# methods for situations where well-posedness criteria hold but global solutions do not exist

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,

• could you fix the subject? it does not parse... Apr 11, 2015 at 12:12
• Having performed an edit, I wonder if I got it right. My apologies if not. Apr 11, 2015 at 23:41
• An Example for question (I): $X=\mathbb{R}$, $u'=u^2$, $u(0)=1$.
– gsa
Apr 12, 2015 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)