不可压缩Navier--Stokes方程双参数双曲松弛的声学滤波与压力恢复
Linear Corrector Estimates and Corrected Pressure Convergence for a Two-Parameter Hyperbolic Relaxation of the Incompressible Navier--Stokes Equations
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中文总结 AI 辅助
针对不可压缩Navier--Stokes方程的双参数双曲松弛，通过引入声学-应力校正器并利用补偿变量和Ladyzhenskaya不等式，在特定参数条件下证明了滤波压力的强恢复及误差界。
中文摘要 AI 辅助
我们研究二维环面上不可压缩Navier--Stokes方程的一个双参数一阶双曲松弛近似。现有的导数级大扰动估计恢复了速度，但仅在乘以人工可压缩性参数的平方根后才能控制压力。我们通过引入一个非自治的声学-应力校正器来隔离相应的声学振荡。两个补偿变量揭示了物理空间中的抵消现象，从而得到速度校正器的积分梯度估计。二维Ladyzhenskaya不等式随后给出其自相互作用的二次界。在移除校正器后，剩余的非线性误差具有零初始数据，并在下一阶得到控制。若松弛参数$\epsilon$和$\delta$满足$\delta^2\ll\epsilon\leq\mu_*\delta$，我们证明滤波压力在$L^\infty(0,T;L^2(\mathbb T^2))$中的强恢复。定量地，滤波压力和速度误差分别由$C_T\delta/\sqrt{\epsilon}$和$C_T\delta$界定。该论证还阐明了为何对于准备不充分的声学数据，通常不能期望未滤波压力的强收敛。
英文摘要
We study a two-parameter first-order hyperbolic relaxation approximation of the incompressible Navier--Stokes equations on the two-dimensional torus, under the derivative-level large-perturbation assumptions of Huang, Rohde and Zhang. Building on the auxiliary-system approach used in earlier pressure-convergence results, we introduce a non-autonomous linear corrector that carries the complete initial discrepancy, the linearized convection and the smooth consistency forcing. This corrector describes the full linear response; it is not asserted to be a purely acoustic component. An exact compensated energy identity, including a time integration by parts in the linearized convection term, yields an integrated gradient estimate for the velocity corrector. The two-dimensional Ladyzhenskaya inequality then bounds its quadratic self-interaction by $O(δ^2)$ in squared space-time $L^2$ norm. Consequently, the nonlinear remainder has weighted energy $O(δ^2)$, and its pressure and velocity components satisfy the $L^\infty(0,T;L^2)$ bounds $C_Tδ/\sqrtε$ and $C_Tδ$, respectively. In the sufficient parameter window $δ^2\llε\leqμ_*δ$, the pressure after subtraction of the linear corrector converges strongly to the incompressible pressure. A mean-zero Fourier-mode construction, transferred to the nonlinear system by the remainder estimate, shows that the uncorrected pressure need not converge even on time intervals bounded away from zero.
发表机构
- Mathematics and Science College, Shanghai Normal University（上海师范大学数理学院）
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