关于$p$ -Laplacian型演化系统及其在第二类超导理论Bean模型中的应用
On a $p$-Laplacian type of evolution system and applications to the Bean model in the type-II superconductivity theory
摘要：
研究一类$p$ -Laplacian型发展系统${\mathbf H}_{t}+\g [ | \g {\mathbf H}|^{p-2} \g {\mathbf H}|]={\mathbf F}$的Cauchy问题.该系统控制磁场${\bf H}$的演化，其中电流位移被忽略，电阻率被假设为电流密度的某个幂。建立了系统解的存在性、唯一性和正则性。此外，还证明了极限解作为幂$p\rightarrow \infty$解决了Bean模型在第二类超导理论中的问题。这一结果为超导材料在外力作用下如何成为正常导体和负导体提供了信息。同时也为Bean模型的数值求解提供了一种有效的方法。
英文摘要：
We study the Cauchy problem for an $p$-Laplacian type of evolution system ${\mathbf H}_{t}+\g [ | \g {\mathbf H}|^{p-2} \g {\mathbf H}|]={\mathbf F}$. This system governs the evolution of a magnetic field ${\bf H}$, where the current displacement is neglected and the electrical resistivity is assumed to be some power of the current density. The existence, uniqueness and regularity of solutions to the system are established. Furthermore, it is shown that the limit solution as the power $p\rightarrow \infty$ solves the problem of Bean's model in the type-II superconductivity theory. The result provides us information about how the superconductor material under the external force to become the normal conductor and vice visa. It also provides an effective method to find numerical solutions to Bean's model.
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