发表机构
School of Mathematics, Nanjing University; The Institute of Mathematical Sciences, The Chinese University of Hong Kong; School of Mathematical Sciences, Ocean University of China（南京大学数学系; 香港中文大学数学科学研究所; 中国海洋大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对带小定常外力的三维可压缩Navier--Stokes--Korteweg方程，构造一致小定常解并结合多种估计建立非定常整体强解存在性，得到低马赫数极限的定量收敛率。
AI 中文摘要
本文研究全空间中三维可压缩Navier--Stokes--Korteweg方程在小定常外力作用下的低马赫数极限。我们首先构造了一族关于马赫数ε一致的小定常解，并证明定常密度波动和定常速度的可压缩分量均为ε²阶。针对这些定常解周围准备不足的非定常扰动，我们结合一致高阶能量估计、低频Besov估计和Kawashima型补偿泛函，建立了唯一整体强解的存在性。主要困难在于Korteweg项不仅改变了定常问题的椭圆结构，还修正了声学模态的色散机制。在Korteweg对称变量下，相关谱投影是一致有界的零阶傅里叶乘子，而声学-毛细相位在低频呈类波行为，在高频呈类薛定谔行为。由于定常系数产生的源项通常在时间上不可积，我们根据Duhamel源的时间可积性和频率行为对其进行分解。二进色散估计、高频阻尼估计以及热方程的最大正则性，给出了混合Besov范数L^r(0,∞;Ḃ^s_{p,1})中的整体时间收敛率ε^{min{1/r,1/2−1/p}}。由此，Besov嵌入也给出了混合Lebesgue范数L^r(0,∞;L^p)中的定量收敛结果。
英文摘要
In this paper, we investigate the low Mach number limit for the three-dimensional compressible Navier--Stokes--Korteweg equations in the whole space under a small stationary external force. We first construct a family of small stationary solutions uniformly with respect to the Mach number $ε$ and prove that both the stationary density fluctuation and the compressible component of the stationary velocity are of order $ε^2$. For ill-prepared non-stationary perturbations around these stationary solutions, we establish the existence and uniqueness of global strong solution by combining uniform high-order energy estimates with a low-frequency Besov estimate and a Kawashima-type compensating functional. The main difficulty is that Korteweg tensor not only changes the elliptic structure of the stationary problem, but also modifies the dispersive mechanism of the acoustic modes. In Korteweg-symmetric variables, the associated spectral projections are uniformly bounded zero-order Fourier multipliers, while the acoustic-capillary phase is wave-like at low frequencies and Schrödinger-like at high frequencies. Since the source terms generated by the stationary coefficients are generally not integrable in time, we decompose the Duhamel source according to its time-integrability and frequency behavior. Dyadic dispersive estimates, high-frequency damping estimates, and maximal regularity for the heat equation yield the global-in-time convergence rate $ε^{\min\{1/r,\,1/2-1/p\}}$ in the mixed Besov norms $L^r(0,\infty;\dot B^s_{p,1})$. As a consequence, Besov embeddings also yield quantitative convergence in the mixed Lebesgue norms $L^r(0,\infty;L^p)$.
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