Similarity solutions for high frequency excitation of liquid metal in an antisymmetric magnetic field.

*(English)*Zbl 1120.34009Biler, Piotr (ed.) et al., Self-similar solutions of nonlinear PDE. Selected papers of the conference, Bȩdlewo, Poland, September 5–9, 2005. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Center Publications 74, 41-57 (2006).

The authors consider the third order nonlinear autonomous differential equation

$${f}^{\text{'}\text{'}\text{'}}+\frac{m+1}{2}f{f}^{\text{'}\text{'}}-m{{f}^{\text{'}}}^{2}=0$$

with the boundary conditions

$$f\left(0\right)=a,\phantom{\rule{1.em}{0ex}}{f}^{\text{'}}\left(0\right)=-1,\phantom{\rule{1.em}{0ex}}{f}^{\text{'}}\left(\infty \right)=0,$$

where $m\in \mathbb{R}$ and ${f}^{\text{'}}\left(\infty \right)={lim}_{t\to \infty}{f}^{\text{'}}\left(t\right)$. Several existence results are established in both cases $m<0$ and $m>0$. The solutions of the above problem are similarity solutions of problems related to the phenomenon of high frequency excitation of liquid metal system in an antisymmetric magnetic field, within the framework of boundary layer approximation.

Reviewer: Zaihong Wang (Beijing)

##### MSC:

34B15 | Nonlinear boundary value problems for ODE |

34C11 | Qualitative theory of solutions of ODE: growth, boundedness |

76D10 | Boundary-layer theory, separation and reattachment, etc. (incompressible viscous fluids) |

34C60 | Qualitative investigation and simulation of models (ODE) |