AI 中文总结
本文通过计算机辅助证明，针对特定参数值，证实了三维三次复Ginzburg-Landau方程在爆破后因不稳定特征值导致解的非唯一性。
AI 中文摘要
我们证明，由光滑初值发展而来的三维三次复Ginzburg-Landau方程的奇点可以在爆破时间之后引发解的非唯一性。我们所研究的解始于向后自相似解，而非唯一性的来源是围绕由奇点生成的前向自相似轮廓线性化时的一个不稳定特征值。我们通过计算机辅助证明，针对方程中参数的特定值，确立了该特征值的存在性，从而确立了非唯一性。这项工作部分受到与Navier-Stokes方程推测行为相似性的启发，复Ginzburg-Landau方程与Navier-Stokes方程共享若干重要性质，如能量不等式和尺度对称性。
英文摘要
We show that singularities of the three-dimensional cubic complex Ginzburg-Landau equation developing from smooth initial data can induce non-uniqueness of solutions after the time of blowup. The solutions we study start as backward self-similar solutions, and the source of the non-uniqueness is an unstable eigenvalue of the linearization around the forward self-similar profile generated by the singularity. We establish the existence of this eigenvalue, and hence the non-uniqueness, by a computer-assisted proof for specific values of the parameters in the equation. The work is in part motivated by similarities to conjectured behavior for the Navier-Stokes equation, with which the complex Ginzburg-Landau equation shares several important properties, such as an energy inequality and the scaling symmetry.
Comments61 pages, 4 figures
