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DCDS-B

Over the period 12-23 October 2009, the program "Recent advances in
integrable systems of hydrodynamic type'' was organized by us and
held at the

For more information please click the “Full Text” above.

*Erwin Schrödinger International Institute for Mathematical Physics*(Vienna, Austria).For more information please click the “Full Text” above.

keywords:

DCDS

The surface diffusion flow is the gradient flow of the surface functional of compact hypersurfaces with respect to the inner product of $H^{-1}$ and leads to a nonlinear evolution equation of fourth order. This is an intrinsically difficult problem, due to the lack of an maximum principle and it is known that this flow may drive smoothly embedded uniformly convex initial surfaces in finite time into non-convex surfaces before developing a singularity [15, 16]. On the other hand it also known that singularities may occur in finite time for solutions emerging from non-convex initial data, cf. [10].

Combining tools from harmonic analysis, such as Besov spaces, multiplier results with abstract results from the theory of maximal regularity we present an analytic framework in which we can investigate weak solutions to the original evolution equation. This approach allows us to prove well-posedness on a large (Besov) space of initial data which is in general larger than $C^2$ (and which is in the distributional sense almost optimal). Our second main result shows that the set of all compact embedded equilibria, i.e. the set of all spheres, is an invariant manifold in this phase space which attracts all solutions which are close enough (which respect to the norm of the phase space) to this manifold. As a consequence we are able to construct non-convex initial data which generate global solutions, converging finally to a sphere.

Combining tools from harmonic analysis, such as Besov spaces, multiplier results with abstract results from the theory of maximal regularity we present an analytic framework in which we can investigate weak solutions to the original evolution equation. This approach allows us to prove well-posedness on a large (Besov) space of initial data which is in general larger than $C^2$ (and which is in the distributional sense almost optimal). Our second main result shows that the set of all compact embedded equilibria, i.e. the set of all spheres, is an invariant manifold in this phase space which attracts all solutions which are close enough (which respect to the norm of the phase space) to this manifold. As a consequence we are able to construct non-convex initial data which generate global solutions, converging finally to a sphere.

DCDS-B

We study a moving boundary problem describing the growth of nonnecrotic
tumors in different regimes of vascularisation.
This model
consists of two decoupled Dirichlet problem, one for the rate at which nutrient is added to the tumor domain and one for the pressure
inside the tumor.
These variables are coupled by a relation which describes the dynamic of the boundary.
By
re-expressing the problem as an abstract evolution equation, we prove
local well-posedness in the small Hölder spaces context.
Further on, we use the principle of linearised stability to characterise the stability properties of the unique radially symmetric equilibrium of the problem.

JGM

In this paper, we study the geodesic flow of a right-invariant metric induced by a general Fourier multiplier on the diffeomorphism group of the circle and on some of its homogeneous spaces. This study covers in particular right-invariant metrics induced by Sobolev norms of fractional order. We show that, under a certain condition on the symbol of the inertia operator (which is satisfied for the fractional Sobolev norm $H^{s}$ for $s \ge 1/2$), the corresponding initial value problem is well-posed in the smooth category and that the Riemannian exponential map is a smooth local diffeomorphism. Paradigmatic examples of our general setting cover, besides all traditional Euler equations induced by a local inertia operator, the Constantin-Lax-Majda equation, and the Euler-Weil-Petersson equation.

CPAA

The increased interest in water wave theory over the last decade has been motivated,
arguably, by two themes: rst, noticeable progress in the investigation of the
governing equations for water waves (well-posedness issues, as well as in-depth qualitative
studies of regular wave patterns{see the discussion and the list of references
in [1, 17], respectively in [9]), and secondly, by the derivation and study of various
model equations that, although simpler, capture with accuracy the prominent features
of the governing equations in a certain physical regime. The two themes are
intertwined with one another.

keywords:

JGM

In the following we study the qualitative properties of solutions
to the geodesic flow induced by a higher order two-component Camassa-Holm system.
In particular, criteria to ensure the existence of temporally global solutions are presented. Moreover
in the metric case, and for inertia operators of order higher than three, the flow is shown
to be geodesically complete.

JGM

The geodesic equations of a class of

*right invariant*metrics on the semi-direct product $Diff(\mathbb{S}^1)$Ⓢ$Diff(\mathbb{S}^1)$ are studied. The equations are explicitly described, they have the form of a system of coupled equations of Camassa-Holm type and possess singular (peakon) solutions. Their integrability is further investigated, however no compatible bi-Hamiltonian structures on the corresponding dual Lie algebra $(Vect(\mathbb{S}^1)$Ⓢ$Vect(\mathbb{S}^1))^{*}$ are found.
DCDS

After some remarks on a possible zero-curvature formulation we
first establish local well-posedness for the 2-component
Camassa-Holm equation. Then precise blow-up scenarios for strong
solutions to the system are derived. Finally we present two
blow-up results for strong solutions to the system.

CPAA

We show that the modified Constantin-Lax-Majda equation modeling vortex and quasi-geostrophic dynamics [27] can be recast as the geodesic flow on the subgroup $\mathrm{Diff}_{1}^{\infty}(\mathbb{S})$ of orientation-preserving diffeomorphisms $\varphi \in \mathrm{Diff}^{\infty}(\mathbb{S})$ such that $\varphi(1) = 1$ equipped with the right-invariant metric induced by the homogeneous Sobolev norm $\dot H^{1/2}$. On the extended group of diffeomorphisms of Sobolev class $H^{k}$ with $k\ge 2$, this induces a

*weak Riemannian*structure. We establish that the geodesic spray is smooth and we obtain local existence and uniqueness of the geodesics.
DCDS-B

This paper is concerned with the long time behaviour of a weakly
dissipative Degasperis-Procesi equation. Our analysis discloses the
co-existence of global in time solutions and finite time break down
of strong solutions. Our blow-up criterion for the initial profile
generalizes considerably results obtained earlier in [32].

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