3
$\begingroup$

I have been learning PDEs (more specifically, nonlinear dispersive equations (Schrödinger/wave/ Klein-Gordan equations etc...)) through the harmonic analysis methods. And I have read a couple of papers which handle local well-posedness results (mainly with contraction mapping principle) and global well-posedness results (mainly with conservation laws/persistence of regularity etc...).

Let $X$ be a Banach space of functions (distributions) on $\mathbb R.$ For instance, we consider the nonlinear Schrödinger equation (NLS): $$iu_{t}+\Delta u = F(u), \ u(t_{0}, x)=u_{0}(x)\in X,$$ where $u:\mathbb R \times \mathbb R \to \mathbb C, u_{0}:\mathbb R\to \mathbb C, F:\mathbb C \to \mathbb C$ are functions, and with initial time $t_{0}\in \mathbb R.$

We suppose that the local well-posedness results and blowup criteria hold in $X,$ that is, for the given initial data $u_{0}\in X$ there is a $T^*= T^*(\|u_{0}\|_{X})>t_{0}$ so that the above NLS has unique solution $u\in C([t_{0}, T^*], X);$ and if $T*<\infty,$ then $\limsup_{t\to T^*} \|u(t, \cdot)\|_{X}= \infty.$

My Questions:

(I) Can you give an example of some Banach space $X$ where local well-posedness and blowup criteria hold (in the above sense) but one can not extend to a solution for all time (I mean one cannot get global solution)?

(II) Is there any well-known method, which can show there do not exist global well-posedness results?

[Of course the question highly depends on $X$ but I am just looking for examples where we have local well-posedness but do not have global well-posedness, and keen to know of a method (if it exists) with which one can show global result does not exist. I hope this question makes sense and some proper references will be o.k for me.]

Thanks,

$\endgroup$
  • 2
    $\begingroup$ could you fix the subject? it does not parse... $\endgroup$ – Dima Pasechnik Apr 11 '15 at 12:12
  • 2
    $\begingroup$ Having performed an edit, I wonder if I got it right. My apologies if not. $\endgroup$ – Todd Trimble Apr 11 '15 at 23:41
  • 5
    $\begingroup$ An Example for question (I): $X=\mathbb{R}$, $u'=u^2$, $u(0)=1$. $\endgroup$ – gsa Apr 12 '15 at 15:42
1
$\begingroup$

On the space of $Imm(S^1,\mathbb R^2)$ of immersed closed parameterized plane curves, one can consider the weak Riemannian metric of Sobolev order $H^k$. The geodesic equation for this metric is locally well-posed for $k\ge 1$, but not globally well posed if $k=1$. This can be shown since concentric circles form a geodesic which reaches 0 in finite time. The geodesic equation is even globally well-posed for $k\ge 2$ and non-vanishing coercive term, see:

  • Martins Bruveris, Peter W. Michor, David Mumford: Geodesic Completeness for Sobolev Metrics on the Space of Immersed Plane Curves. Forum of Mathematics, Sigma 2, e19, 38 pages, 2014. (pdf)

  • arXiv:1407.0601

On groups of diffeomorphisms $Diff(M)$ for $M$ a compact manifold or $\mathbb R^n$, the right invariant weak Riemannian metric of Sobolev order $H^k$ has a geodesic equation which is locally well-posed for $k\ge 1$, and it is globally well-posed if $k> \frac{\dim(M)}2+1$,

But on diffeomorphism groups the situation is not so clear cut, because sometimes there exists a geodesic completion, like in the case of of the homogeneous $\dot H^1$-metric on an extension of the group $Diff(\mathbb R)$, see:

  • Martin Bauer, Martins Bruveris, Peter W. Michor: Homogeneous Sobolev metric of order one on diffeomorphism groups on the real line. Journal of Nonlinear Science 24, 5 (2014), 769-808. (pdf)

Even for Burgers equation (= geodesic equation for $H^0$-metric on $Diff(\mathbb R)$) with its well known hysteresis breaking behavior there exists a geodesic completion, a subset of the space of plane curves, see:

  • Boris Khesin, Peter W. Michor: The flow completion of Burgers' equation. In: Infinite dimensional groups and manifolds. Editor: Tilmann Wurzbacher. IRMA Lectures in Mathematics and Theoretical Physics 5. De Gruyter, Berlin, 2004. pp. 17-26. (pdf)

More papers with results on geodesically complete and incomplete weak Riemannian manifolds of immersions and shapes are collected in the references of the overview paper

  • Martin Bauer, Martins Bruveris, Peter W. Michor: Overview of the Geometries of Shape Spaces and Diffeomorphism Groups. Journal of Mathematical Imaging and Vision, 50, 1-2, 60-97, 2014. (pdf)
| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.