# Weak and classical solutions of equations of motion for third grade fluids

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 33, Issue: 6, page 1091-1120
- ISSN: 0764-583X

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topBernard, Jean Marie. "Weak and classical solutions of equations of motion for third grade fluids." ESAIM: Mathematical Modelling and Numerical Analysis 33.6 (2010): 1091-1120. <http://eudml.org/doc/197589>.

@article{Bernard2010,

abstract = {
This paper shows that the decomposition method with special
basis, introduced by Cioranescu and Ouazar, allows one to
prove global existence in time of the weak solution for the third
grade fluids, in three dimensions, with small data. Contrary to the
special case where $\vert\alpha_1+\alpha_2\vert\le(24\nu\beta)^\{1/2\}$,
studied by Amrouche and Cioranescu, the H1 norm of the
velocity is not bounded for all data. This fact, which led others
to think, in contradiction to
this paper, that the method of decomposition could not apply
to the general case of third grade, complicates substantially the
proof of the existence of the solution. We also prove further
regularity results by a method similar to
that of Cioranescu and Girault for second grade fluids. This
extension to the third grade fluids is not straightforward, because of a
transport equation which is much more complex.
},

author = {Bernard, Jean Marie},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Galerkin method; special basis; energy estimates.; decomposition method; weak solution; classical solution; third grade fluid; global existence of solution; small initial data; regularity; energy estimate},

language = {eng},

month = {3},

number = {6},

pages = {1091-1120},

publisher = {EDP Sciences},

title = {Weak and classical solutions of equations of motion for third grade fluids},

url = {http://eudml.org/doc/197589},

volume = {33},

year = {2010},

}

TY - JOUR

AU - Bernard, Jean Marie

TI - Weak and classical solutions of equations of motion for third grade fluids

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

PB - EDP Sciences

VL - 33

IS - 6

SP - 1091

EP - 1120

AB -
This paper shows that the decomposition method with special
basis, introduced by Cioranescu and Ouazar, allows one to
prove global existence in time of the weak solution for the third
grade fluids, in three dimensions, with small data. Contrary to the
special case where $\vert\alpha_1+\alpha_2\vert\le(24\nu\beta)^{1/2}$,
studied by Amrouche and Cioranescu, the H1 norm of the
velocity is not bounded for all data. This fact, which led others
to think, in contradiction to
this paper, that the method of decomposition could not apply
to the general case of third grade, complicates substantially the
proof of the existence of the solution. We also prove further
regularity results by a method similar to
that of Cioranescu and Girault for second grade fluids. This
extension to the third grade fluids is not straightforward, because of a
transport equation which is much more complex.

LA - eng

KW - Galerkin method; special basis; energy estimates.; decomposition method; weak solution; classical solution; third grade fluid; global existence of solution; small initial data; regularity; energy estimate

UR - http://eudml.org/doc/197589

ER -

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