TY  - CHAP
T1  - On Hausdorff Dimension of Blow-Up Times Relevant to Weak Solution of Generalized Navier-Stokes Fluids
A1  - Bulíček, Miroslav
A1  - Málek, Josef
A1  - Terasawa, Yutaka
TI  - Mathematical Analysis on the Navier-Stokes Equations and Related Topics, Past and Future  In memory of Professor Tetsuro Miyakawa
T3  - Mathematical Sciences and Applications
Y1  - 2011
VL  - 35
SP  - 116
EP  - 129
PB  - Gakuto International Series
N1  - Preprint NCMM no. 2010-032
UR  - http://www.karlin.mff.cuni.cz/ncmm/preprints/10320230720pr32.pdf
KW  - Generalized Navier-Stokes fluid
KW  - Hausdorf measure
KW  - Power-law fluid
KW  - singular set
KW  - times of blow-up
KW  - weak solution
N2  - We consider an initial and spatially periodic problem for °ows
of generalized Navier-Stokes °uids (of power-law type). We study qualitative
properties of weak solutions, which are known to exist for large-data and on
an arbitrary time interval, in the case where the weak solution itself is not
an admissible test function in the balance of linear momentum. We focus on
establishing the upper estimates of the Hausdor® dimension of the possible
times at which the singularity can occur - the L2
-norm of the velocity gradient
can blow up. We provide a simple method that improves the estimates known
before in two ways: the estimates are valid for larger range of model parameters
and the estimates are sharper. For some values of model parameters, we
establish new results concerning uniqueness of the strong solution in the class
of weak solutions satisfying the energy inequality.
ER  -