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The condensed formulation of a [[Cauchy problem|Cauchy problem]] (as phrased by J. Hadamard) in an infinite-dimensional [[Topological vector space|topological vector space]]. While it seems to have arisen between the two World Wars (F. Browder in [[#References|[a2]]], Foreword), it was apparently introduced as such by E. Hille in 1952, [[#References|[a2]]], Sec. 1.7.
 
The condensed formulation of a [[Cauchy problem|Cauchy problem]] (as phrased by J. Hadamard) in an infinite-dimensional [[Topological vector space|topological vector space]]. While it seems to have arisen between the two World Wars (F. Browder in [[#References|[a2]]], Foreword), it was apparently introduced as such by E. Hille in 1952, [[#References|[a2]]], Sec. 1.7.
  
Narrowly, but loosely speaking, the abstract Cauchy problem consists in solving a linear abstract differential equation (cf. also [[Differential equation, abstract|Differential equation, abstract]]) or abstract [[Evolution equation|evolution equation]] subject to an initial condition. More precise explanations slightly differ from textbook to textbook [[#References|[a2]]], [[#References|[a5]]]. Following A. Pazy [[#References|[a5]]], given a [[Linear operator|linear operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120040/a1200401.png" /> on a [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120040/a1200402.png" /> with domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120040/a1200403.png" /> and given an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120040/a1200404.png" />, one tries to solve
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Narrowly, but loosely speaking, the abstract Cauchy problem consists in solving a linear abstract differential equation (cf. also [[Differential equation, abstract|Differential equation, abstract]]) or abstract [[Evolution equation|evolution equation]] subject to an initial condition. More precise explanations slightly differ from textbook to textbook [[#References|[a2]]], [[#References|[a5]]]. Following A. Pazy [[#References|[a5]]], given a [[Linear operator|linear operator]] $A$ on a [[Banach space|Banach space]] $X$ with domain $D ( A )$ and given an element $x _ { 0 } \in X$, one tries to solve
  
<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/a120040/a1200405.png" /></td> </tr></table>
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\begin{equation*} x ^ { \prime } ( t ) = A x ( t ) , t &gt; 0 ; \quad x ( 0 ) = x 0, \end{equation*}
  
i.e., one looks for a [[Continuous function|continuous function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120040/a1200406.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120040/a1200407.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120040/a1200408.png" /> is differentiable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120040/a1200409.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120040/a12004010.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120040/a12004011.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120040/a12004012.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120040/a12004013.png" />.
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i.e., one looks for a [[Continuous function|continuous function]] $x$ on $[ 0 , \infty )$ such that $x$ is differentiable on $( 0 , \infty )$, $x ( t ) \in D ( A )$ for all $t &gt; 0$, and $( d / d t ) x ( t ) = A x ( t )$ for all $t \in ( 0 , \infty )$.
  
Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120040/a12004014.png" /> is required to be continuous at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120040/a12004015.png" />, the Cauchy problem can only be solved for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120040/a12004016.png" />.
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Since $x$ is required to be continuous at $0$, the Cauchy problem can only be solved for $x _ { 0 } \in \overline { D ( A ) }$.
  
A Cauchy problem is called correctly set if the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120040/a12004017.png" /> is uniquely determined by the initial datum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120040/a12004018.png" />. It is called well-posed (properly posed) if, in addition, the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120040/a12004019.png" /> depends continuously on the initial datum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120040/a12004020.png" />, i.e., for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120040/a12004021.png" /> there exists some constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120040/a12004022.png" /> (independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120040/a12004023.png" />) such that
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A Cauchy problem is called correctly set if the solution $x$ is uniquely determined by the initial datum $x _ { 0 }$. It is called well-posed (properly posed) if, in addition, the solution $x$ depends continuously on the initial datum $x _ { 0 }$, i.e., for every $\tau &gt; 0$ there exists some constant $c &gt; 0$ (independent of $x _ { 0 }$) such 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/a120040/a12004024.png" /></td> </tr></table>
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\begin{equation*} \| x ( t ) \| \leq c \| x _ { 0 } \| \text { for all } \, t \in [ 0 , \tau ], \end{equation*}
  
and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120040/a12004025.png" /> for which a solution exists. Sometimes it is also required that solutions exist for a subspace of initial data which is large enough in an appropriate sense, e.g., dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120040/a12004026.png" />.
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and all $x _ { 0 }$ for which a solution exists. Sometimes it is also required that solutions exist for a subspace of initial data which is large enough in an appropriate sense, e.g., dense in $X$.
  
 
The notion of a Cauchy problem can be extended to non-autonomous evolution equations [[#References|[a2]]], [[#References|[a5]]] and to semi-linear [[#References|[a5]]], quasi-linear [[#References|[a5]]], or fully non-linear evolution equations [[#References|[a1]]], [[#References|[a4]]]. In this process it may become necessary to replace classical solutions by more general solution concepts (mild solutions [[#References|[a1]]], limit solutions [[#References|[a4]]], integral solutions (in the sense of Ph. Bénilan; [[#References|[a4]]]) in order to keep the problem meaningful. See [[#References|[a1]]] and the references therein.
 
The notion of a Cauchy problem can be extended to non-autonomous evolution equations [[#References|[a2]]], [[#References|[a5]]] and to semi-linear [[#References|[a5]]], quasi-linear [[#References|[a5]]], or fully non-linear evolution equations [[#References|[a1]]], [[#References|[a4]]]. In this process it may become necessary to replace classical solutions by more general solution concepts (mild solutions [[#References|[a1]]], limit solutions [[#References|[a4]]], integral solutions (in the sense of Ph. Bénilan; [[#References|[a4]]]) in order to keep the problem meaningful. See [[#References|[a1]]] and the references therein.
  
Well-posedness of linear Cauchy problems is intimately linked to the existence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120040/a12004027.png" />-semi-groups of linear operators (cf. also [[Semi-group of operators|Semi-group of operators]]), strongly continuous evolution families [[#References|[a2]]], [[#References|[a5]]] and related more general concepts like distribution semi-groups, integrated semi-groups, convoluted semi-groups, and regularized semi-groups, while the well-posedness of non-linear Cauchy problems is linked to the existence of non-linear semi-groups (the Crandall–Liggett theorem and its extensions) or (semi-) dynamical systems [[#References|[a1]]], [[#References|[a4]]], and to (evolutionary) processes and skew product flows [[#References|[a3]]].
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Well-posedness of linear Cauchy problems is intimately linked to the existence of $C _ { 0 }$-semi-groups of linear operators (cf. also [[Semi-group of operators|Semi-group of operators]]), strongly continuous evolution families [[#References|[a2]]], [[#References|[a5]]] and related more general concepts like distribution semi-groups, integrated semi-groups, convoluted semi-groups, and regularized semi-groups, while the well-posedness of non-linear Cauchy problems is linked to the existence of non-linear semi-groups (the Crandall–Liggett theorem and its extensions) or (semi-) dynamical systems [[#References|[a1]]], [[#References|[a4]]], and to (evolutionary) processes and skew product flows [[#References|[a3]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Benilan,  P. Wittbold,  "Nonlinear evolution equations in Banach spaces: Basic results and open problems"  K.D. Bierstedt (ed.)  A. Pietsch (ed.)  W.M. Ruess (ed.)  D. Vogt (ed.) , ''Functional Analysis'' , ''Lecture Notes Pure Appl. Math.'' , '''150''' , M. Dekker  (1994)  pp. 1–32</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.O. Fattorini,  "The Cauchy problem" , Addison-Wesley  (1983)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.K. Hale,  "Asymptotic behavior of dissipative systems" , Amer. Math. Soc.  (1988)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  V. Lakshmikantham,  S. Leela,  "Nonlinear differential equations in abstract spaces" , Pergamon  (1981)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A. Pazy,  "Semigroups of linear operators and applications to partial differential equations" , Springer  (1983)</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  P. Benilan,  P. Wittbold,  "Nonlinear evolution equations in Banach spaces: Basic results and open problems"  K.D. Bierstedt (ed.)  A. Pietsch (ed.)  W.M. Ruess (ed.)  D. Vogt (ed.) , ''Functional Analysis'' , ''Lecture Notes Pure Appl. Math.'' , '''150''' , M. Dekker  (1994)  pp. 1–32</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  H.O. Fattorini,  "The Cauchy problem" , Addison-Wesley  (1983)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  J.K. Hale,  "Asymptotic behavior of dissipative systems" , Amer. Math. Soc.  (1988)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  V. Lakshmikantham,  S. Leela,  "Nonlinear differential equations in abstract spaces" , Pergamon  (1981)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  A. Pazy,  "Semigroups of linear operators and applications to partial differential equations" , Springer  (1983)</td></tr></table>

Revision as of 16:59, 1 July 2020

The condensed formulation of a Cauchy problem (as phrased by J. Hadamard) in an infinite-dimensional topological vector space. While it seems to have arisen between the two World Wars (F. Browder in [a2], Foreword), it was apparently introduced as such by E. Hille in 1952, [a2], Sec. 1.7.

Narrowly, but loosely speaking, the abstract Cauchy problem consists in solving a linear abstract differential equation (cf. also Differential equation, abstract) or abstract evolution equation subject to an initial condition. More precise explanations slightly differ from textbook to textbook [a2], [a5]. Following A. Pazy [a5], given a linear operator $A$ on a Banach space $X$ with domain $D ( A )$ and given an element $x _ { 0 } \in X$, one tries to solve

\begin{equation*} x ^ { \prime } ( t ) = A x ( t ) , t > 0 ; \quad x ( 0 ) = x 0, \end{equation*}

i.e., one looks for a continuous function $x$ on $[ 0 , \infty )$ such that $x$ is differentiable on $( 0 , \infty )$, $x ( t ) \in D ( A )$ for all $t > 0$, and $( d / d t ) x ( t ) = A x ( t )$ for all $t \in ( 0 , \infty )$.

Since $x$ is required to be continuous at $0$, the Cauchy problem can only be solved for $x _ { 0 } \in \overline { D ( A ) }$.

A Cauchy problem is called correctly set if the solution $x$ is uniquely determined by the initial datum $x _ { 0 }$. It is called well-posed (properly posed) if, in addition, the solution $x$ depends continuously on the initial datum $x _ { 0 }$, i.e., for every $\tau > 0$ there exists some constant $c > 0$ (independent of $x _ { 0 }$) such that

\begin{equation*} \| x ( t ) \| \leq c \| x _ { 0 } \| \text { for all } \, t \in [ 0 , \tau ], \end{equation*}

and all $x _ { 0 }$ for which a solution exists. Sometimes it is also required that solutions exist for a subspace of initial data which is large enough in an appropriate sense, e.g., dense in $X$.

The notion of a Cauchy problem can be extended to non-autonomous evolution equations [a2], [a5] and to semi-linear [a5], quasi-linear [a5], or fully non-linear evolution equations [a1], [a4]. In this process it may become necessary to replace classical solutions by more general solution concepts (mild solutions [a1], limit solutions [a4], integral solutions (in the sense of Ph. Bénilan; [a4]) in order to keep the problem meaningful. See [a1] and the references therein.

Well-posedness of linear Cauchy problems is intimately linked to the existence of $C _ { 0 }$-semi-groups of linear operators (cf. also Semi-group of operators), strongly continuous evolution families [a2], [a5] and related more general concepts like distribution semi-groups, integrated semi-groups, convoluted semi-groups, and regularized semi-groups, while the well-posedness of non-linear Cauchy problems is linked to the existence of non-linear semi-groups (the Crandall–Liggett theorem and its extensions) or (semi-) dynamical systems [a1], [a4], and to (evolutionary) processes and skew product flows [a3].

References

[a1] P. Benilan, P. Wittbold, "Nonlinear evolution equations in Banach spaces: Basic results and open problems" K.D. Bierstedt (ed.) A. Pietsch (ed.) W.M. Ruess (ed.) D. Vogt (ed.) , Functional Analysis , Lecture Notes Pure Appl. Math. , 150 , M. Dekker (1994) pp. 1–32
[a2] H.O. Fattorini, "The Cauchy problem" , Addison-Wesley (1983)
[a3] J.K. Hale, "Asymptotic behavior of dissipative systems" , Amer. Math. Soc. (1988)
[a4] V. Lakshmikantham, S. Leela, "Nonlinear differential equations in abstract spaces" , Pergamon (1981)
[a5] A. Pazy, "Semigroups of linear operators and applications to partial differential equations" , Springer (1983)
How to Cite This Entry:
Abstract Cauchy problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Abstract_Cauchy_problem&oldid=50297
This article was adapted from an original article by H. Thieme (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article