The mysterious $\dot{H}^{-1}$ notation.

Question

I have encountered the $\dot{H}^{-1}$ notation in one of the SIAM Journal on Mathematical Analysis articles. It appears to be standard (or at least not uncommon) to use this one in the field, since there is no formal introduction or definition of $\dot{H}^{-1}$ in the article itself, and because of the fact that it first appears in the abstract. Authors seem to be confident using expressions like

• $\dot{H}^{-1}$ norm,
• $\dot{H}^{-1} $ distance,
• $\dot{H}^{-1} $ gradient flow of energy,

and others.

I tried my best searching for it online and checking the article references, but could not figure it out. Can anyone give me a hint on where to find a definition of either of $\dot{H}^{-1}$ objects mentioned above?

The article is devoted to the stability analysis of a certain type of solutions of a nonlinear PDE.

Answer 1

Here is the answer provided by the author of the original article, where I saw this notation first:

$\dot{H^{-1}}$ is the homogeneous $H^{-1}$ norm. It is the dual space of $\dot{H^{1}}$, the homogeneous $H^1$ norm. The homog $H^1$ norm of $g$ is $\int |\nabla g|^2 dx$. i.e., no zeroth order term. (So it is not really a norm, but a seminorm. The constants are allowed.)

You can also think about the $H^k$ spaces via Fourier. The homogeneous negative norms have terms like $\int (f_k/ k)^2 dk$.

On $\mathbb{R}$ (as in our paper that you referred to), you can see—by integration by parts—that the norm of $f$ in $\dot{H^{-1}}$ will turn out to be the $L^2$ norm of an antiderivative of $f$. At the same time, you see that if $f$ has an antiderivative $F$ that is in $L^2$, it only has ONE SUCH antiderivative (since constants are not in $L^2(\mathbb{R})$).

I hope that helps a bit. I don't know of a nice source that discusses these things. I think Evans only discusses the dual of $H_0^1(\Omega)$ for nice, bounded $\Omega$ (if I recall correctly). I saw some lecture notes online, but nothing that I would really recommend.

Hope it will be helpful for someone.

Answer 2

$\dot H^{-1}$ is dual homogeneous Sobolev space http://sma.epfl.ch/~wwywong/papers/sobolevnotes.pdf