Zihao Song

We present a new derivation for the asymptotic behaviors for solutions of the compressible Korteweg system with derivatives of arbitrary high order in general L r ( R d ) ( d ⩾ 2 , r ∈ [ 1 , ∞ ] ). It would be shown that solutions decay at the rates of heat kernels for 2 ⩽ r ⩽ ∞ while decay slower, asymptotically behaving as so called ‘diffusion waves’, for 1 ⩽ r < 2 . By introducing an initial space with negative regularity B ˙ 1 , ∞ − σ , we obtain the uniform decay for solutions within L 1 framework, compared to the possible growth in classical results for compressible models. The approach is so-called Gevrey method, based on the parabolic mechanics of the linearized system, which enables us to transform high order derivatives’ estimates into decay rates and low-order analytic estimates. In order to control the growth of the radius of analyticity under the particular regularity, especially in case 1 ⩽ r < 2 , we shall treat the momentum form of the perturbed system, establish necessary linear analysis based on the semi-group and Gevrey product estimates in sub-analytic families.

We investigate the global well-posedness of the compressible Euler system with damping in R^d (d ≥ 1) and its relaxation limit toward the porous medium equation. In [12], the first author and Danchin studied these two problems in hybrid Besov spaces, where the high-frequency components of the solution are bounded in L2 -based norms, while the lowfrequency components are controlled in L p -based norms with p ∈ [2, max{4, 2d/(d -2)}]. Motivated by the observation that the limit system is well-posed in Lp -based spaces for p ∈ [2, ∞), we extend the low-frequency analysis to this full range, thereby providing a more unified framework for studying such relaxation limits.The core of our proof consists in establishing refined product and commutator estimates describing sharply the interactions between the high, medium, and low-frequency regimes. A key observation underlying our analysis is that the product of two functions localized at low frequencies generates only interactions between low and medium frequencies, never purely high-frequency ones. Consequently, for a suitable choice of frequency threshold, the high-frequency projection of the product of two functions localized low frequencies vanishes.

We investigate the global well-posedness of the compressible Euler system with damping in Rd (d\geq1) and its relaxation limit toward the porous medium equation. In [12], the first author and Danchin studied these two problems in hybrid Besov spaces, where the high-frequency components of the solution are bounded in L2-based norms, while the low-frequency components are controlled in Lp-based norms with p\in[2,\max{4,\frac{2d}{d-2}}]. Motivated by the observation that the limit system is well-posed in Lp-based spaces for p\in[2, \infty), we extend the low-frequency analysis to this full range, thereby providing a more unified framework for studying such relaxation limits. The core of our proof consists in establishing refined product and commutator estimates describing sharply the interactions between the high, medium, and low-frequency regimes. A key observation underlying our analysis is that the product of two functions localized at low frequencies generates only interactions between low and medium frequencies, never purely high-frequency ones. Consequently, for a suitable choice of frequency threshold, the high-frequency projection of the product of two functions localized low frequencies vanishes.