February 26, 2026
In this paper, we study the Cauchy’s problem of the compressible Euler system with damping and establish the global-in-time well-posedness in \(L^p\)-type critical Besov spaces for \(1\leq p<2\). To achieve it, a new product estimate is established in \(L^2\)-\(L^p\) hybrid Besov spaces.
This work studies the compressible Euler equations with linear damping, a model describing gas flow through a porous medium where the solid matrix exerts a frictional force proportional to and opposite to the fluid momentum. The governing system is given by \[\begin{align} \label{1461} \left\{\aligned &\partial_t \rho+\rm div\;\!(\rho {\boldsymbol{u} })=0, \\ &\partial_t (\rho {\boldsymbol{u} })+\nabla \cdot (\rho {\boldsymbol{u} }\otimes {\boldsymbol{u} })+\nabla P(\rho)+\alpha\rho {\boldsymbol{u} }=0 \endaligned\right. \end{align}\tag{1}\] for \(t\geq0,\;\boldsymbol{x}\in \mathbb{R}^d(d\geq 1)\) and a damping coefficient \(\alpha>0\) . Here \(\rho\in\mathbb{R}^{+}\), \({\boldsymbol{u} }=(u^1, u^2,\cdots, u^d)^\top\) (\(\top\) represents transpose) denote the density and velocity of fluid flow, respectively. The pressure function \(P=P(\rho)\) is assumed to be smooth around the constant reference density \(\bar{\rho}>0\).
There are lots of mathematical results about the existence and asymptotic behavior of system 1 in sobolev space (see [1]–[6] ). In critical Besov space, Fang and Xu [7] (with improvements in [8]) studied the existence and asymptotic behavior of classical solutions. Later, Xu and Wang [9] justified the relaxation convergence from 1 to the porous medium equation. And also, there are some results on hyperbolic system for balance laws including 1 (see [10], [11]). Recently, Crin-Barat and Danchin [12]–[14] improved those results [9]–[11] in the \(L^2\)-\(L^p\) hybrid Besov spaces with \(2\leq p\leq\max\{4,\frac{2d}{d+2}\}\), where the low frequencies are bounded in \(L^p\)-type spaces and the high frequencies in \(L^2\)-type spaces with a specific linearity assumption. And then, the first author and Xu [15] remove the assumption and to show that the results of [12], [13] hold true for the general pressure function. Very recently, Crin-Barat and Song [16] extend the work of [12], [13], [15] to the case \(p\in[2,\infty)\). And a natural question is how about \(p<2\)?
Introducing a new unknown called “enthalpy" \(n(\rho)\triangleq \int_1^{\rho} \frac{P'(s)}{s} ds.\) System 1 can be rewritten as \[\begin{align} \label{1462} \left\{\aligned &\partial_t n + {\boldsymbol{v} }\cdot \nabla n + P'(\bar{\rho})\rm div\;\!{\boldsymbol{v} }+ G(n)\rm div\;\!{\boldsymbol{v} }= 0 \\ &\partial_t {\boldsymbol{v} }+ {\boldsymbol{v} }\cdot \nabla {\boldsymbol{v} }+\nabla n= -\frac{1}{\varepsilon} {\boldsymbol{v} } \endaligned\right. \end{align}\tag{2}\] with the initial data \[\begin{align} \label{1462initial} (n,{\boldsymbol{v} })|_{t=0}=(n_0, {\boldsymbol{v} }_0). \end{align}\tag{3}\] Here \(P'(\bar{\rho})>0\) for \(\bar{\rho}>0\), \(\bar{\rho}=P'(\bar{\rho})=1\) and the composite function \(G(n)\) is smooth.
Define \(J_\varepsilon=-[\log_2\varepsilon]+k\) being the threshold between high and low frequencies with a suitable integer \(k\) is to be determined later, and the space \(E_T^{J_\varepsilon}\) as follows \[\begin{align} &E_T^{J_\varepsilon}=\bigg\{(n,{\boldsymbol{v} })|(n^{\ell,J_\varepsilon},{\boldsymbol{v} }^{\ell,J_\varepsilon})\in {\mathcal{C}}_b([0,T];{\dot{B}}_{p,1}^{\frac{d}{p}}),\; \varepsilon n^{\ell,J_\varepsilon}\in \widetilde{L}_T^1({\dot{B}}_{p,1}^{\frac{d}{p}+2}),\; {\boldsymbol{v} }^{\ell,J_\varepsilon}\in \widetilde{L}_T^1({\dot{B}}_{p,1}^{\frac{d}{p}+1}), \varepsilon^{-\frac{1}{2}} {\boldsymbol{v} }\in \widetilde{L}_T^2({\dot{B}}_{p,1}^{\frac{d}{p}}),\\ &\qquad\qquad (\varepsilon n^{h,J_\varepsilon}, \varepsilon{\boldsymbol{v} }^{h,J_\varepsilon})\in {\mathcal{C}}_b([0,T];{\dot{B}}_{2,1}^{\frac{d}{2}+1}),\; (n^{h,J_\varepsilon},{\boldsymbol{v} }^{h,J_\varepsilon}) \in \widetilde{L}_T^1({\dot{B}}_{2,1}^{\frac{d}{2}+1}), \varepsilon^{-1}{\boldsymbol{v} }+\nabla n\in \widetilde{L}_T^1({\dot{B}}_{p,1}^{\frac{d}{p}}) \bigg\}. \end{align}\] When \(T=\infty\) we also use \(E_\infty^{J_\varepsilon}\) for convenience. For the reader’s convenience, We introduce the notations \(\|\cdot\|_{{\dot{B}}_{q_1,1}^s}^{h,J_{\varepsilon}}\) and \(\|\cdot\|_{{\dot{B}}_{q_2,1}^s}^{\ell,J_{\varepsilon}}\) to denote Besov semi-norms with respect to the threshold \(J_{\varepsilon}\), that is, \[\label{DefHLB} \|f\|_{{\dot{B}}_{q_1,1}^{s_1}}^{h,J_{\varepsilon}}\triangleq \sum_{j\geq J_{\varepsilon}} 2^{s_1j}\|\dot{\Delta}_jf\|_{q_1}\quad and\quad \|f\|_{{\dot{B}}_{q_2,1}^{s_2}}^{\ell,J_{\varepsilon}}\triangleq \sum_{j\leq J_{\varepsilon}-1} 2^{s_2j}\|\dot{\Delta}_jf\|_{q_2}.\tag{4}\] It is not difficult to deduce that for all \(\sigma_0>0\), \[\label{HLEst} \|f\|_{{\dot{B}}_{q_1,1}^{s_1}}^{h,J_{\varepsilon}} \leq 2^{-\sigma_0J_{\varepsilon}}\|f\|_{{\dot{B}}_{q_1,1}^{s_1+\sigma_0}}^{h,J_{\varepsilon}}\quad and\quad \|f\|_{{\dot{B}}_{q_1,1}^{s_1}}^{\ell,J_{\varepsilon}} \leq 2^{\sigma_0J_{\varepsilon}}\|f\|_{{\dot{B}}_{q_1,1}^{s_1-\sigma_0}}^{\ell,J_{\varepsilon}}.\tag{5}\]
The main goal of this paper is to broaden the assumption on \(p\) and to show that the results of [12], [13], [15] hold true for the case \(p<2\). Our main result is stated as follows.
Theorem 1. Assume \(1\leq p <2\), \(d\geq 1\) and the pressure \(P\) satisfies \(P'(\bar{\rho})>0\) for \(\bar{\rho}>0\). There is a small constant \(\delta_1>0\) such that if \(\|(n_0,{\boldsymbol{v} }_0)\|_{{\dot{B}}_{p,1}^{\frac{d}{p}}}^{\ell,J_\varepsilon} +\varepsilon\|(n_0,{\boldsymbol{v} }_0)\|_{{\dot{B}}_{2,1}^{\frac{d}{2}+1}}^{h,J_\varepsilon} \leq \delta_1,\) then the Cauchy problem 2 3 admits a uniform global unique solution in \(E_\infty^{J_\varepsilon}\) for all \(\varepsilon>0\).
Remark 2. Note that \(\dot{B}_{p,1}^{\frac{d}{p}}\hookrightarrow \dot{B}_{2,1}^{\frac{d}{2}}\) when \(p<2\), that means the existence and uniqueness of solutions hold in a smaller space rather than escaping to the complement \(\dot{B}_{2,1}^{\frac{d}{2}}\setminus \dot{B}_{p,1}^{\frac{d}{p}}\). On the other hand, 4 implies that a smaller space for \(p\) corresponds to more singular and concentrated data, which naturally arises in many physical problems (such as point sources and vortex filaments).
Remark 3. Based on Theorem 1, the relaxation limit for the case \(p<2\) can be performed by the same procedure as [13].
The major difficulty of proof of Theorem 1 lies in dealing with the nonlinear terms. To achieve it, a new product estimation will be developed, see Lemma 1 below. As a matter of fact, the new tool could be applied to investigate other systems, such as the hyperbolic-parabolic chemotaxis system (see [17]) in \(L^2-L^p\) framework and Navier-Stokes equations (see [18] ) in \(L^p-L^2\) framework for \(p<2\).
In this section, we only give the key a priori estimate for the case \(p<2\), which lead to the global existence and uniqueness of solutions in \(E_\infty^{J_\varepsilon}\). See [19] for more details. For convenience, we use \(f\lesssim g\) to denote that there exists a generic constant \(C>0\) independent on \(\varepsilon\) such that \(f\leq C g\) in this section.
First of all, we establish the following product estimation in \(L^p-L^2\) hybrid Besov spaces.
Lemma 1. Let \(0<s_1<\frac{d}{2}\), \(1\leq p<2\). Then, we have the following inequality \[\begin{align} \label{newproduct} \|ab\|_{{\dot{B}}_{p,1}^{s_1}}^{\ell,J_\varepsilon} \leq C \left(\|b\|_{{\dot{B}}_{p,1}^{s_1}}^{\ell,J_\varepsilon}+ 2^{(s_1-\frac{d}{p})J_\varepsilon}\|b\|_{{\dot{B}}_{2,1}^{\frac{d}{2}}}^{h,J_\varepsilon} \right)\|a\|_{{\dot{B}}_{2,1}^{\frac{d}{2}}} \end{align}\tag{6}\] with \(C\) independent on \(\varepsilon\).
Proof. Using Bony’s paraproduct decomposition, we have \[\nonumber \begin{align} ab=T_{a}b+R[a,b]+T_{b}a\quad with\quad T_{a}b\triangleq\sum_{j'\in \mathbb{Z}} \dot{S}_{j'-1}a \dot{\Delta}_{j'}b\quadand\quad R[a,b]\triangleq\sum_{|j'-j''|\leq 1} \dot{\Delta}_{j''} a \dot{\Delta}_{j'}b. \end{align}\]
First, we bound \(T_{a}b\). It is clear that \[\nonumber \begin{align} \|T_{a} b\|_{{\dot{B}}_{p,1}^{s_1}}^{\ell,J_\varepsilon} \leq \sum_{j\leq J-2\atop |j-j'|\leq 1}2^{s_1 j} \|\dot{S}_{j'-1} a \dot{\Delta}_j\dot{\Delta}_{j'}b\|_{L^p} +\sum_{j\leq J-2\atop |j-j'|\leq 4} 2^{s_1 j}\|[\dot{\Delta}_j, \dot{S}_{j'-1} a] \dot{\Delta}_{j'}b\|_{L^p}. \end{align}\] The embedding \(\dot{B}_{p,1}^{\frac{d}{p}}\hookrightarrow L^{\infty}\) leads to \[\nonumber \begin{align} \sum_{j\leq J-2\atop |j-j'|\leq 1} 2^{s_1j}\|\dot{S}_{j'-1} a \dot{\Delta}_j\dot{\Delta}_{j'}b\|_{L^p} \lesssim \|\dot{S}_{j'-1} a\|_{L^\infty}\sum_{j\leq J-2} 2^{s_1j} \|\dot{\Delta}_jb\|_{L^p} \lesssim \|a\|_{{\dot{B}}_{2,1}^{\frac{d}{2}}}\|b\|_{{\dot{B}}_{p,1}^{s_1}}^{\ell,J_\varepsilon}. \end{align}\] Note that \[\nonumber \begin{align} \sum_{j\leq J-2\atop |j-j'|\leq 4} 2^{s_1 j}\|[\dot{\Delta}_j, \dot{S}_{j'-1} a] \dot{\Delta}_{j'}b\|_{L^p} \lesssim\Big(\sum\limits_{j'\leq J-2}2^{s_1j'} + \sum\limits_{J -2 \leq j'\leq J+2} \Big)2^{s_1j'} \|[\dot{\Delta}_j, S_{j'-1} a] \Delta_{j'} b\|_{L^p}. \end{align}\] The Young-like inequality ensures that the commutator estimate in [20] also holds for \([\dot{\Delta}_j, S_{j'-1} a] \Delta_{j'} b\) and therefore we have \[\nonumber \begin{align} \sum\limits_{j'\leq J-2}2^{s_1j'} \|[\dot{\Delta}_j, S_{j'-1} a] \Delta_{j'} b\|_{L^p} \lesssim \|\nabla a\|_{{\dot{B}}_{\infty,1}^{-1}} \|b\|_{{\dot{B}}_{p,1}^{s_1}}^{\ell,J_\varepsilon} \lesssim \|a\|_{{\dot{B}}_{2,1}^{\frac{d}{2}}} \|b\|_{{\dot{B}}_{p,1}^{s_1}}^{\ell,J_\varepsilon}. \end{align}\] Similarly, as \(\frac{2p}{2-p}\leq 2\) when \(1\leq p\leq 2\), one has \[\nonumber \begin{align} &\sum\limits_{J -2 \leq j'\leq J+2}2^{s_1j'} \|[\dot{\Delta}_j, S_{j'-1} a] \Delta_{j'} b\|_{L^p} \lesssim 2^{(s_1-\frac{d}{p})J_\varepsilon} \|b\|_{{\dot{B}}_{2,1}^{\frac{d}{2}}}^{h,J_\varepsilon} \|a\|_{{\dot{B}}_{\frac{2p}{2-p},1}^{\frac{d}{p}-\frac{d}{2}}}^{h,J_\varepsilon}\lesssim 2^{(s_1-\frac{d}{p})J_\varepsilon} \|b\|_{{\dot{B}}_{2,1}^{\frac{d}{2}}}^{h,J_\varepsilon} \|a\|_{{\dot{B}}_{2,1}^{\frac{d}{2}}}^{h,J_\varepsilon}. \end{align}\] Hence, it follows that \[\nonumber \begin{align} \|T_{a} b\|_{{\dot{B}}_{p,1}^{s_1}}^{\ell,J_\varepsilon} \lesssim \|a\|_{{\dot{B}}_{2,1}^{\frac{d}{2}}}\|b\|_{{\dot{B}}_{p,1}^{s_1}}^{\ell,J_\varepsilon} +2^{(s_1-\frac{d}{p})J_\varepsilon}\|b\|_{{\dot{B}}_{2,1}^{\frac{d}{2}}}^{h,J_\varepsilon} \|a\|_{{\dot{B}}_{2,1}^{\frac{d}{2}}}^{h,J_\varepsilon} \lesssim \left(\|b\|_{{\dot{B}}_{p,1}^{s_1}}^{\ell,J_\varepsilon}+ 2^{(s_1-\frac{d}{p})J_\varepsilon}\|b\|_{{\dot{B}}_{2,1}^{\frac{d}{2}}}^{h,J_\varepsilon} \right)\|a\|_{{\dot{B}}_{2,1}^{\frac{d}{2}}}. \end{align}\]
For \(T_{b} a\), because \(s_1-\frac{d}{2}<0\), it follows the classical paraproduct estimate (see [18]) that \[\nonumber \begin{align} \|T_{b} a\|_{{\dot{B}}_{p,1}^{s_1}}^{\ell,J_\varepsilon} \lesssim \|b\|_{{\dot{B}}_{\frac{2p}{2-p},1}^{s_1-\frac{d}{2}}}\|a\|_{{\dot{B}}_{2,1}^{\frac{d}{2}}} \lesssim \|b\|_{{\dot{B}}_{2,1}^{s_1+\frac{d}{2}-\frac{d}{p}}}\|a\|_{{\dot{B}}_{2,1}^{\frac{d}{2}}} \lesssim \left(\|b\|_{{\dot{B}}_{p,1}^{s_1}}^{\ell,J_\varepsilon}+ 2^{(s_1-\frac{d}{p})J_\varepsilon}\|b\|_{{\dot{B}}_{2,1}^{\frac{d}{2}}}^{h,J_\varepsilon} \right)\|a\|_{{\dot{B}}_{2,1}^{\frac{d}{2}}}. \end{align}\]
Finally, by classical remainder estimates (see [20]) we can directly obtain \[\nonumber \begin{align} \|R[a,b]\|_{{\dot{B}}_{p,1}^{s_1}}^{\ell,J_\varepsilon} \lesssim \|b\|_{{\dot{B}}_{\frac{2p}{2-p},1}^{s_1-\frac{d}{2}}}\|a\|_{{\dot{B}}_{2,1}^{\frac{d}{2}}} \lesssim \left(\|b\|_{{\dot{B}}_{p,1}^{s_1}}^{\ell,J_\varepsilon}+ 2^{(s_1-\frac{d}{p})J_\varepsilon}\|b\|_{{\dot{B}}_{2,1}^{\frac{d}{2}}}^{h,J_\varepsilon} \right)\|a\|_{{\dot{B}}_{2,1}^{\frac{d}{2}}}. \end{align}\] Adding above three inequality together, we can finally obtain 6 . ◻
For simplicity, we define \({\mathcal{X}}(T):=\|(n,{\boldsymbol{v} })\|_{E_T^{J\varepsilon}}\). The proof of Theorem 1 reduces to establishing a global-in-time a priori estimate. Specifically, we claim that if \[\begin{align} \|n\|_{L^\infty}+\|{\boldsymbol{v} }\|_{L^\infty} \ll 1 \quad \text{on} \quad [0,T], \end{align}\] then there exists a constant \(C\), independent of \(T\) and \(\varepsilon\), such that \[\label{APrioriEstimate} {\mathcal{X}}(T)\leq C\big( {\mathcal{X}}(0)+{\mathcal{X}}^2(T)\big) \qquad\text{for all } \varepsilon>0.\tag{7}\]
The proof of the inequality 7 is divided into two steps.
Step 1: The low-frequency estimate in the \(L^p\) framework
Defining the effective velocity as \({\boldsymbol{z} }\triangleq \varepsilon^{-1}{\boldsymbol{v} }+\nabla n\) (cf. the analogous case for Navier-Stokes equations in [21]), and operating on (2 ) with \(\dot{\Delta}_j\), yields \[\label{2461} \begin{align} \left\{\aligned &\partial_t \dot{\Delta}_jn - \varepsilon\Delta \dot{\Delta}_jn =- \varepsilon\rm div\;\!\dot{\Delta}_j{\boldsymbol{z} }- \dot{\Delta}_j({\boldsymbol{v} }\cdot \nabla n)-\dot{\Delta}_j(G(n)\rm div\;\!{\boldsymbol{v} })\\ &\partial_t\dot{\Delta}_j{\boldsymbol{z} }+\frac{1}{\varepsilon} \dot{\Delta}_j{\boldsymbol{z} }= H \endaligned\right. \end{align}\tag{8}\] with \[H=\varepsilon \left(\nabla\Delta \dot{\Delta}_jn-\nabla \rm div\;\!\dot{\Delta}_j{\boldsymbol{z} }\right) - \dot{\Delta}_j\nabla({\boldsymbol{v} }\cdot \nabla n) -\nabla\dot{\Delta}_j(G(n)\rm div\;\!{\boldsymbol{v} }) -\frac{1}{\varepsilon}\dot{\Delta}_j({\boldsymbol{v} }\cdot\nabla{\boldsymbol{v} }).\] It is clear that the equation on \(n\) is a heat equation and the equation on \({\boldsymbol{z} }\) is a damped equation, hence the standard estimate (see [15], [19]) implies that
\[\label{LowEst3} \begin{align} &\|(n,\varepsilon {\boldsymbol{z} })\|_{\widetilde{L}_T^\infty(\dot{B}_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon} +\bigg(\varepsilon\|n\|_{\widetilde{L}_T^1(\dot{B}_{p,1}^{\frac{d}{p}+2})}^{\ell,J_\varepsilon} +\|{\boldsymbol{z} }\|_{\widetilde{L}_T^1(\dot{B}_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}\bigg)\\ &\quad\leq C\bigg(\|(n_0, \varepsilon {\boldsymbol{z} }_0)\|_{\dot{B}_{p,1}^{\frac{d}{p}}}^{\ell,J_\varepsilon} +(1+2^{k})\|({\boldsymbol{v} }\cdot \nabla n,G(n)\rm div\;\!{\boldsymbol{v} },{\boldsymbol{v} }\cdot \nabla{\boldsymbol{v} })\|_{\widetilde{L}_T^1({\dot{B}}_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}\bigg) \end{align}\tag{9}\] with \(C\) depending only on \(d,p,k\).
Next, we will deal with the nonlinear part. For the first nonlinear term, it follows the Lemma 1 that \[\begin{align} \|{\boldsymbol{v} }\cdot \nabla n\|_{\widetilde{L}_T^1({\dot{B}}_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon} &\lesssim \left( \|\nabla n\|_{\widetilde{L}_T^2({\dot{B}}_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon} + \|\nabla n\|_{\widetilde{L}_T^2({\dot{B}}_{2,1}^{\frac{d}{2}})}^{h,J_\varepsilon} \right) \|{\boldsymbol{v} }\|_{\widetilde{L}_T^2({\dot{B}}_{2,1}^{\frac{d}{2}})}. \end{align}\] Then the interpolation inequalities (see Lemma 3.2 in [15]) implies \[\label{NL1} \begin{align} \|{\boldsymbol{v} }\cdot \nabla n\|_{\widetilde{L}_T^1({\dot{B}}_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon} &\lesssim \left(\left(\| n\|_{\widetilde{L}_T^\infty({\dot{B}}_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon} \|n\|_{\widetilde{L}_T^1({\dot{B}}_{p,1}^{\frac{d}{p}+2})}^{\ell,J_\varepsilon}\right)^{\frac{1}{2}} +\left(\|n\|_{\widetilde{L}_T^\infty({\dot{B}}_{2,1}^{\frac{d}{2}+1})}^{h,J_\varepsilon} \|n\|_{\widetilde{L}_T^1({\dot{B}}_{2,1}^{\frac{d}{2}+1})}^{h,J_\varepsilon}\right)^{\frac{1}{2}} \right)\\ &\quad \times\left(\|{\boldsymbol{v} }\|_{\widetilde{L}_T^2({\dot{B}}_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon} +\varepsilon\left(\|{\boldsymbol{v} }\|_{\widetilde{L}_T^\infty({\dot{B}}_{2,1}^{\frac{d}{2}+1})}^{h,J_\varepsilon} \|{\boldsymbol{v} }\|_{\widetilde{L}_T^1({\dot{B}}_{2,1}^{\frac{d}{2}+1})}^{h,J_\varepsilon}\right)^{\frac{1}{2}}\right)\\ &\lesssim {\mathcal{X}}^2(T)+{\mathcal{X}}(T)\varepsilon^{-\frac{1}{2}}\|{\boldsymbol{v} }\|_{\widetilde{L}_T^2({\dot{B}}_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}. \end{align}\tag{10}\] And similarly, for the convection term we can deduce \[\label{NL2} \begin{align} \|{\boldsymbol{v} }\cdot \nabla {\boldsymbol{v} }\|_{\widetilde{L}_T^1({\dot{B}}_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon} \lesssim {\mathcal{X}}^2(T). \end{align}\tag{11}\] For the last nonlinear term, the inequality 6 implies \[\begin{align} \|G(n)\rm div\;\!{\boldsymbol{v} }\|_{\widetilde{L}_T^1({\dot{B}}_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon} &\lesssim \|G(n)\|_{\widetilde{L}_T^\infty({\dot{B}}_{2,1}^{\frac{d}{2}})} \left(\|\rm div\;\!{\boldsymbol{v} }\|_{\widetilde{L}_T^1({\dot{B}}_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon} + \|\rm div\;\!{\boldsymbol{v} }\|_{\widetilde{L}_T^1({\dot{B}}_{2,1}^{\frac{d}{2}})}^{h,J_\varepsilon}\right). \end{align}\] By the classical composition estimate (see [20]), we can obtain \[\label{NL3} \begin{align} \|G(n)\rm div\;\!{\boldsymbol{v} }\|_{\widetilde{L}_T^1({\dot{B}}_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon} \lesssim \|n\|_{\widetilde{L}_T^\infty({\dot{B}}_{2,1}^{\frac{d}{2}})} \left(\|{\boldsymbol{v} }\|_{\widetilde{L}_T^1({\dot{B}}_{p,1}^{\frac{d}{p}+1})}^{\ell,J_\varepsilon} +\|{\boldsymbol{v} }\|_{\widetilde{L}_T^1({\dot{B}}_{2,1}^{\frac{d}{2}+1})}^{h,J_\varepsilon}\right) \lesssim {\mathcal{X}}^2(T). \end{align}\tag{12}\]
Inserting 10 12 into 9 , we can obtain \[\begin{align} &\|(n, \varepsilon {\boldsymbol{z} })\|_{\widetilde{L}_T^\infty(\dot{B}_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon} +\bigg(\varepsilon\|n\|_{\widetilde{L}_T^1(\dot{B}_{p,1}^{\frac{d}{p}+2})}^{\ell,J_\varepsilon} +\|{\boldsymbol{z} }\|_{\widetilde{L}_T^1(\dot{B}_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}\bigg)\\ &\quad\leq C\bigg( \|(n_0, \varepsilon {\boldsymbol{z} }_0)\|_{\dot{B}_{p,1}^{\frac{d}{p}}}^{\ell,J_\varepsilon} +{\mathcal{X}}^2(T)+{\mathcal{X}}(T)\varepsilon^{-\frac{1}{2}} \|{\boldsymbol{v} }\|_{\widetilde{L}_T^2({\dot{B}}_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}\bigg) , \end{align}\] which eventually leads to \[\label{LowEst4} \begin{align} &\|n\|_{\widetilde{L}_T^\infty(\dot{B}_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon} +\|{\boldsymbol{v} }\|_{\widetilde{L}_T^\infty(\dot{B}_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon} +\bigg(\varepsilon\|n\|_{\widetilde{L}_T^1(\dot{B}_{p,1}^{\frac{d}{p}+2})}^{\ell,J_\varepsilon}+\|\varepsilon^{-1}{\boldsymbol{v} }+\nabla n\|_{\widetilde{L}_T^1(\dot{B}_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon} +\|{\boldsymbol{v} }\|_{\widetilde{L}_T^1(\dot{B}_{p,1}^{\frac{d}{p}+1})}^{\ell,J_\varepsilon} +\varepsilon^{-\frac{1}{2}} \|{\boldsymbol{v} }\|_{\widetilde{L}_T^2({\dot{B}}_{p,1}^{\frac{d}{p}})}^{\ell,J_\varepsilon}\bigg)\\ &\quad\lesssim {\mathcal{X}}(0)+{\mathcal{X}}^2(T). \end{align}\tag{13}\]
It is known that most hyperbolic systems with no \(0\)-order terms are ill-posed in \(L^p\) spaces with \(p\neq2\) (see [22]), and the the \(0\)-th order damping plays a key role in above Low-frequency analysis. Next, we turn to the high-frequency analysis. In that case, the formal eigenvalue analysis (see [12]) implies that it is only suitable within the \(L^2\) framework. Therefore, we use the energy method which is a classical method in dissipative system (such as [10], [23]–[25])
Step 2: The high-frequency estimates in the \(L^2\) framework
First, we localize system (2 ) by applying the operator \(\dot{\Delta}_j\) and obtaining \[\begin{align} \left\{\aligned &\partial_t \dot{\Delta}_jn + \rm div\;\!\dot{\Delta}_j{\boldsymbol{v} }+ G(n)\rm div\;\!\dot{\Delta}_j{\boldsymbol{v} }+ {\boldsymbol{v} }\cdot \nabla \dot{\Delta}_jn = R_j^1+R_j^2 \\ &\partial_t\dot{\Delta}_j{\boldsymbol{v} }+\nabla \dot{\Delta}_jn+ {\boldsymbol{v} }\cdot \nabla \dot{\Delta}_j{\boldsymbol{v} }+\frac{1}{\varepsilon} \dot{\Delta}_j{\boldsymbol{v} }= R_j^3 \endaligned\right. \end{align}\] with the commutators defined as \[\begin{align} \begin{aligned} R_j^1:=-\dot{\Delta}_j({\boldsymbol{v} }\cdot \nabla n)+{\boldsymbol{v} }\cdot \nabla \dot{\Delta}_jn,\; R_j^2:=-\dot{\Delta}_j(G(n)\rm div\;\!{\boldsymbol{v} })+ G(n)\rm div\;\!\dot{\Delta}_j{\boldsymbol{v} },\; R_j^3:=-\dot{\Delta}_j({\boldsymbol{v} }\cdot \nabla{\boldsymbol{v} })+{\boldsymbol{v} }\cdot \nabla \dot{\Delta}_j{\boldsymbol{v} }. \end{aligned} \end{align}\]
By using the \(L^2\) weighted energy method, as in [12], [15], one can deduce that \[\label{dissi4} \begin{align} &\frac{1}{2}\frac{d}{dt}\varepsilon{\mathcal{L}}_j^2+\|(\nabla\dot{\Delta}_jn,\nabla\dot{\Delta}_j{\boldsymbol{v} })\|_{L^2}^2\\ &\quad\lesssim \varepsilon\|(\partial_tG(n),\nabla G(n), \nabla {\boldsymbol{v} })\|_{L^\infty}{\mathcal{L}}_j^2 +{\mathcal{L}}_j \varepsilon \sum_{i=1}^32^j\|R_j^i\|_{L^2} \end{align}\tag{14}\] with the Lyapunov functional \({\mathcal{L}}_j^2\) \[\varepsilon{\mathcal{L}}_j^2:= 2^{2j}\varepsilon\|(\dot{\Delta}_jn, \dot{\Delta}_j{\boldsymbol{v} })\|_{L^2}^2+2\tilde{c} \int \dot{\Delta}_j{\boldsymbol{v} }\cdot\nabla\dot{\Delta}_jn dx \approx 2^{2j}\varepsilon\|(\dot{\Delta}_jn, \dot{\Delta}_j{\boldsymbol{v} })\|_{L^2}^2\quadfor\quad \tilde{c}2^{-k}<1.\] And then it follows from the Lemma 5.1. in [13] that \[\label{HighEst} \begin{align} &\varepsilon\|(n, {\boldsymbol{v} })\|_{\widetilde{L}_T^\infty({\dot{B}}_{2,1}^{\frac{d}{2}+1})}^{h,J_\varepsilon} +\|(n, {\boldsymbol{v} })\|_{\widetilde{L}_T^1({\dot{B}}_{2,1}^{\frac{d}{2}+1})}^{h,J_\varepsilon}\\ &\quad\lesssim {\mathcal{X}}(0) +\varepsilon \|(\partial_t G(n), \nabla G(n), \nabla {\boldsymbol{v} })\|_{L_T^\infty(L^\infty)}{\mathcal{X}}(T) +\varepsilon\int_0^T \sum_{i=1}^3\sum_{j\geq J_\varepsilon}2^{(\frac{d}{2}+1)j}\|R_j^i\|_{L^2}dt. \end{align}\tag{15}\]
Since \(2^{J_\varepsilon}\approx 2^{k}\varepsilon^{-1}\), by the spatial embedding \({\dot{B}}_{p,1}^{\frac{d}{p}}(\mathbb{R}^d)\hookrightarrow {\dot{B}}_{2,1}^{\frac{d}{2}}(\mathbb{R}^d)\hookrightarrow L^\infty(\mathbb{R}^d)\), the classical estimation on smooth functions (see Corollary 2.65 in [20]) and 5 we can obtain that \[\label{HNonLinEst1} \begin{align} \varepsilon\|(\nabla G(n),\nabla {\boldsymbol{v} })\|_{L_T^\infty(L^\infty)} \lesssim \varepsilon\|(n, {\boldsymbol{v} })\|_{\widetilde{L}_T^\infty({\dot{B}}_{2,1}^{\frac{d}{2}+1})} \lesssim{\mathcal{X}}(T). \end{align}\tag{16}\] By the continuity equation, it is not difficult to get \[\label{HNonLinEst2} \begin{align} \varepsilon\|\partial_t G(n)\|_{L_T^\infty(L^\infty)} &\lesssim \varepsilon \|\rm div\;\!{\boldsymbol{v} }+{\boldsymbol{v} }\cdot \nabla n + G(n)\rm div\;\!{\boldsymbol{v} }\|_{\widetilde{L}_T^\infty({\dot{B}}_{2,1}^{\frac{d}{2}})}\\ &\lesssim \varepsilon\|{\boldsymbol{v} }\|_{\widetilde{L}_T^\infty({\dot{B}}_{2,1}^{\frac{d}{2}+1})} +\|{\boldsymbol{v} }\|_{\widetilde{L}_T^\infty({\dot{B}}_{2,1}^{\frac{d}{2}})}\varepsilon \|\nabla n\|_{\widetilde{L}_T^\infty({\dot{B}}_{2,1}^{\frac{d}{2}})}\; +\|n\|_{\widetilde{L}_T^\infty({\dot{B}}_{2,1}^{\frac{d}{2}})}\varepsilon \|\rm div\;\!{\boldsymbol{v} }\|_{\widetilde{L}_T^\infty({\dot{B}}_{2,1}^{\frac{d}{2}})}\\ &\lesssim {\mathcal{X}}(T)+{\mathcal{X}}^2(T). \end{align}\tag{17}\]
Next, we bound the commutator \(R_j^i\) (\(i=1,2,3\)), by the Lemma 2.100 in [20] we can deduce \[\label{HNonLinEst3} \begin{align} &\varepsilon\int_0^T \sum_{i=1}^3\sum_{j\geq J_\varepsilon}2^{(\frac{d}{2}+1)j} \|R_j^i\|_{L^2}dt \lesssim \varepsilon\|{\boldsymbol{v} }\|_{\widetilde{L}_T^1({\dot{B}}_{2,1}^{\frac{d}{2}+1})} \left(\|n\|_{\widetilde{L}_T^\infty({\dot{B}}_{2,1}^{\frac{d}{2}+1})} +\|{\boldsymbol{v} }\|_{\widetilde{L}_T^\infty({\dot{B}}_{2,1}^{\frac{d}{2}+1})} \right) \lesssim {\mathcal{X}}^2(T). \end{align}\tag{18}\]
Substitute 16 18 into 15 we can finally obtain \[\label{HighEst2} \begin{align} &\varepsilon\|(n, {\boldsymbol{v} })\|_{\widetilde{L}_T^\infty({\dot{B}}_{2,1}^{\frac{d}{2}+1})}^{h,J_\varepsilon} +\|(n, {\boldsymbol{v} })\|_{\widetilde{L}_T^1({\dot{B}}_{2,1}^{\frac{d}{2}+1})}^{h,J_\varepsilon} +\varepsilon^{-\frac{1}{2}}\|{\boldsymbol{v} }\|_{\widetilde{L}_T^2({\dot{B}}_{p,1}^{\frac{d}{p}})}^{h,J_\varepsilon}\\ &\quad+|\varepsilon^{-1}{\boldsymbol{v} }+\nabla n\|_{\widetilde{L}_T^1(\dot{B}_{p,1}^{\frac{d}{p}})}^{h,J_\varepsilon} \lesssim {\mathcal{X}}(0)+ {\mathcal{X}}^2(T)+{\mathcal{X}}^3(T). \end{align}\tag{19}\]
The estimate 7 follows immediately from a combination of 19 and 13 . Applying a standard bootstrap argument, as detailed in [19], establishes the global well-posedness of the solution. Consequently, Theorem 1 is proved.
J. Z. Zhang was supported by the Natural Science Foundation of Shandong Province , China (ZR2024QA003); X. L. Li was supported by the National Natural Science Foundation of China (12571178).