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.]


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

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)
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