Why we study abstract differential equations in Banach and Hilbert spaces?

Question

We know that Abstract differential equation is a differential equation in which unknown function takes value in Abstract spaces such as Hilbert and Banach spaces, I know they are complete inner product and normed spaces respectively. What is the benefit of studying differential equations in abstract spaces.

Answer 1

These are interesting because they allow to treat certain cases of partial differential equations with methods equal or similar to ordinary differential equations. With PDE it is often not even certain that a solution exists uniformly, or globally in the space variables, in forward time. However if one can mimic the conditions of the Picard-Lindelöf theorem, then this type of solution is automatic.

Note that the general idea of the ODE existence theorems is to transform the differential equation into an integral equation, so that considered as an fixed-point iteration it continually smooths the iterates.

The difficulty with applying this to the "abstract" situation is that the operator $A$ in $\frac{du}{dt}=Au+f$ will be some differential operator, so that applying the same transformation to an integral equation $$u(t)=u(0)+\int_0^t[Au(s)+f(s)]\,ds$$ at a first glance the "input" of the right side is first degraded in terms of space differentiability and then improved in time differentiability. Now if one could balance these trends over the dimensions,... which leads to the consideration of the semi-groups of propagators that generalize the idea of a fundamental matrix that solve the homogeneous equation, etc.

Answer 2

It is very useful for PDEs. For instance, consider the heat equation $$\partial_t u(t,x) = \Delta u(t,x),\quad t > 0,\ x \in \mathbb R^n$$ $$u(t=0,x) = u_0(x)$$ where $u$ is a function of $t$ and $x$. For a given $t>0$, $x \in \mathbb R^n \mapsto u(t,x)$ belongs to an abstract space $E$, for instance $$E = \{ v \in L^2(\mathbb R^n) \colon \Delta u \in L^2(\mathbb R^n) $$

Now the heat equation can be written under the abstract differential equation $$u^\prime(t) = A u(t),\quad t>0$$ $$ u(0) = u_0 $$ where $u$ is now consider as a function of $t$ but the values now belongs to $E$: $u(t) \in E$ and where $A$ is now an operator from $E$ to $L^2(\mathbb R^n)$ defined by $$Av = \Delta v$$