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[23] NOVOTNÝ, A. Weak solutions for a bi-fluid model for a mixture of two compressible non interacting fluids. *Science China Mathematics* **63**, 12 (2020), 2399-2414.

[24] NOVOTNÝ, A., AND POKORNÝ, M. Weak solutions for some compressible multicomponent fluid models. *Archive for Rational Mechanics and Analysis* **235**, 1 (2020), 355-403.

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**B.3. Final step.** We now turn to the application of the Banach's fixed-point theorem in the following vector space

$$E := \left\{ v \in \mathcal{C}([0, T], H^1(\mathbb{R}^d)) : \sup_{t \in [0, T]} e^{-\beta t} \|v(t)\|_{H^1(\mathbb{R}^d)} < \infty \right\}$$

endowed with the norm:

$$\forall v \in E, \quad \|v\|_E = \sup_{t \in [0, T]} e^{-\beta t} \|v(t)\|_{H^1(\mathbb{R}^d)}.$$

Above, $\beta$ is a large number such that (B.6)-(B.8) hold true. It is obvious that $(E, \|\cdot\|_E)$ is a Banach space and, in particular the closed ball $E_T = B(0, KR_T)$ is complete with respect to the norm $\|\cdot\|_E$. On the first hand, Lemma B.3 ensures that $\Phi$ maps $E_T$ into itself and the Lemma B.4 ensures that $\Phi$ is a contracting map. Hence, by Banach's fixed-point theorem, $\Phi$ admits a unique fixed-point $v$ in $E_T$ that solves the equation (B.1). Furthermore, $v$ verifies

$$A_k v = \mathcal{T}(kt)A v_0 + \int_0^t \mathcal{T}(k(t-s))A_k f(s)ds + \int_0^t \mathcal{T}(k(t-s))A_k \operatorname{div}(b_0 \cdot v)(s)ds$$

and [6, Lemma 2.2] implies $v_t, \nabla^2 v \in L^2((0, T) \times \mathbb{R}^d)$.

**Data availability.** There are no data sets generated or analyzed in this study, therefore data sharing does not apply.

**Conflict of interest.** The author declares that they have no conflict of interest.

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