Title: 2010_10_08_Czubak_AnalysisApplied Author: James Colliander Email: colliand@math.toronto.edu Web: http://www.math.toronto.edu/colliand Date: 2010-08-31 Affiliation: Department of Mathematics, University of Toronto image: http://wiki.math.toronto.edu/TorontoMathWiki/images/2/20/Cherry.png BibTex: /Users/colliand/Documents/Master.bib BibliographyStyle: alpha XHTML XSLT: xhtml-toc-h2.xslt
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Nonuniqueness of the Leray-Hopf Solutions of Navier-Stokes in the Hyperbolic Setting

Background References

(joint work w. Chi Hin Chan [IMA])

Let’s look at the equation on Euclidean space first.

On $R^n$, we have the NS equation

$$ \partial_t u = \Delta u + u \cdot \nabla u + \nabla p = 0, ~ \nabla \cdot u = 0.$$

Leray (1934) and Hopf (1951) showed existence of at least one $ u \in L^\infty (0,\infty; L^2 ) \cap L^2(0,\infty; H^1)$, for $n \geq 2,$ solution for $t \geq 0$ which satisfies the energy inequality $$ \int |u(t,x)|^2 dx + 2 \int_0^t \int |\nabla u|^2 dx ds \leq \int |u_0|^2 dx $$

Question: How does the geometry affect the solution to NS?

The question has been refined further in this study to How does the geometry affect the uniqueness of the L-H solution?

Strategy for showing nonuniqueness

Motivation: Consider $F: R^n \rightarrow R$ satisfying $\Delta F = 0$? Then, if we let $u(t,x) = \psi (t) \nabla F$ can we make this thing solve NS? We can calculate the time derivative. We are certain that the components of the gradient $\nabla F$ are also harmonic. We also see that this expression is divergence free. We compute $u \cdot \nabla u$ and we find it equals $\psi^2 \frac{1}{2}\nabla |\nabla F|^2$. So, if we let $p = - \partial_t \psi F - \frac{1}{2} \psi^2 |\nabla F |^2.$ We then encounter some cancellations. This motivates us to choose this $p$ and this $u$. If we take $u(0) = \nabla F$ and we pick $\psi(0) =1$. If we had all this, we could construct nonunique solutions. We would like, at a very minimum, that $u \in L^2$ in order to have a L-H solution. If we could do all this, we could construct nonunique solutions on $R^n$. On $R^n$, this strategy will fail because the only bounded harmonic functions on $R^n$ are constants.

But what happens when we require $u \in L^\infty (L^2_x)$? On $R^n$, we lose.

Theorem (S.T. Yau 1976): Let $p \in (1, \infty), M$ any complete Riemannian manifold. If $u$ is a positive subharmonic function such that $u \in L^p (M)$ then $u = constant$.

There are counterexamples associated with $q=1$.

So, for example, on $R^n$, if $\nabla F \in L^2$ is nontrivial, we would find that $F \in L^6$ so that would contradict the theorem of Yau.

What happens on the hyperbolic space? There are nice things happening for us, but bad things for NS.

Yau 1975: Any bounded harmonic function on a manifold which has positive Ricci curvature.

Anderson-Sullivan 1983: Let $M$ be a simply connected $n$-dimensional complete Riemannian manifold with sectional curvature $-b^2 \leq K_M \leq - a^2 < 0$. Then there exists a unique solution $F \in C^\infty (M) \cap C^0 (M)$ to $\Delta F=0$ such that $F$ restricts to the sphere at $\infty$ to a function $\phi \in C^0 (S(\infty))$.

Thus, there exists a nontrivial bounded harmonic function on manifolds with negative (but bounded) sectional curvature. A simplified proof has been given in the Schoen-Yau book. There is also a nice gradient estimate for $F$.

Replace $- \Delta$ by the Hodge laplacian. We can recast the equation as a condition on a one form rather than on a vector using the natural identification through the metric. Using this machinery, we can write the NS equation on a manifold

$$ \partial_t {U} - \Delta {U} + {\overline{\nabla_u}} {U} - 2 Ric ({U}) + dp = 0, ~{dstar} {U} = 0. $$

This equation was written down in 1970. In 1994 Priebe has a paper on compact manifolds with no-slip boundary conditions. Avez-Baumberger…Q.S. Zhang on noncompact manifolds 2006 has a nonuniqueness result….other references….

Theorem: Let $a>0$. Consider $NS( \mathbb{H}^2 (- a^2))$ is ill-posed in the following sense: There exists (infinitely many) $u_0 \in L^2 (\mathbb{H}^2 (-a^2))$ such that there exist infinitely many smooth solutions which have finite energy, finite dissipation and saitisfy the global energy inequality.

Corollary: Let $n \geq 2$. Let $a >0$. There exist nontrivial bounded solutions of Navier-Stokes on $\mathbb{H}^2 (-a^2)$.

The pressure formula we gave in the motivation discussion, has an extra term related to the Ricci curvature.

2010-10-12 Still working on the talk comments…but I had some small ideas.

The “Liouville Property” discussed in the KNSS 2007 paper and mentioned in Kenig-Koch 2009 captures a certain “rigidity” in the ancient solutions of Navier-Stokes.

The Liouville Property is a global property of the Stokes problem on the Euclidean domain. A local property that can sometimes be exploited to prove the Liouville Property is the Unique Continuation idea, which is often implemented using Carleman Estimates.

Carleman Estimates are exponential weighted inequalities.

I wonder if the exponential weights in the Carleman Estimates and the volume growth function in the manifold setting are geometrically intertwined. Can you show the geometric failure of some (true in Euclidean space) Carlemen estimate on the hyperbolic domain?