A local existence result for a system of viscoelasticity with physical viscosity
Marta Lewicka Piotr B. Mucha
Evolution Equations & Control Theory 2013, 2(2): 337-353 doi: 10.3934/eect.2013.2.337
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.
keywords: maximal regularity well posedness. viscoelasticity Local existence viscosity
The surface diffusion flow on rough phase spaces
Joachim Escher Piotr B. Mucha
Discrete & Continuous Dynamical Systems - A 2010, 26(2): 431-453 doi: 10.3934/dcds.2010.26.431
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.
keywords: maximal regularity centre manifold stability Besov spaces Surface diffusion flow free boundary problem. global existence geometric evolution equation
Raphaël Danchin Piotr B. Mucha
Discrete & Continuous Dynamical Systems - S 2013, 6(5): 1163-1172 doi: 10.3934/dcdss.2013.6.1163
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.
keywords: generalizations boundary. low regularity Divergence
Limit of kinetic term for a Stefan problem
Piotr B. Mucha
Conference Publications 2007, 2007(Special): 741-750 doi: 10.3934/proc.2007.2007.741
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.
keywords: nonlocal parabolic systems curvature optimal regularity. Besov spaces Stefan problems vanishing kinetic effects
3D steady compressible Navier--Stokes equations
Milan Pokorný Piotr B. Mucha
Discrete & Continuous Dynamical Systems - S 2008, 1(1): 151-163 doi: 10.3934/dcdss.2008.1.151
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).
keywords: slip boundary conditions steady compressible Navier--Stokes equations existence and regularity of the solution.

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