Abstract
We investigate a generic compressible two-fluid model with common pressure (\(P^+=P^-\)) in \(\mathbb {R}^3\). Under some smallness assumptions, Evje–Wang–Wen [Arch Rational Mech Anal 221:1285–1316, 2016] obtained the global solution and its optimal decay rate for the 3D compressible two-fluid model with unequal pressures \(P^+\ne P^-\). More precisely, the capillary pressure \(f(\alpha ^-\rho ^-)=P^+-P^-\ne 0\) is taken into account, and is assumed to be a strictly decreasing function near the equilibrium. As indicated by Evje–Wang–Wen, this assumption played a key role in their analysis and appeared to have an essential stabilization effect on the model. However, global well–posedness of the 3D compressible two-fluid model with common pressure has been a challenging open problem due to the fact that the system is partially dissipative and its nonlinear structure is very terrible. In the present work, by exploiting the dissipation structure of the model and making full use of several key observations, we prove global existence and large time behavior of classical solutions to the 3D compressible two-fluid model with common pressure. To the best of our knowledge, we establish the first result on the global existence of classical solutions to the 3D compressible two-fluid model with common pressure and without capillary effects. The method relies upon careful analysis of the linearized system, exploitation of the algebraic structure of the nonlinear system, and the introduction of an auxiliary velocity \(v=(2\mu ^++\lambda ^+)u^+-(2\mu ^-+\lambda ^-)u^-\) which plays the role of the effective viscous flux (since in this system \(P^+= P^-\)) in the single phase case: such velocity has better regularity than phase velocities \(u^\pm \).
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Acknowledgements
Guochun Wu’s research was partially supported by National Natural Science Foundation of China \(\#\)12271114, and Natural Science Foundation of Fujian Province \(\#\)2022J01304. Lei Yao’s research was partially supported by National Natural Science Foundation of China \(\#\)12171390, \(\#\)11931013, and Natural Science Basic Research Plan for Distinguished Young Scholars in Shaanxi Province of China (Grant No. 2019JC-26). Yinghui Zhang’ research is partially supported by National Natural Science Foundation of China \(\#\)12271114, Science and Technology Project of Guangxi \(\#\)GuikeAD21220114, and Key Laboratory of Mathematical and Statistical Model (Guangxi Normal University), Education Department of Guangxi Zhuang Autonomous Region.
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Analytic tools
Analytic tools
We recall the Sobolev interpolation of the Gagliardo–Nirenberg inequality.
Lemma A.1
(Gagliardo–Nirenberg interpolation, general case). Let \(1 \le \) \(p, q \le \infty , m \in \mathbb {N}, k \in \mathbb {N}_0\), with \(0 \le k<m\), and let a, r be such that
and
Then there exists a constant \(c=c(m, p, q, a, k)>0\) such that
for every \(f \in L^q\left( \mathbb {R}^3\right) \cap \dot{W}^{m, p}\left( \mathbb {R}^3\right) \)(where the homogeneous Sobolev space \(\dot{W}^{m, p}\left( \mathbb {R}^3\right) \) is the space of all functions \(f\in L_{loc}^1(\mathbb {R})^3\) whose \(\alpha \)-th weak derivative \(\partial ^\alpha f\) belongs to \(L^p(\mathbb {R}^3)\) with \(|\alpha |=m\)), with the following exceptional cases:
(i) If \(k=0, m p<3\), and \(q=\infty \), we assume that f vanishes at infinity.
(ii) If \(1<p<\infty \) and \(m-k-3 / p\) is a nonnegative integer, then (A.1) only holds for \(0<a \le 1-k / m\).
Proof
See [12, pp. 403, Theorem 12.87] and [16, pp. 125, Theorem]. \(\square \)
Lemma A.2
([6, 10, 13]). For any integer \(k\ge 1\), we have
and
where \(p, p_1, p_{2}, p_{3}, p_{4} \in [1, \infty ]\) and
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Wu, G., Yao, L. & Zhang, Y. Global well-posedness and large time behavior of classical solutions to a generic compressible two-fluid model. Math. Ann. (2023). https://doi.org/10.1007/s00208-023-02732-5
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DOI: https://doi.org/10.1007/s00208-023-02732-5