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Recently I have come across a method known as "variational method" in which we try to establish weak solutions of various boundary value problems involving ordinary derivatives, partial derivatives, and fractional derivatives. The use of Sobolev spaces is the key to finding such solutions in which the Lax Milgram theorem is applied to show the existence and uniqueness of solutions. The general approach can be seen in the 8th chapter of the book "Functional analysis, Sobolev spaces, and partial differential equations" (publisher link).

Now I was wondering if a similar approach can be applied to check the weak solutions of an abstract evolution equation whose solutions are Banach space-valued functions, i.e, bounded linear operators. The abstract Cauchy problem is:

$$u'(t)=Au(t), \text{for}\hspace{3pt} t\geq 0$$ $$u(0)=x$$ where $A:D(A)\subset X \rightarrow X$ is the linear operator and the generator of a semigroup and $X$ is a Banach space. So far I am aware of two types of solutions namely classical solution in which $u(0)=x \in D(A)$ and a mild/integral solution in which $u(0)=x\in X$.

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  • $\begingroup$ Have a look at this Q&A and the references cited therein. Tonti's method is applicable not only to elliptic boundary value problems, but also to general evolution problems. $\endgroup$ Jul 8 '20 at 16:38
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Another possible reference is Theorem 3.1.7 in

Curtain, Ruth F.; Zwart, Hans, An introduction to infinite-dimensional linear systems theory, Texts in Applied Mathematics. 21. New York, NY: Springer-Verlag. xviii, 698 p. (1995). ZBL0839.93001.

where the equivalence of mild and weak solutions is formulated.

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See the book of A. Bensoussane et al. for different notions of solutions in section: 3 Nonhomogeneous linear evolution equations, especially Proposition 3.1 and 3.2 where it is proved that some of these notions coincide.

A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, Birkhäuser, 2007.

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