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### Open Access Journals

EECT

We prove the local in time existence of
regular solutions to the system of equations of isothermal
viscoelasticity with clamped boundary conditions.
We deal with a general form of viscous stress tensor
$\mathcal{Z}(F,\dot F)$, assuming a Korn-type condition on its
derivative $D_{\dot F}\mathcal{Z}(F, \dot F)$. This condition
is compatible with the balance of angular momentum, frame invariance
and the Claussius-Duhem inequality. We give examples of linear and
nonlinear (in $\dot F$) tensors $\mathcal{Z}$ satisfying these
required conditions.

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-S

This note is dedicated to a few questions related to the divergence equation which
have been motivated by recent studies concerning the Neumann problem for the Laplace
equation or the (evolutionary) Stokes system in domains of $\mathbb{R}^n.$
For simplicity, we focus on the classical Sobolev spaces framework in bounded domains, but our results
have natural and simple extensions to the Besov spaces framework in more general domains.

PROC

We investigate solutions to the one-phase quasi-stationary Stefan problem with the surface tension and kinetic term. Main results show existence of unique regular solutions with a suitable bound which enables to obtain the limit as the kinetic term is vanishing. Our problem is considered in anisotropic
Besov spaces locally in time.

DCDS-S

We study the steady compressible Navier--Stokes equations in a bounded smooth
three-dimensional domain,
together with the slip boundary conditions. We show that for a certain
class of the pressure laws,
there exists a weak solution with bounded density
(in $L^\infty$ up to boundary).

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