On nonlinear stability of linear pinch

Abstract

Nonlinear stability of cylindrical configurations with free boundaries is studied for electro-conducting fluids. First, the initial boundary value problem is correctly set for the full system governing flows of incompressible, electrically conducting fluids in Ω _t with unknown free surface. Notice that when the domain exterior to the fluid is a dielectric or the vacuum, also the electromagnetic field in becomes an unknown parameter of the problem. Second a nonlinear stability criterion for the rest state of an electro-conducting, incompressible fluid confined in a section of a right cylinder with lateral free boundary in the vacuum is proposed. This criterion proposes an alternative definition of perturbation, and is deeply relied to the unknown motion of the boundary. Third, if the fluid has non-significant magnetic susceptibility, in the presence of not too large surface currents, for large initial data, nonlinear stability is proved in a suitable class of global regular solutions. Kinematic viscosity, magnetic diffusivity, surface tension may be only non-negative in the absence of surface currents.

Keywords:

• electro-conducting fluids
• free-boundary
• capillarity
• nonlinear stability

AMS Subject Classifications::

• 76E05
• 76W05
• 35Q30

Acknowledgements

Padula thanks 60% MURST, the GNFM of the italian CNR-INDAM, and 40% MIUR-COFIN 2006-2008 ‘La matematica dei processi di crescita tumorali e trasporto nelle applicazioni biomediche e industriali’. He thanks Prof V.A. Solonnikov for his careful reading of this article and his wise corrections, and Prof G. Bizhanova for stimulating discussions on the subject. The author wishes sincerely to thank the referees of this article, who have stimulated resolution of the problem, when surface currents are present. I feel indebted with the referees for their deep suggestions which have lead to this version of this article.

Notes

Notes

1. It has been shown in 17 that the surface tension must be function of temperature, thus more elaborate computations arise. This is not considered in this article, however we allow for zero surface tension.

2. We recall that the Poynting vector is defined for magnetic and electric fields as . This vector is nonlinear, hence what we are considering here is the pure nonlinear part of , and by the abuse of notations we continue to call it Poynting vector.

3. This means that reversal flows are excluded.

4. It is worth of notice that even though goes to zero as it is not summable.

5. To give an example, think for the vector function defined only for . By we mean the same function extended to the domain .

6. In linear case, for general field Φ these equations are reduced to equations on using the expansion of functions in terms of the normal direction to , cf. [5, Section 5.2],

7. Different definitions of perturbation can be proposed. For example, we may work using the induction magnetic field as unknown field. In this case we find

The definition of perturbation strongly depends on the geometrical properties of the surface.

8. If one uses the perturbation of B, then one can make an analogous comparison.

9. The reader interested to compute the energy equation for classical perturbations may refer to 31.