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Either a differential equation in some abstract space (a Hilbert space, a Banach space, etc.) or a differential equation with operator coefficients. The classical abstract differential equation which is most frequently encountered is the equation
 
Either a differential equation in some abstract space (a Hilbert space, a Banach space, etc.) or a differential equation with operator coefficients. The classical abstract differential equation which is most frequently encountered is the 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/d/d031/d031900/d0319001.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
Lu  =
 +
\frac{\partial  u }{\partial  t }
 +
- Au  = f ,
 +
$$
 +
 
 +
where the unknown function  $  u = u ( t) $
 +
belongs to some function space  $  X $,
 +
0 \leq  t \leq  T \leq  \infty $,
 +
and  $  A: X \rightarrow X $
 +
is an operator (usually a linear operator) acting on this space. If the operator  $  A $
 +
is a bounded operator or a constant (does not depend on  $  t $),
 +
the formula
 +
 
 +
$$
 +
u ( t)  = e  ^ {tA} u _ {0} + \int\limits _ { 0 } ^ { t }  e ^ {( t - \tau ) A }
 +
f ( \tau ) d \tau
 +
$$
  
where the unknown function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d0319002.png" /> belongs to some function space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d0319003.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d0319004.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d0319005.png" /> is an operator (usually a linear operator) acting on this space. If the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d0319006.png" /> is a bounded operator or a constant (does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d0319007.png" />), the formula
+
yields the unique solution of equation (1) satisfying the condition  $  u ( 0) = u _ {0} $.  
 +
For a variable operator  $  A ( t) $,
 +
$  e ^ {( t - \tau ) A } $
 +
is replaced by the [[Evolution operator|evolution operator]]  $  U ( t , \tau ) $(
 +
cf. also [[Cauchy operator|Cauchy operator]]). If the operator  $  A $
 +
is unbounded, the solutions of the Cauchy problem  $  u ( 0) = u _ {0} $
 +
need not exist for some  $  u _ {0} $,  
 +
need not be unique and may break off for  $  t < T $.
 +
An exhaustive treatment of the homogeneous ( $  f \equiv 0 $)
 +
equation (1) with a constant operator is given by the theory of semi-groups, while the problems of existence and uniqueness are solved in terms of the resolvent of  $  A $[[#References|[1]]], [[#References|[5]]]. The same method is also applicable to a variable operator, if it depends smoothly on  $  t $.
 +
Another method of study of equation (1), which usually gives less accurate results, but which is applicable to wider classes of equations (even including non-linear equations in some cases), is the use of energy inequalities: $  \| u \| \leq  c  \| Lu \| $,
 +
which are also obtained if certain assumptions are made regarding  $  A $.  
 +
For a Hilbert space  $  X $
 +
one usually postulates different positivity properties of the scalar product  $  ( Au, u ) $[[#References|[2]]]. All the above can be extended, to a certain extent, to more general abstract differential equations
  
<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/d/d031/d031900/d0319008.png" /></td> </tr></table>
+
$$ \tag{2 }
  
yields the unique solution of equation (1) satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d0319009.png" />. For a variable operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d03190010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d03190011.png" /> is replaced by the [[Evolution operator|evolution operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d03190012.png" /> (cf. also [[Cauchy operator|Cauchy operator]]). If the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d03190013.png" /> is unbounded, the solutions of the Cauchy problem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d03190014.png" /> need not exist for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d03190015.png" />, need not be unique and may break off for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d03190016.png" />. An exhaustive treatment of the homogeneous (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d03190017.png" />) equation (1) with a constant operator is given by the theory of semi-groups, while the problems of existence and uniqueness are solved in terms of the resolvent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d03190018.png" /> [[#References|[1]]], [[#References|[5]]]. The same method is also applicable to a variable operator, if it depends smoothly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d03190019.png" />. Another method of study of equation (1), which usually gives less accurate results, but which is applicable to wider classes of equations (even including non-linear equations in some cases), is the use of energy inequalities: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d03190020.png" />, which are also obtained if certain assumptions are made regarding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d03190021.png" />. For a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d03190022.png" /> one usually postulates different positivity properties of the scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d03190023.png" /> [[#References|[2]]]. All the above can be extended, to a certain extent, to more general abstract differential equations
+
\frac{d ^ {2} u }{d t  ^ {2} }
 +
+ A _ {1}
 +
\frac{d u }{d t }
 +
+ A _ {2} u  = f ,
 +
$$
  
<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/d/d031/d031900/d03190024.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
studied under the condition  $  u ( 0) = u _ {0} $,
 +
$  u _ {t}  ^  \prime  ( 0) = u _ {1} $.
 +
Very often, the study of equation (2) by various methods (reduction to a set of equations of the first order, a substitution  $  u = \int _ {0}  ^ {t} v ( \tau )  d \tau $,
 +
subdivision of the left-hand side into a product of two operators of the first order, etc.) really amounts to the study of equation (1). A principal reason for the existing interest in abstract differential equations is that the so-called mixed problems in cylindrical domains for classical parabolic and hyperbolic equations of the second order can be reduced to equations of the form (1) or (2): The function  $  u ( t , x _ {1} \dots x _ {n} ) $
 +
is regarded as a function of  $  t $
 +
with values in the corresponding space of functions in  $  x $,
 +
while the operators  $  A $,
 +
$  A _ {k} $
 +
are generated by differentiations with respect to  $  x $,
 +
subject to the boundary conditions on the side surfaces of the cylinder (the generatrices of which are parallel to the  $  t $-
 +
axis). Equations (1), (2), in which the postulated properties of the operators  $  A $,
 +
$  A _ {k} $
 +
coincide with those obtained in the situation described above, are known as parabolic or hyperbolic. Abstract elliptic operators are considered less often.
  
studied under the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d03190025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d03190026.png" />. Very often, the study of equation (2) by various methods (reduction to a set of equations of the first order, a substitution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d03190027.png" />, subdivision of the left-hand side into a product of two operators of the first order, etc.) really amounts to the study of equation (1). A principal reason for the existing interest in abstract differential equations is that the so-called mixed problems in cylindrical domains for classical parabolic and hyperbolic equations of the second order can be reduced to equations of the form (1) or (2): The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d03190028.png" /> is regarded as a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d03190029.png" /> with values in the corresponding space of functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d03190030.png" />, while the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d03190031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d03190032.png" /> are generated by differentiations with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d03190033.png" />, subject to the boundary conditions on the side surfaces of the cylinder (the generatrices of which are parallel to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d03190034.png" />-axis). Equations (1), (2), in which the postulated properties of the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d03190035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d03190036.png" /> coincide with those obtained in the situation described above, are known as parabolic or hyperbolic. Abstract elliptic operators are considered less often.
+
Problems in scattering theory [[#References|[3]]] in the interval  $  - \infty < t < + \infty $
 +
are often formulated in terms of semi-groups and equation (1). The reduction of problems in partial differential equations to problems (1) and (2) in abstract differential equations are very convenient in developing approximate (e.g. difference [[#References|[4]]]) methods of solution and in the study of asymptotic methods ( "small" and  "large" parameters). General abstract differential equations with operator coefficients
  
Problems in scattering theory [[#References|[3]]] in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d03190037.png" /> are often formulated in terms of semi-groups and equation (1). The reduction of problems in partial differential equations to problems (1) and (2) in abstract differential equations are very convenient in developing approximate (e.g. difference [[#References|[4]]]) methods of solution and in the study of asymptotic methods ( "small" and  "large" parameters). General abstract differential equations with operator coefficients
+
$$
 +
\sum _ {k= 0 } ^ { m }  A _ {k}
 +
\frac{^ {k} }{d t ^ {k} }
  
<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/d/d031/d031900/d03190038.png" /></td> </tr></table>
+
$$
  
and boundary conditions on both ends of the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d03190039.png" /> for unbounded operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d03190040.png" /> can be meaningfully studied only if very special assumptions concerning <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d03190041.png" /> are made. For bounded <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031900/d03190042.png" /> there is no difficulty in extending the theory of ordinary differential equations in an appropriate manner.
+
and boundary conditions on both ends of the interval $  ( 0 , T ) $
 +
for unbounded operators $  A _ {k} $
 +
can be meaningfully studied only if very special assumptions concerning $  A _ {k} $
 +
are made. For bounded $  A _ {k} $
 +
there is no difficulty in extending the theory of ordinary differential equations in an appropriate manner.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Hille,  R.S. Phillips,  "Functional analysis and semi-groups" , Amer. Math. Soc.  (1957)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.L. Lions,  "Equations différentielles operationelles et problèmes aux limites" , Springer  (1961)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P.D. Lax,  R.S. Philips,  "Scattering theory" , Acad. Press  (1967)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.A. Samarskii,  "Theorie der Differenzverfahren" , Akad. Verlagsgesell. Geest u. Portig K.-D.  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S.G. Krein,  "Linear differential equations in Banach space" , ''Transl. Math. Monogr.'' , '''29''' , Amer. Math. Soc.  (1971)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Hille,  R.S. Phillips,  "Functional analysis and semi-groups" , Amer. Math. Soc.  (1957)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.L. Lions,  "Equations différentielles operationelles et problèmes aux limites" , Springer  (1961)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P.D. Lax,  R.S. Philips,  "Scattering theory" , Acad. Press  (1967)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.A. Samarskii,  "Theorie der Differenzverfahren" , Akad. Verlagsgesell. Geest u. Portig K.-D.  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S.G. Krein,  "Linear differential equations in Banach space" , ''Transl. Math. Monogr.'' , '''29''' , Amer. Math. Soc.  (1971)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 17:33, 5 June 2020


Either a differential equation in some abstract space (a Hilbert space, a Banach space, etc.) or a differential equation with operator coefficients. The classical abstract differential equation which is most frequently encountered is the equation

$$ \tag{1 } Lu = \frac{\partial u }{\partial t } - Au = f , $$

where the unknown function $ u = u ( t) $ belongs to some function space $ X $, $ 0 \leq t \leq T \leq \infty $, and $ A: X \rightarrow X $ is an operator (usually a linear operator) acting on this space. If the operator $ A $ is a bounded operator or a constant (does not depend on $ t $), the formula

$$ u ( t) = e ^ {tA} u _ {0} + \int\limits _ { 0 } ^ { t } e ^ {( t - \tau ) A } f ( \tau ) d \tau $$

yields the unique solution of equation (1) satisfying the condition $ u ( 0) = u _ {0} $. For a variable operator $ A ( t) $, $ e ^ {( t - \tau ) A } $ is replaced by the evolution operator $ U ( t , \tau ) $( cf. also Cauchy operator). If the operator $ A $ is unbounded, the solutions of the Cauchy problem $ u ( 0) = u _ {0} $ need not exist for some $ u _ {0} $, need not be unique and may break off for $ t < T $. An exhaustive treatment of the homogeneous ( $ f \equiv 0 $) equation (1) with a constant operator is given by the theory of semi-groups, while the problems of existence and uniqueness are solved in terms of the resolvent of $ A $[1], [5]. The same method is also applicable to a variable operator, if it depends smoothly on $ t $. Another method of study of equation (1), which usually gives less accurate results, but which is applicable to wider classes of equations (even including non-linear equations in some cases), is the use of energy inequalities: $ \| u \| \leq c \| Lu \| $, which are also obtained if certain assumptions are made regarding $ A $. For a Hilbert space $ X $ one usually postulates different positivity properties of the scalar product $ ( Au, u ) $[2]. All the above can be extended, to a certain extent, to more general abstract differential equations

$$ \tag{2 } \frac{d ^ {2} u }{d t ^ {2} } + A _ {1} \frac{d u }{d t } + A _ {2} u = f , $$

studied under the condition $ u ( 0) = u _ {0} $, $ u _ {t} ^ \prime ( 0) = u _ {1} $. Very often, the study of equation (2) by various methods (reduction to a set of equations of the first order, a substitution $ u = \int _ {0} ^ {t} v ( \tau ) d \tau $, subdivision of the left-hand side into a product of two operators of the first order, etc.) really amounts to the study of equation (1). A principal reason for the existing interest in abstract differential equations is that the so-called mixed problems in cylindrical domains for classical parabolic and hyperbolic equations of the second order can be reduced to equations of the form (1) or (2): The function $ u ( t , x _ {1} \dots x _ {n} ) $ is regarded as a function of $ t $ with values in the corresponding space of functions in $ x $, while the operators $ A $, $ A _ {k} $ are generated by differentiations with respect to $ x $, subject to the boundary conditions on the side surfaces of the cylinder (the generatrices of which are parallel to the $ t $- axis). Equations (1), (2), in which the postulated properties of the operators $ A $, $ A _ {k} $ coincide with those obtained in the situation described above, are known as parabolic or hyperbolic. Abstract elliptic operators are considered less often.

Problems in scattering theory [3] in the interval $ - \infty < t < + \infty $ are often formulated in terms of semi-groups and equation (1). The reduction of problems in partial differential equations to problems (1) and (2) in abstract differential equations are very convenient in developing approximate (e.g. difference [4]) methods of solution and in the study of asymptotic methods ( "small" and "large" parameters). General abstract differential equations with operator coefficients

$$ \sum _ {k= 0 } ^ { m } A _ {k} \frac{d ^ {k} }{d t ^ {k} } $$

and boundary conditions on both ends of the interval $ ( 0 , T ) $ for unbounded operators $ A _ {k} $ can be meaningfully studied only if very special assumptions concerning $ A _ {k} $ are made. For bounded $ A _ {k} $ there is no difficulty in extending the theory of ordinary differential equations in an appropriate manner.

References

[1] E. Hille, R.S. Phillips, "Functional analysis and semi-groups" , Amer. Math. Soc. (1957)
[2] J.L. Lions, "Equations différentielles operationelles et problèmes aux limites" , Springer (1961)
[3] P.D. Lax, R.S. Philips, "Scattering theory" , Acad. Press (1967)
[4] A.A. Samarskii, "Theorie der Differenzverfahren" , Akad. Verlagsgesell. Geest u. Portig K.-D. (1984) (Translated from Russian)
[5] S.G. Krein, "Linear differential equations in Banach space" , Transl. Math. Monogr. , 29 , Amer. Math. Soc. (1971) (Translated from Russian)

Comments

For elliptic problems see [a3].

References

[a1] H. Tanabe, "Equations of evolution" , Pitman (1979) (Translated from Japanese)
[a2] A. Pazy, "Semigroups of linear operators and applications to partial differential equations" , Springer (1983)
[a3] S. Agmon, "Unicité convexité dans problèmes différentiels" , Univ. Montréal (1966)
How to Cite This Entry:
Differential equation, abstract. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Differential_equation,_abstract&oldid=46666
This article was adapted from an original article by A.A. Dezin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article