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Discrete & Continuous Dynamical Systems - S

2010 , Volume 3 , Issue 3

Special issue
on evolutionary PDEs in fluid mechanics

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Preface
Eduard Feireisl, Josef Málek and Mirko Rokyta
2010, 3(3): i-ii doi: 10.3934/dcdss.2010.3.3i +[Abstract](387) +[PDF](38.2KB)
Abstract:
This special volume consists of survey contributions based on a series of lectures delivered at the Eleventh International School on Mathematical Theory in Fluid Mechanics, held in a small village of Kácov in the central part of Bohemia. The main speakers who delivered series of four lectures were Martin Oberlack , Technical University Darmstadt, Steve Shkoller, University of California at Davis, Endre Süli, Mathematical Institute of Oxford University, Roman Shvydkoy, University of Illinois at Chicago, and Alexis Vasseur, University of Texas at Austin. Details concerning the scope and program of the school are available at the web-page

www.karlin.mff.cuni.cz/paseky-fluid/2009/

Martin Oberlack and Andreas Rosteck in New statistical symmetries of the multipoint equations and its importance for turbulent scaling laws show that the infinite set of multi-point correlation equations, which are direct statistical implications of the Navier-Stokes equations, admit a vast set of Lie symmetry groups. In particular, a new scaling group and translational groups of vectors and all independent variables are discovered. These new statistical groups provide important implications for understanding turbulent scaling laws.

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Existence of global weak solutions to Fokker-Planck and Navier-Stokes-Fokker-Planck equations in kinetic models of dilute polymers
John W. Barrett and Endre Süli
2010, 3(3): 371-408 doi: 10.3934/dcdss.2010.3.371 +[Abstract](699) +[PDF](461.6KB)
Abstract:
This survey paper reviews recent developments concerning the existence of global weak solutions to Fokker-Planck equations with unbounded drift terms, and coupled Navier-Stokes-Fokker-Planck systems of partial differential equations, that arise in finitely extensible nonlinear elastic (FENE) type kinetic models of incompressible dilute polymeric fluids in the case of general noncorotational flow.
The De Giorgi method for regularity of solutions of elliptic equations and its applications to fluid dynamics
Luis A. Caffarelli and Alexis F. Vasseur
2010, 3(3): 409-427 doi: 10.3934/dcdss.2010.3.409 +[Abstract](726) +[PDF](223.9KB)
Abstract:
This paper is dedicated to the application of the De Giorgi-Nash-Moser kind of techniques to regularity issues in fluid mechanics. In a first section, we recall the original method introduced by De Giorgi to prove $C^\alpha$-regularity of solutions to elliptic problems with rough coefficients. In a second part, we give the main ideas to apply those techniques in the case of parabolic equations with fractional Laplacian. This allows, in particular, to show the global regularity of the Surface Quasi-Geostrophic equation in the critical case. Finally, a last section is dedicated to the application of this method to the 3D Navier-Stokes equation.
A simple proof of well-posedness for the free-surface incompressible Euler equations
Daniel Coutand and Steve Shkoller
2010, 3(3): 429-449 doi: 10.3934/dcdss.2010.3.429 +[Abstract](732) +[PDF](244.7KB)
Abstract:
The purpose of this this paper is to present a new simple proof for the construction of unique solutions to the moving free-boundary incompressible 3-D Euler equations in vacuum. Our method relies on the Lagrangian representation of the fluid, and the anisotropic smoothing operation that we call horizontal convolution-by-layers. The method is general and can be applied to a number of other moving free-boundary problems.
New statistical symmetries of the multi-point equations and its importance for turbulent scaling laws
Martin Oberlack and Andreas Rosteck
2010, 3(3): 451-471 doi: 10.3934/dcdss.2010.3.451 +[Abstract](1027) +[PDF](256.0KB)
Abstract:
We presently show that the infinite set of multi-point correlation equations, which are direct statistical consequences of the Navier-Stokes equations, admit a rather large set of Lie symmetry groups. This set is considerable extended compared to the set of groups which are implied from the original set of equations of fluid mechanics. Specifically a new scaling group and translational groups of the correlation vectors and all independent variables have been discovered. These new statistical groups have important consequences on our understanding of turbulent scaling laws to be exemplarily revealed by two examples. Firstly, one of the key foundations of statistical turbulence theory is the universal law of the wall with its essential ingredient is the logarithmic law. We demonstrate that the log-law fundamentally relies on one of the new translational groups. Second, we demonstrate that the recently discovered exponential decay law of isotropic turbulence generated by fractal grids is only admissible with the new statistical scaling symmetry. It may not be borne from the two classical scaling groups implied by the fundamental equations of fluid motion and which has dictated our understanding of turbulence decay since the early thirties implicated by the von-Kármán-Howarth equation.
Lectures on the Onsager conjecture
Roman Shvydkoy
2010, 3(3): 473-496 doi: 10.3934/dcdss.2010.3.473 +[Abstract](581) +[PDF](334.0KB)
Abstract:
These lectures give an account of recent results pertaining to the celebrated Onsager conjecture. The conjecture states that the minimal space regularity needed for a weak solution of the Euler equation to conserve energy is $1/3$. Our presentation is based on the Littlewood-Paley method. We start with quasi-local estimates on the energy flux, introduce Onsager criticality, find a positive solution to the conjecture in Besov spaces of smoothness $1/3$. We illuminate important connections with the scaling laws of turbulence. Results for dyadic models and a complete resolution of the Onsager conjecture for those is discussed, as well as recent attempts to construct dissipative solutions for the actual equation.
   The article is based on a series of four lectures given at the 11th school "Mathematical Theory in Fluid Mechanics" in Kácov, Czech Republic, May 2009.
On the motion of incompressible inhomogeneous Euler-Korteweg fluids
Miroslav Bulíček, Eduard Feireisl, Josef Málek and Roman Shvydkoy
2010, 3(3): 497-515 doi: 10.3934/dcdss.2010.3.497 +[Abstract](712) +[PDF](246.2KB)
Abstract:
We study a system of equations governing evolution of incompressible inhomogeneous Euler-Korteweg fluids that describe a class of incompressible elastic materials. A local well-posedness theory is developed on a bounded smooth domain with no-slip boundary condition on velocity and vanishing gradient of density. The cases of open space and periodic box are also considered, where the local existence and uniqueness of solutions is shown in Sobolev spaces up to the critical smoothness $\frac{n}{2}+1$.

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