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Well-posedness of the two-dimensional Abels–Garcke–Grün model for two-phase flows with unmatched densities

Abstract

We study the Abels–Garcke–Grün (AGG) model for a mixture of two viscous incompressible fluids with different densities. The AGG model consists of a Navier–Stokes–Cahn–Hilliard system characterized by a (non-constant) concentration-dependent density and an additional flux term due to interface diffusion. In this paper we address the well-posedness problem in the two-dimensional case. We first prove the existence of local strong solutions in general bounded domains. In the space periodic setting we show that the strong solutions exist globally in time. In both cases we prove the uniqueness and the continuous dependence on the initial data of the strong solutions. Lastly, we show a stability result for the strong solutions to the AGG model and the model H in terms of the density values.

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Notes

  1. In comparison with [5, 6], we have set the mobility \(m(\phi )\) and the energy coefficient \(a(\phi )\) equal to one in (1.1).

  2. It is worth mentioning that the strong solutions herein proved in Theorems 3.1 and 3.3 are such that \(\phi \) takes values in the physical range \([-1,1]\) entailing that \(\rho (\phi )\) defined in (1.2) remains positive for all times.

  3. A different approximation leading to a concentration \(\phi _m\) with values outside the interval \([-1,1]\) may need a suitable extension of \(\rho (\cdot )\) outside the interval \([-1,1]\), and, in general, it may happen that \(\rho '(\phi )\ne \frac{\rho _1-\rho _2}{2}\).

  4. In contrast to the classical periodic setting for the incompressible Navier–Stokes [cf. [33]], we do not require that \({\widehat{{{\varvec{u}}}}}_0=0\) in the definition of \({\mathbb {H}}_\sigma \) and \({\mathbb {V}}_\sigma \). This is due to the fact that \({\overline{{{\varvec{u}}}}}=\frac{1}{|\Omega |}\int _{\Omega } {{\varvec{u}}}\, \mathrm{d}x\) is not conserved by the flow of (1.1).

  5. The estimate (4.12) is derived from [1, Lemma 3] and [22, Lemma 5.1]. More precisely, [22, Eq. (5.6)] yields

    $$\begin{aligned}&\frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}t}\Vert \nabla A^{-1} \partial _t^h \phi _m\Vert _{L^2(\Omega )}^2 + \frac{1}{2} \Vert \nabla \partial _t^h \phi _m\Vert _{L^2(\Omega )}^2 \le C\Vert \nabla A^{-1} \partial _t^h \phi _m|_{L^2(\Omega )}^2 \\&\qquad + ({{\varvec{v}}}(t+h)\partial _t^h \phi _m, \nabla A^{-1}\partial _t^h \phi _m) +(\partial _t^h {{\varvec{v}}}\phi _m, \nabla A^{-1}\partial _t^h \phi _m), \end{aligned}$$

    for \(t\in (0,T-h\), where \(\partial _t^h f (\cdot )= \frac{1}{h}(f(\cdot +h) -f(\cdot ))\) and \(A^{-1}\) is the inverse of the Laplace operator with Neumann boundary conditions. Since \(\phi _m\) is bounded [cf. (4.8)] and following [22, Lemma 5.1], we obtain (cf. also [1, Eq. (3.19)])

    $$\begin{aligned}&\frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}t}\Vert \nabla A^{-1} \partial _t^h \phi _m\Vert _{L^2(\Omega )}^2 + \frac{1}{4} \Vert \nabla \partial _t^h \phi _m\Vert _{L^2(\Omega )}^2 \\&\quad \le C(1+ \Vert {{\varvec{v}}}(t+h)\Vert _{L^2(\Omega )}^2) \Vert \nabla A^{-1} \partial _t^h \phi _m|_{L^2(\Omega )}^2+ C \Vert \partial _t^h {{\varvec{v}}}\Vert _{L^2(\Omega )}^2, \end{aligned}$$

    where C is independent of h. Thanks to \(\Vert \nabla A^{-1} \partial _t^h \phi (0)\Vert _L^2(\Omega )\le C(1+ \Vert \nabla \mu _0\Vert _{L^2(\Omega )}+ \Vert {{\varvec{v}}}\Vert _{L^\infty (0,T; L^2(\Omega ))})\) [cf. [1, Eq. (3.18)] and [14, Eq. 5.8]], we conclude from the Gronwall lemma that (4.12) holds replacing \(\partial _t \phi _m\) with \(\partial _t^h \phi _m\). Then, passing to the limit \(h\rightarrow 0\) as in [1, Lemma 3], we arrive at (4.12).

  6. Let us consider the Lipschitz truncation of \(\phi _m\) defined by \(\phi _m^k=h_k(\phi _m)\), where \(h_k(s)=s\) for \(s\in (-1+\frac{1}{k},1-\frac{1}{k})\), \(h_k(s)= 1-\frac{1}{k}\) for \(s\in [1-\frac{1}{k},1)\), and \(h_k(s)=-1+\frac{1}{k}\) for \(s \in (-1, -1 +\frac{1}{k}]\). Thanks to (4.8), for almost every \(t\in (0,T)\), \(\phi _m^k \rightarrow \phi _m\) almost everywhere in \(\Omega \), thereby \(F'(\phi _m^k) \rightarrow F'(\phi _m)\), \(F''(\phi _m^k) \rightarrow F''(\phi _m)\) almost everywhere in \(\Omega \). Since \(|F'(\phi _m^k)|\le |F'(\phi _m)|\) and \(F'(\phi _m)\in L^\infty (0,T;L^2(\Omega ))\), we infer that \(F'(\phi _m^k) \rightarrow F'(\phi _m)\) in \(L^2(\Omega )\) for almost every \(t \in (0,T)\). Then, for any test function \(\varphi \in C_c^\infty (\Omega )\)

    $$\begin{aligned} \int _{\Omega } F'(\phi _m) \partial _{x_i} \varphi \, \mathrm{d}x&= -\lim _{k\rightarrow \infty } \int _{\Omega }F'(\phi ^k_m) \partial _{x_i} \varphi \, \mathrm{d}x = \lim _{k \rightarrow \infty } \int _{\Omega } F''(\phi ^k_m)\partial _{x_i} \phi ^k_m \, \varphi \, \mathrm{d}x\\&= \lim _{k \rightarrow \infty } \int _{\Omega } F''(\phi ^k_m) \chi _{\lbrace \phi _m \in (-1+\frac{1}{k}, 1-\frac{1}{k})\rbrace } (\phi _m) \partial _{x_i} \phi _m \, \varphi \, \mathrm{d}x = \int _{\Omega } F''(\phi _m) \partial _{x_i} \phi _m \, \varphi \, \mathrm{d}x, \end{aligned}$$

    where the last limit follows from Lebesgue’s dominated convergence theorem since \(|F''(\phi ^k_m) \chi _{\lbrace \phi _m \in (-1+\frac{1}{k}, 1-\frac{1}{k})\rbrace } (\phi _m)|\le |F''(\phi _m)|\) and \(F''(\phi _m) \in L^\infty (0,T;L^2(\Omega ))\).

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Correspondence to Andrea Giorgini.

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Giorgini, A. Well-posedness of the two-dimensional Abels–Garcke–Grün model for two-phase flows with unmatched densities. Calc. Var. 60, 100 (2021). https://doi.org/10.1007/s00526-021-01962-2

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Mathematics Subject Classification

  • 35D35
  • 35Q35
  • 76D03
  • 76D45
  • 76T06