# Derivative of Banach-space valued functions, and weak solutions

• zmhl0910
In summary, a weak solution of a parabolic PDE is defined as a Banach-space valued function u that satisfies certain conditions. The weak derivative of u, denoted by u', is defined using Bochner integrals and can be identified with a function in H_0^1(\Omega) through the self-duality of the space. This is necessary to make sense of the expression and ensure the existence and uniqueness of solutions. The distinction between L^2\left(0,T; H_0^1(\Omega)\right) and L^2\left(0,T; H^{-1}(\Omega)\right) is important in guaranteeing that the solution u belongs to a suitable function space for the
zmhl0910
Hello everyone,

I'm new to this forum and this is my first post. I hope this is the right place to ask the following.

I've been reading up a little on PDE theory lately, and the following point has been causing me a bit of confusion.

In the definition of the weak solution of a parabolic PDE, one considers Banach-space valued functions

$u\in L^2\left(0,T; H_0^1(\Omega)\right)$​

such that

$u' \in L^2\left(0,T; H^{-1}(\Omega)\right)$​

Here $T>0$ and $\Omega\subset\mathbb{R}^n$ ($n\ge 3$) is bounded open.

Does $u'$ here refer to the weak derivative of $u$, i.e. the function defined by

$\int_0^T u'(t)\phi(t) \;\mathrm{d}t := -\int_0^T u(t) \phi'(t) \;\mathrm{d}t \quad \mbox{for all \phi\in C^\infty_c(0,T)}$​

(Here the integrals are Bochner integrals.)

If so, how does one make sense of the above expression? Specifically, my gripe is that the left-hand side technically belongs to $H^{-1}(\Omega)$ while the right-hand side belongs to $H_0^1(\Omega)$. Is one supposed to identify both sides via the self-duality of $H_0^1(\Omega)$ as a Hilbert space?

In which case, what is the rationale for distinguishing between $L^2\left(0,T; H_0^1(\Omega)\right)$ and $L^2\left(0,T; H^{-1}(\Omega)\right)$ (for example, in the definition of the weak solution to a parabolic PDE) in the first place?

Thank you for your question. Yes, u' refers to the weak derivative of u in the definition of the weak solution of a parabolic PDE. To make sense of the expression, one can use the self-duality of H_0^1(\Omega) as a Hilbert space, as you mentioned. This allows us to identify both sides and treat them as equal. The rationale for distinguishing between L^2\left(0,T; H_0^1(\Omega)\right) and L^2\left(0,T; H^{-1}(\Omega)\right) is to ensure that the solution u belongs to a suitable function space that allows for the existence and uniqueness of solutions to the parabolic PDE. By requiring u' to be in L^2\left(0,T; H^{-1}(\Omega)\right), we ensure that the weak derivative of u is a well-defined function that belongs to a suitable space for the PDE. I hope this helps clarify your confusion. Keep reading and learning, and don't hesitate to ask more questions on the forum. Best of luck in your studies!

## 1. What is a Banach space?

A Banach space is a complete normed vector space, meaning that it is a space of objects (called vectors) with a defined distance function (called a norm) that satisfies certain properties. In simpler terms, a Banach space is a mathematical object that allows us to measure the size and direction of its elements.

## 2. What is the derivative of a Banach-space valued function?

The derivative of a Banach-space valued function is a linear operator that measures the rate of change of the function at a given point. In other words, it tells us how much the function changes as we move along a certain direction in its domain.

## 3. How is the derivative of a Banach-space valued function defined?

The derivative of a Banach-space valued function is defined using a limit process, similar to the definition of derivatives for real-valued functions. However, since Banach spaces are more complex mathematical objects, the definition involves more technical details and notation.

## 4. What are weak solutions in the context of Banach-space valued functions?

Weak solutions are a type of solution to a differential equation that may not satisfy the equation in the traditional sense, but still satisfies a modified version of the equation that involves a weaker notion of differentiation. In the context of Banach-space valued functions, weak solutions are often used to study more general and abstract problems.

## 5. Why are Banach spaces important in the study of derivative and weak solutions?

Banach spaces are important in the study of derivative and weak solutions because they provide a rich and general framework for analyzing mathematical problems that involve functions with more complicated properties. In particular, Banach spaces allow us to define and study derivatives of functions that take values in these spaces, and provide a natural setting for understanding weak solutions to differential equations.

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