Abstract Cauchy problem

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)