Some boundary problems of the spin systems and the systems of ferro- magnetic chain. I: Nonlinear boundary problems. (English) Zbl 0644.35060

Some nonlinear boundary problems for the system of ferro-magnetic chain \[ (1)\;z_ t = z\times z_{xx}+f(x,t,z) \] and the corresponding spin system \[ (2)\;z_ t = \epsilon z_{xx}+z\times z_{xx}+f(x,t,z) \] are considered in the rectangular \(Q_ T=\{0\leq x\leq \ell\); \(0\leq t\leq T\}\) with the nonlinear boundary conditions \[ (3)\;z_ x(0,t)=\text{grad}\psi_ 0(t,z(0,t)),\quad -z_ x(\ell,t)=\text{grad}\psi_ 1(t,z(\ell,t)) \] and the initial condition \((4)\quad z(x,0)=\phi(x),\) where \(z=(u,v,w)\), \(f(x,t,z)\) and \(\phi(x)\) are the three-dimensional vector functions; \(\psi_ 0(t,z)\) and \(\psi_ 1(t,z)\) are scalar functions, “\(\times\)” denotes the cross product operator of two three-dimensional vectors and “grad” denotes the gradient operator with respect to the vector variable z. The system (1) contains the so-called Landau-Lifschitz equation of isotropic Heisenberg ferro-magnetic chain as the special case. The three-dimensional generalized global vector solution z(x,t) of the nonlinear boundary problem (3) and (4) for the spin system (2) is established by the method of discretization of space variable \(x\in [0,\ell]\). The generalized solution z(x,t) satisfies the spin system (2) in generalized sense and the nonlinear boundary conditions (3) and the initial condition (4) in classical sense. By the limiting process \(\epsilon\to 0\), the weak global solution \(z(x,t)\in L_{\infty}(0,T;H^ 1(0,\ell))\cap C^{(1/2,1/4)}(Q_ T)\) of the nonlinear boundary problem (3) and (4) is obtained for the strongly degenerate system of ferro-magnetic chain (1).
Reviewer: B.L.Guo


35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35Q99 Partial differential equations of mathematical physics and other areas of application
35L65 Hyperbolic conservation laws