发表机构
School of Mathematics, Jilin University; Department of Mathematical Sciences, Seoul National University（吉林大学数学学院; 首尔大学数学科学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文证明三维相对论量子流体动力学系统在小光滑局部扰动下全局经典解存在，通过结构抵消和正规形变换克服二次共振，获得能量界与衰减，并证明散射到线性化系统。
AI 中文摘要
我们建立了三维空间中相对论量子流体动力学系统在常数非真空平衡态附近足够小、光滑且局部化的扰动的全局存在性、衰减和散射。在对数振幅和相位变量中，方程组构成一个耦合波动方程的半线性系统。时间导数之间的反对称耦合在能量恒等式中相互抵消，从而产生自然的导数能量。线性化系统有两个色散分支。在低频下，慢分支表现出薛定谔型色散，而快分支具有谱隙；在高频下，两个分支都呈波动状。主要的非线性困难源于具有非平凡时间和时空共振的二次相互作用。我们证明了每个活跃二次相互作用的符号都包含相应的相互作用相位作为精确因子。这种结构抵消消除了共振分母，并允许我们通过非奇异正规形变换消除二次项。将该变换与色散和能量估计相结合，我们获得了导数能量的一致时间界以及一阶导数在$L^\infty$中的$\langle t\rangle^{-3/2}$衰减。我们进一步证明了非线性解在高阶导数能量范数下散射到线性化系统的解。
英文摘要
We establish global existence, decay, and scattering for sufficiently small, smooth, and localized perturbations of a constant non-vacuum equilibrium of a relativistic quantum hydrodynamic system in three space dimensions. In logarithmic-amplitude and phase variables, the equations form a semilinear system of coupled wave equations. The skew-symmetric coupling between the time derivatives cancels in the energy identity, yielding a natural derivative energy. The linearized system has two dispersion branches. At low frequencies, the slow branch exhibits Schrödinger-type dispersion, while the fast branch has a spectral gap; at high frequencies, both branches are wave-like. The main nonlinear difficulty arises from quadratic interactions with nontrivial time and space-time resonances. We show that the symbol of every active quadratic interaction contains the corresponding interaction phase as an exact factor. This structural cancellation removes the resonant denominator and allows us to eliminate the quadratic terms by a nonsingular normal-form transformation. Combining this transformation with dispersive and energy estimates, we obtain uniform-in-time bounds for the derivative energy and $\langle t\rangle^{-3/2}$ decay of the first derivatives in $L^\infty$. We further prove that the nonlinear solution scatters to a solution of the linearized system in the high-order derivative energy norm.
