具有结构化Lipschitz系数的方程的Strichartz估计
Strichartz estimates for equations with structured Lipschitz coefficients
AI总结：
本文针对具有结构化Lipschitz系数的Schrödinger和波动方程证明了尖锐Strichartz估计，利用Phillips泛函演算替代Fourier逆变换，并推广至有界变差系数，应用于波动方程和求解非线性Schrödinger方程。
摘要：
本文证明了满足附加结构假设的Lipschitz系数的Schrödinger方程和波动方程的尖锐Strichartz估计。我们使用Phillips泛函演算作为Fourier逆变换的替代，这表明色散性质是如何从常系数情形继承而来的。如果系数的导数可积，则全局Strichartz估计成立。这些估计可推广到有界变差的结构化系数。作为应用，我们推导了具有Hölder连续系数的波动方程的带有额外导数损失的Strichartz估计，并求解了非线性Schrödinger方程。最后，我们记录了谱乘子估计，这些估计由Strichartz估计通过众所周知的方法得出。
英文摘要：
Sharp Strichartz estimates are proved for Schrödinger and wave equations with Lipschitz coefficients satisfying additional structural assumptions. We use Phillips functional calculus as a substitute for Fourier inversion, which shows how dispersive properties are inherited from the constant coefficient case. Global Strichartz estimates follow provided that the derivatives of the coefficients are integrable. The estimates extend to structured coefficients of bounded variations. As applications we derive Strichartz estimates with additional derivative loss for wave equations with Hölder-continuous coefficients and solve nonlinear Schrödinger equations. Finally, we record spectral multiplier estimates, which follow from the Strichartz estimates by well-known means.
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