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Information Center for Mathematical Science

### 논문검색

Information Center for Mathematical Science

#### 논문검색

Communications in Partial Differential Equations
( Vol. 19 NO.11 / (1994))
Young Measure-Valued Solutions for Non-Newtonian Incompressible Fluids

Pages. 1763-1803
Abstract For the model of a nonlinear bipolar fluid, in which the highest order viscosity vanishes, and the viscous part of the stress tensor satisfies a growth condition of the form $| au_{ij}(mathbf e)|leq C(1+|mathbf e|)^{p-1}, C>0$, $mathbf e$ the rate of strain tensor, we demonstrate the existence of Young-measure valued solutions for $p>1$ (in dim $n=2$) and for $p> 6/5$ (in dim $n=3$); these solutions are proven to be weak solutions for $3/2 < p< 2$ (in dim $n=2$) and for $9/5 < p <11/5$ (in dim $n=3$) and unique regular weak solutions for $pleq 2$ (in dim $n=2$) and for $pgeq 11/5$ (in dim $n=3$). Much of the analysis deals with the associated space periodic problems. 1. Introduction 2. The incompressible bipolar fluid 3. Young measure-valued solutions 4. Young measures are Dirac and the weak solutions are regular for $p\geq 2$, $n=2$ 5. Young measures are Dirac for $3/2< p < 2$, $n=2$ 6. Young measures are Dirac and the weak solutions are regular for $7/3 \leq p< 2$, $n=3$ 7. Young measures are Dirac and the weak solutions are regular for $11/5 \leq p< 7/3$, $n=3$ 8. Young measures are Dirac for $2\leq p < 11/5$, $n=3$ 9. Young measures are Dirac for $9/5< p < 2$, $n=3$