论文标题
五分之四阶的不良性schrödinger方程
Ill-posedness of quintic fourth order Schrödinger equation
论文作者
论文摘要
我们证明，与Quintic第四阶非线性Schrödinger方程相关的解决方案图在超临界规律性的Sobolev空间中的每个点都表现出规范通胀现象。实际上，我们在负面和正规性的情况下分别证明了这一结果。在负的规律性情况下，我们证明了散热和聚焦方程的结果：在一维情况下，相关的解决方案图显示了超临界规律性的Sobolev空间中的范数（包括关键指数）；在较高维度的情况下，解决方案图在负正常的空间中表现出相同的现象。同时，在正常定期的情况下，我们证明了在尺寸中进行散焦方程的结果$ d = 3,4,5 $。我们的证明分别基于“高低”和“低到高”频率级联。
We prove that the solution map, associated to the quintic fourth order nonlinear Schrödinger equation, exhibits the norm inflation phenomenon at every point in the Sobolev spaces of super-critical regularity. Indeed, we prove this result separately in the cases of negative and of positive regularity. In the negative regularity case, we prove the result for both the defocusing and focusing equations: in the one dimensional case, the associated solution map exhibits norm inflation in Sobolev spaces of super-critical regularity (including the critical index); in the higher dimensional case, the solution map exhibits the same phenomenon in spaces of negative regularity. Meanwhile, in the case of positive regularity, we prove the result for the defocusing equation in dimensions $d=3,4,5$. Our proofs are based on the "high-to-low" and "low-to-high" frequency cascades respectively.
