欧氏空间上非线性Schrödinger方程Sobolev范数的增长

Growth of Sobolev Norms for Nonlinear Schrödinger Equation in Euclidean Space

摘要

通过构造修正能量泛函，研究二维欧氏空间中非线性 Schrödinger 方程（NLS）高阶 Sobolev 范数的时间增长性。基于三次非线性项结果，建立了任意高阶非线性项的多项式界 $\left(\sup _{t \in(0, T)}\|\boldsymbol{u}\|_{H^{m}} \leqslant C \max \{1, T\}^{m-1+t}\right) $ ，所得结果完善了 NLS 高阶正则性演化理论。

Abstract

By constructing a modified energy functional，we investigated the temporal growth of higher-order Sobolev norms for the nonlinear Schrödinger equation（NLS）in two-dimensional Euclidean spaces．Based on results for cubic nonlinearities，we established a polynomial bound applicable to arbitrary higher-order nonlinearities $\left(\sup _{t \in(0, T)}\|\boldsymbol{u}\|_{H^{m}} \leqslant C \max \{1, T\}^{m-1+t}\right) $ ．The obtained results improved the theory of higher-order regularity evolution for NLS．

关键词

非线性 Schrödinger 方程 / Sobolev 范数 / 修正能量 / Strichartz 估计

Key words

nonlinear Schrödinger equation / Sobolev norm / modified energy / Strichartz estimate

参考文献

[1] BOURGAIN J. Global Solutions of Nonlinear Schrödinger Equations[M]. Providence, RI: American Mathematical Society, 1999: 28-33.

[2] BOURGAIN J. On Growth of Sobolev Norms in Linear Schrödinger Equations with Smooth Time Dependent Potential[J]. Journal D'Analyse Mathématique, 1999, 77(1): 315-348.

[3] CHEN Q L, DENG M M. On the Growth of High Sobolev Norms of the Fourth-Order Schrödinger Equation[J]. Differential and Integral Equations, 2023, 36(7/8): 661-678.

[4] BOURGAIN J. Fourier Transform Restriction Phenomena for Certain Lattice Subsets and Applications to Nonlinear Evolution Equations. II. The KdV-Equation[J]. Geometric and Functional Analysis, 1993, 3(3): 209-262.

[5] DUCA A, NERSESYAN V. Bilinear Control and Growth of Sobolev Norms for the Nonlinear Schrödinger Equation[J]. Journal of the European Mathematical Society, 2025, 27(6): 2603-2622.

[6] CHENG X, GUO C Y, ZHENG Y R. The Limit Theory of Energy-Critical Complex Ginzburg-Landau Equation[J]. Acta Mathematica Sinica (English Series), 2025, 41(12): 3003-3019.

[7] GUARDIA M, HAUS E, PROCESI M. Growth of Sobolev Norms for the Analytic NLS on T { }^{2}[J]. Advances in Mathematics, 2016, 301: 615-692.

[8] LIU Y Y, ZHAO L G. Normalized Solutions for Schrödinger Equations with General Nonlinearities on Bounded Domains[J]. Journal of Geometric Analysis, 2025, 35(2) : 54-1-54-32.

[9] BURQ N, GÉRARD P, TZVETKOV N. Strichartz Inequalities and the Nonlinear Schrödinger Equation on Compact Manifolds[J]. American Journal of Mathematics, 2004, 126(3): 569-605.

[10] ZHONG S. The Growth in Time of Higher Sobolev Norms of Solutions to Schrödinger Equations on Compact Riemannian Manifolds[J]. Journal of Differential Equations, 2008, 245(2): 359-376.

[11] KOCH H, TATARU D. Conserved Energies for the Cubic Nonlinear Schrödinger Equation in One Dimension[J]. Duke Mathematical Journal, 2018, 167(17): 3207-3313.

基金资助

国家自然科学基金(U2340221)

江苏省自然科学基金(BK20230026)

江苏省自然科学基金(BK20221497)