An L²-maximal regularity result for the evolutionary Stokes–Fourier system

Abstract

We establish an L ²-regularity result for a weak solution of the evolutionary Stokes–Fourier system. Although this system does not contain the convective terms, the fact that the viscosity depends on the temperature makes the considered system of partial differential equations nonlinear. The result holds for a class of the viscosities that includes the Arrhenius formula as a special case. For simplicity, we restrict ourselves to a spatially periodic setting in this study.

Keywords:

• regularity
• Stokes–Fourier system
• temperature-dependent viscosity

AMS Subject Classifications::

• primary 35Q10
• secondary 35B10, 35B65

Acknowledgements

M. Bulíček is supported by Nečas Center for Mathematical Modeling, project LC06052 financed by MŠMT. The contribution of P. Kaplický and J. Málek to this work is a part of the research project MSM 0021620839 financed by MŠMT. They thank the Czech Science Foundation, the projects GAČR 201/09/0917 (P. Kaplický) and GAČR 201/08/0315 (J. Málek), for its support.

Notes

Notes

1. Neglecting the convective term in (4) one obtains The Equation (7) is then achieved if one subtracts from the above equation the equation that is a result of a scalar multiplication of (6) by v . While for the evolutionary Stokes–Fourier system, the above equation and (7) are equivalent, the similar procedure adopted to the complete Navier–Stokes–Fourier system is not in general feasible if one deals with the concept of weak solutions. In such cases, the form (4) should be used, see 3,4 for details.

2. Note that at this level of approximation such procedure is a possible that is a consequence of standard parabolic regularity results.

3. Note that (54) directly implies at least local in time regularity even for non-smooth initial data.