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**Stability for static walls in ferromagnetic nanowires.**
*(English)*
Zbl 1220.82163

This paper analyzes the time asymptotic stability of 1-D Bloch walls in ferromagnetic nanowires. First, it is considered the Landau-Lifshitz equation (LLE), describing variations of the magnetic moment \(u = (u_1, u_2, u_3)\). The LLE is simplified, by considering the 1-D model and using the presentation of the demagnetizing field in the canonical basis \((e_1, e_2, e_3)\) of \(\mathbb R^3\), when the diameter of the wire tends to zero.

The aim of the paper is to study the stability of a static wall profile \(M_0\), which separates the domain in which \(u = -e_1\) (in the neighborhood of \(-\infty\)) and the domain in which \(u = +e_1\) (in the neighborhood of \(+\infty\)). The reduced LLE is invariant by translation in the variable \(x\) and by rotation around \(e_1\). Then, the perturbations of \(M_0\) are described in the mobile frame, on the base of equivalent formulation of reduced LLE with unknown \(r\) in the form \(\frac{\partial r}{\partial t}= Lr+F(x,r,\frac{\partial r}{\partial x},\frac{\partial^2r}{\partial x^2})\), where \(Lr\) denotes the linear part. The stability of \(M_0\) for the reduced LLE is equivalent to the stability of the zero solution for the pointed equation.

The two-parameter family of static solutions for the reduced LLE induces in the new coordinates a two-parameter family of static solutions for the above equation for \(r\). Further, the solution \(r\) is decomposed by a rather classical decomposition to study the static solution stability for semilinear parabolic equations. The main difficulty is that the above equation for \(r\) is quasilinear, i.e. the nonlinear term \(F\) also depends on \(\frac{\partial^2r}{\partial x^2}\). Then, variational estimates for the nonlinear part combined with more classical linear estimates on the operator \(L\), are used. This leads to the proof of the main theorem on the stability of the static Bloch wall profile.

The aim of the paper is to study the stability of a static wall profile \(M_0\), which separates the domain in which \(u = -e_1\) (in the neighborhood of \(-\infty\)) and the domain in which \(u = +e_1\) (in the neighborhood of \(+\infty\)). The reduced LLE is invariant by translation in the variable \(x\) and by rotation around \(e_1\). Then, the perturbations of \(M_0\) are described in the mobile frame, on the base of equivalent formulation of reduced LLE with unknown \(r\) in the form \(\frac{\partial r}{\partial t}= Lr+F(x,r,\frac{\partial r}{\partial x},\frac{\partial^2r}{\partial x^2})\), where \(Lr\) denotes the linear part. The stability of \(M_0\) for the reduced LLE is equivalent to the stability of the zero solution for the pointed equation.

The two-parameter family of static solutions for the reduced LLE induces in the new coordinates a two-parameter family of static solutions for the above equation for \(r\). Further, the solution \(r\) is decomposed by a rather classical decomposition to study the static solution stability for semilinear parabolic equations. The main difficulty is that the above equation for \(r\) is quasilinear, i.e. the nonlinear term \(F\) also depends on \(\frac{\partial^2r}{\partial x^2}\). Then, variational estimates for the nonlinear part combined with more classical linear estimates on the operator \(L\), are used. This leads to the proof of the main theorem on the stability of the static Bloch wall profile.

Reviewer: I. A. Parinov (Rostov-na-Donu)

### MSC:

82D40 | Statistical mechanics of magnetic materials |

82D80 | Statistical mechanics of nanostructures and nanoparticles |

35Q82 | PDEs in connection with statistical mechanics |

34D10 | Perturbations of ordinary differential equations |

34D05 | Asymptotic properties of solutions to ordinary differential equations |