399–423</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  K. Kobayashi,  N. Sanekata,  "A method of iterations for quasi-linear evolution equations in nonreflexive Banach spaces"  ''Hiroshima Math. J.'' , '''19'''  (1989)  pp. 521–540</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  T. Kato,  "Abstract evolution equations, linear and quasilinear, revisited"  J. Komatsu (ed.) , ''Funct. Anal. and Rel. Topics. Proc. Conf. in Memory of K. Yoshida (RIMS, 1991)'' , ''Lecture Notes Math.'' , '''1540''' , Springer  (1991)  pp. 103–125</td></tr></table>

Latest revision as of 05:03, 7 March 2022

Consider the Cauchy problem for the symmetric hyperbolic system (cf. also Hyperbolic partial differential equation)

$$ \begin{cases} \frac { \partial u } { \partial t } + \sum _ { j = 1 } ^ { m } a _ { j } ( x ) \frac { \partial u } { \partial x _ { j } } + c ( x ) u = f ( x , t ) , \\ { ( x , t ) \in \Omega \times [ 0 , T ] }, \\ { u ( x , 0 ) = u _ { 0 } ( x ) , \quad x \in \Omega, } \end{cases} $$ with the boundary conditions

\begin{equation*} u ( x , t ) \in P ( x ) , \quad ( x , t ) \in \partial \Omega \times [ 0 , T ]. \end{equation*}

Here, $\Omega \subset \mathbf{R} ^ { m }$ is a bounded domain with smooth boundary $\partial \Omega$ (when $\Omega = \mathbf{R} ^ { m }$, no boundary conditions are necessary), and $a_{j} ( x )$, $j = 1 , \ldots , m$, and $c ( x )$ are smooth functions on $\Omega$ with as values real matrices in $L ( \mathbf{R} ^ { p } )$, the $a_{j} ( x )$ being symmetric. It is assumed that the boundary matrix $b ( x ) = \sum _ { j = 1 } ^ { m } n _ { j } ( x ) a _ { j } ( x )$, $x \in \partial \Omega$, is non-singular, where $n = ( n _{1} , \ldots , n _ { m } )$ is the unit outward normal vector to $\partial \Omega$. Also, $P ( x )$ denotes the maximal non-negative subspace of $\mathbf{R} ^ { p }$ with respect to $b ( x )$, i.e. $( b ( x ) u , u ) \geq 0$, $u \in P ( x )$, and $P ( x )$ is not a proper subset of any other subspace of $\mathbf{R} ^ { p }$ with this property. The function $u = ( u _ { 1 } , \ldots , u _ { p } )$ is the unknown function.

One can handle this problem as the Cauchy problem for an evolution equation in a Banach space (cf. also Linear differential equation in a Banach space). Indeed, let $A$ be the smallest closed extension in $X = [ L ^ { 2 } ( \Omega ) ] ^ { p }$ of the operator $\mathcal{A}$ defined by

\begin{equation*} {\cal A} u = \sum _ { j = 1 } ^ { m } a _ { j } ( x ) \frac { \partial u } { \partial x _ { j } } + c ( x ) u \end{equation*}

with domain

\begin{equation*} D ( \mathcal{A} ) = \left\{ u \in [ H ^ { 1 } ( \Omega )] ^ { p } : u ( x ) \in P ( x ) \text { a.e. on } \partial \Omega \right\}. \end{equation*}

Then $A$ is the negative generator of a $C _ { 0 }$ semi-group on $X$ (cf. [a1], [a2]; see also Semi-group of operators). Hence, the Hille–Yoshida theorem proves the existence of a unique solution $u \in C ( [ 0 , T ] ; D ( \mathcal{A} ) ) \cap C ^ { 1 } ( [ 0 , T ] ; X )$ to the Cauchy problem

\begin{equation*} \frac { d u } { d t } + A u = f ( t ) , t \in [ 0 , T ], \end{equation*}

which is given in the form

\begin{equation*} u ( t ) = e ^ { - t A } u _ { 0 } + \int _ { 0 } ^ { t } e ^ { - ( t - s ) A } f ( s ) d s, \end{equation*}

\begin{equation*} u ( 0 ) = u _ { 0 } \in D ( \mathcal{A} ) , f \in C ( [ 0 , T ] ; D ( A ) ). \end{equation*}

Next to this idea of an abstract formulation for hyperbolic systems, the study of the linear evolution equation

\begin{equation*} \left\{ \begin{array} { l l } { \frac { d u } { d t } + A ( t ) u = f ( t ) , } & { t \in [ 0 , T ], } \\ { u ( 0 ) = u _ { 0, } } \end{array} \right. \end{equation*}

was originated by T. Kato, and was developed by him and many others (cf. [a3], Chap. 7). Here, $A ( t )$ denotes a given function with values in the space of closed linear operators acting in a Banach space $X$; $f ( t )$ and $u_{0}$ are the initial data, and $u = u ( t )$ is the unknown function with values in $X$.

Among others, Kato's theorem in [a4] is fundamental: Suppose that

I) $A ( t )$ is a stable family on $X$, in the sense that

\begin{equation*} \left\| ( \lambda + A ( t _ { k } ) ) ^ { - 1 } \ldots ( \lambda + A ( t _ { 1 } ) ) ^ { - 1 } \right\| _ { L ( X ) } \leq \frac { M } { ( \lambda - \beta ) ^ { k } } \end{equation*}

for any $0 \leq t _ { 1 } \leq t _ { k } \leq T$ and any $\lambda > \beta$ with some fixed $M$ and $\beta$.

II) There is a second Banach space, $Y$, such that $Y \subset D ( A ( t ) )$, and $A ( t )$ is a continuous function of $t$ with values in $L ( X , Y )$.

III) There is an isomorphism $S$ from $Y$ onto $X$ such that $S A ( t ) S ^ { - 1 } = A ( t ) + B ( t )$, with $B ( t )$ a strongly continuous function of $t$ with values in $L ( X )$. Then there is a unique solution $u \in C ( [ 0 , T ] ; Y ) \cap C ^ { 1 } ( [ 0 , T ] ; X )$, and it is given by

\begin{equation*} u ( t ) = U ( t , 0 ) u _ { 0 } + \int _ { 0 } ^ { t } U ( t , s ) f ( s ) d s, \end{equation*}

$u _ { 0 } \in Y$, $f \in C ( [ 0 , T ] ; Y )$, where $U ( t , s )$ is a unique evolution operator. It is easily seen that III) implies, in particular, the stability of $A ( t )$ on $Y$. When $X$ and $Y$ are Hilbert spaces, III) can be replaced by the simpler condition [a5]:

III') There exists a positive-definite self-adjoint operator $S$ on $X$ with $D ( S ) = Y$ such that $| \operatorname { Re } ( A ( t ) u , S ^ { 2 } u ) _ { X } | \leq \gamma \| S u \| _ { X } ^ { 2 }$ for any $u \in D ( S ^ { 2 } )$, with some constants $\gamma$.

The Cauchy problem for the quasi-linear differential equation

\begin{equation*} \left\{ \begin{array} { l } { \frac { d u } { d t } + A ( t , u ) u = f ( t , u ) , \quad t \in [ 0 , T ], } \\ { u ( 0 ) = u _ { 0 }, } \end{array} \right. \end{equation*}

has been studied by several mathematicians on the basis of results for linear problems, [a7]. Here, $A ( t , u )$ depends also on the unknown function $u$. In [a6], [a7], $A ( t , u )$, defined for $( t , u ) \in [ 0 , T ] \times W$, where $W \subset Y$ is a bounded open set, is assumed to satisfy conditions similar to I)–III) and a Lipschitz condition $\| A ( t , u ) - A ( t , u ^ { \prime } ) \| _ { L ( Y , X ) } \leq \mu \| u - u ^ { \prime } \| _ { X }$ with respect to $u$. Under such conditions, the existence and uniqueness of a local solution, continuous dependence on the initial data and applications to quasi-linear hyperbolic systems have been given.

References

[a1] E. Hille, R.S. Phillips, "Functional analysis and semi-groups" , Amer. Math. Soc. (1957)
[a2] K. Yoshida, "Functional analysis" , Springer (1957)
[a3] H. Tanabe, "Functional analytic methods for partial differential equations" , M. Dekker (1997)
[a4] T. Kato, "Linear evolution equations of "hyperbolic" type" J. Fac. Sci. Univ. Tokyo , 17 (1970) pp. 241–248
[a5] N. Okazawa, "Remarks on linear evolution equations of hyperbolic type in Hilbert space" Adv. Math. Sci. Appl. , 8 (1998) pp. 399–423
[a6] K. Kobayashi, N. Sanekata, "A method of iterations for quasi-linear evolution equations in nonreflexive Banach spaces" Hiroshima Math. J. , 19 (1989) pp. 521–540
[a7] T. Kato, "Abstract evolution equations, linear and quasilinear, revisited" J. Komatsu (ed.) , Funct. Anal. and Rel. Topics. Proc. Conf. in Memory of K. Yoshida (RIMS, 1991) , Lecture Notes Math. , 1540 , Springer (1991) pp. 103–125
How to Cite This Entry:
Abstract hyperbolic differential equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Abstract_hyperbolic_differential_equation&oldid=15200
This article was adapted from an original article by A. Yagi (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article