Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 72, pp. 1–22. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS FOR A NON-ISOTHERMAL NAVIER-STOKES-ALLEN-CAHN SYSTEM WITH THERMO-INDUCED COEFFICIENTS JULIANA HONDA LOPES, GABRIELA PLANAS Abstract. This article aims to study the existence of solutions for a non- isothermal Navier-Stokes-Allen-Cahn system with thermo-induced coefficients. The system can be considered as a model describing the motion of a mixture of two viscous incompressible fluids with viscosity, thermal conductivity and interfacial thickness being temperature-dependent. This is a more general sys- tem than the previous ones considered in the literature, involving temperature dependence on all main coefficients. The strong non-linear couplings between those equations because of the temperature dependence brings new mathemat- ical difficulties that only allows working in two dimensions. 1. Introduction We consider a non-isothermal Navier-Stokes-Allen-Cahn system where certain coefficients are temperature-dependent. The system can be considered as a model describing the motion of a mixture of two viscous incompressible fluids with viscos- ity, thermal conductivity and interfacial thickness being temperature-dependent, and matched densities. This system consists of Navier-Stokes equations coupled with a phase-field equation given by a convective Allen-Cahn equation and an en- ergy transport equation. More precisely, we investigate the existence of solutions to the problem ut + u · ∇u−∇ · (ν(θ)Du) +∇p = λ ( −∇ · (ε(θ)∇φ) + 1 ε(θ) F ′(φ) ) ∇φ− α∆θ∇θ, (1.1) ∇ · u = 0, (1.2) φt + u · ∇φ = γ ( ∇ · ( ε(θ)∇φ ) − 1 ε(θ) F ′(φ) ) , (1.3) θt + u · ∇θ = ∇ · (k(θ)∇θ), (1.4) 2020 Mathematics Subject Classification. 35K51, 35A02, 35Q35. Key words and phrases. Nonlinear system; Navier-Stokes-Allen-Cahn system; non-isothermal phase-field. ©2022. This work is licensed under a CC BY 4.0 license. Submitted May 20, 2022. Published October 25, 2022. 1 2 J. H. LOPES, G. PLANAS EJDE-2022/72 in Ω× (0,∞), where Ω is a bounded domain of R2 with smooth boundary ∂Ω. We complement the system with the initial and boundary conditions u(0) = u0, φ(0) = φ0, θ(0) = θ0, in Ω, u = 0, ∂φ ∂η = 0, ∂θ ∂η = 0, on ∂Ω× (0,∞), (1.5) where η stands for the exterior normal to the boundary ∂Ω. Here, u, p and θ denote the mean velocity of the fluid mixture, the pressure, and the temperature; the phase-field variable φ represents the volume fraction of the two components. Du = 1 2 ( ∇u+ (∇u)T ) corresponds to the symmetric part of the velocity gradient. ν > 0 is the viscosity of the mixture, λ > 0 is the surface tension parameter, α > 0 is associated to the thermal expansion coefficient, ε > 0 is a parameter related to the interfacial thickness, F (φ) is the potential energy density, γ > 0 is the relaxation time of the interface and k > 0 is the thermal conductivity. We assume that ν, ε and k are temperature-dependent. Isothermal Navier-Stokes-Allen-Cahn systems have been extensively investigated in the literature, see, for instance, [13, 25, 14, 11, 19] and references therein. Con- cerning non-isothermal Navier-Stokes-Allen-Cahn systems, there are few theoretical results available. This is because it is not a trivial question to include temperature dependence in a way such that the obtained models are at the same time thermo- dynamically consistent and mathematically tractable. For instance, [23] introduced a non-isothermal model for a mixture of two fluids with thermo-induced Marangoni effects. The mathematical analysis of that system was carried out in [27, 28]. See [6] for the well-posedness of the one-dimensional non-isentropic compressible Navier- Stokes-Allen-Cahn system with temperature-dependent heat conductivity and [26] for the case with phase variable dependent viscosity. It is worth mentioning the works [8, 9, 10, 15] where a non-isothermal Navier-Stokes-Cahn-Hilliard system was analysed. The system under consideration was previously studied by the authors in two different situations. When only the viscosity coefficient is temperature-dependent, the problem was considered in [16]. The well-posedness of the model was established showing the existence of global weak solutions in dimensions two and three, and existence and uniqueness of global strong solutions in dimension two, and local strong solutions in dimension three. No restriction on the size of the initial data was required. Later, in [17], it was also allowed that the thermal conductivity to be temperature-dependent. The existence and uniqueness of local strong solutions in two and three dimensions for any initial data were proved. Moreover, the existence of global weak solutions and the existence and uniqueness of global strong solutions in dimension two were obtained when the initial temperature is suitably small. As far as we know, there are no studies about the phase-field equation with an interfacial thickness that develops with temperature. However, there are some physical situations where the thickness of the interface between fluid phases could depend on the temperature. We can mention [3], where is presented an atomistically informed parametrization of a phase-field model to describe the anisotropic mobility of liquid-solid interfaces in silicon. In this model, the interfacial mobility parameter of the phase-field describing the relaxation dynamics of the interface depends on temperature and also on interface orientation. See also [4], whereby using the method of molecular dynamics, the thickness of the interface between the fluid phases is determined as a function of temperature. EJDE-2022/72 NON-ISOTHERMAL NAVIER-STOKES-ALLEN-CAHN SYSTEM 3 From the mathematical point of view, a closer study about a interfacial thickness variation is the analysis of sharp interface limit for phase-field models. In those cases, the thickness of the diffuse interface tends to zero. We can mention [20] for the Allen-Cahn equation, [2] for the Stokes-Allen-Cahn system, [5] for the Cahn- Hilliard equation, and [1] for the Navier-Stokes-Cahn-Hilliard system. The goal of this article is to consider the general system (1.1)-(1.5) involving temperature dependence on all main coefficients. The strong non-linear couplings between those equations due to the temperature dependence brings new mathemat- ical challenges that only allow working in dimension two. Moreover, the existence of solutions will be local in time; however, no restriction on the size of initial condi- tions is imposed. Observe that lower and higher order estimates cannot be obtained in a separate way. In order to prove the existence of local solutions, a novel higher order differential inequality for the shifted functions is constructed and combined with a small time argument, which was inspired by [18]. This article is organized as follows. In Section 2, some notation, assumptions, and the main theorem of this paper are introduced. In Section 3, the phase-field equation with thermo-induced interfacial thickness is analysed. In Section 4, the main theorem about existence of local in time solutions to (1.1)-(1.5) is proved. 2. Notation and main result Before we state the main result of this paper, we fix the notation used through the text. Let Ω ⊂ R2 be a bounded domain with smooth boundary ∂Ω and let |Ω| denote the measure of the set Ω. By Lp = Lp(Ω), with 1 ≤ p ≤ ∞, we denote the standard Lebesgue spaces and Hm = Wm,2(Ω), 0 ≤ m < +∞, the Sobolev spaces. H1 0 = H1 0 (Ω) is the closure of C∞0 (Ω) in the H1-norm. The norms in L2 or (L2)2 will be denoted by ‖ · ‖ and the inner product in R2 and the tensor product in R22 are denoted by (·, ·); the norms on Hm and (Hm)2 are indicated by ‖ · ‖Hm . We write u ∈ L2, even when u is a vector field, meaning that all of its components are in L2, and so on, since no confusion arises. Given two Banach spaces X,Y , we define the norm in X ∩ Y by ‖ · ‖X∩Y = ‖ · ‖X + ‖ · ‖Y . For the mean velocity, we introduce the standard functional spaces of divergence- free vector fields V(Ω) = {v ∈ (C∞0 (Ω))2 : ∇ · v = 0 in Ω}, H = V(Ω) (L2)2 , V = V(Ω) (H1 0 )2 . The duality product between V and V ′ will be denoted by 〈·, ·〉, and by Poincaré and Korn inequalities, ‖∇v‖ and ‖Dv‖ are equivalent norms in V . Let P be the orthogonal projection from (L2)2 onto H. We shall denote by wi and λi, respec- tively, the eigenfunctions and eigenvalues of the Stokes operator A = −P∆ defined in V ∩ H2. We observe that wi are orthogonal and complete in the spaces H, V and V ∩H2, see [24]. Let us highlight some inequalities that will be frequently used in the text. We recall that we are considering the two-dimensional case. The Ladyzhenskaya in- equality ‖v‖2L4 ≤ √ 2‖∇v‖ ‖v‖, ∀v ∈ H1 0 , (2.1) 4 J. H. LOPES, G. PLANAS EJDE-2022/72 and the Gagliardo-Nirenberg interpolation inequality ‖Djv‖Lp ≤ C‖v‖αWm,r‖v‖1−αLq , ∀v ∈ Lq ∩Wm,r, 1 ≤ q, r ≤ ∞ (2.2) where 1 p = j 2 + α (1 r − m 2 ) + (1− α) 1 q , j m ≤ α ≤ 1, for some constant C > 0 with the only exception: if 1 < r < ∞ and m − j − 2 r is a non-negative integer, then the above inequality holds only for j m ≤ α < 1 (see, e.g., [24, 12, 21]). For the potential energy, we consider the following conditions: F ∈ C2(R) and satisfies lim s→±∞ F ′(s) = ±∞, lim s→±∞ F ′′(s) = +∞, (2.3) F ′(1) ≥ 0, F ′(−1) ≤ 0. (2.4) We observe that the classical double-well potential F (s) = 1 4 (s2 − 1)2 satisfies (2.3)-(2.4). We note that (2.3) implies the existence of some positive constants Ci, i = 1, 2, 3 such that − C1 ≤ F ′′(s), −C2 ≤ F (s) ≤ F ′(s)s+ C3 for all s ∈ R, (2.5) where F (s) = ∫ s 0 F ′(r)dr. Moreover, assumption (2.4) guarantees the maximum principle for the phase-field equation, and with this, F, F ′, F ′′ will be bounded. We assume that the viscosity ν ∈ C1(R), the interfacial thickness ε ∈ C2(R), and the thermal conductivity k ∈ C2(R) satisfy 0 < ν(s), 0 < ε(s), 0 < k(s), ∀s ∈ R. However, as the maximum principle for the temperature will be true, we can trans- form equations (1.1), (1.3), and (1.4) into equivalent ones. This can be done by suitably modifying ν, ε, and k outside the interval [−‖θ0‖L∞ , ‖θ0‖L∞ ], see, e.g., [18, 28]. Therefore, we can assume that ν, ε, and k satisfy 0 < ν0 ≤ ν(s), 0 < ε0 ≤ ε(s), 0 < k0 ≤ k(s) for all s ∈ R, ν, ε, k, ν′, ε′, k′, ε′′, k′′ are bounded for all s ∈ R. From now on, without loss of generality, we assume the constants λ = α = γ = 1. Throughout this paper, C will denote a positive constant which may vary from line to line. Now we state the main result of this paper. Theorem 2.1. Given u0 ∈ V , φ0, θ0 ∈ H2, with ∂φ0 ∂η = ∂θ0 ∂η = 0 on ∂Ω and ‖φ0‖L∞ ≤ 1, there exists T ∗ > 0 such that problem (1.1)-(1.4) with initial and boundary conditions (1.5) has at least one solution (u, φ, θ) satisfying u ∈ L∞(0, T ∗;H) ∩ L2(0, T ∗;V ), ut ∈ L 4 3 (0, T ∗;V ′), φ, θ ∈ L∞(0, T ∗;H1 ∩ L∞) ∩ L2(0, T ∗;H2), φt, θt ∈ L2(0, T ∗;L2), |φ| ≤ 1, |θ| ≤ ‖θ0‖L∞ a.e. Ω× (0, T ∗), 〈ut, v〉+ (u · ∇u, v) + (ν(θ)Du,Dv) = (( −∇ · (ε(θ)∇φ) + 1 ε(θ) F ′(φ) ) ∇φ, v ) − (∆θ∇θ, v),∀v ∈ V, φt + u · ∇φ = ∇ · (ε(θ)∇φ)− 1 ε(θ) F ′(φ), EJDE-2022/72 NON-ISOTHERMAL NAVIER-STOKES-ALLEN-CAHN SYSTEM 5 θt + u · ∇θ = ∇ · (k(θ)∇θ), for a.e. (0, T ∗) × Ω, and it fulfills the boundary conditions ∂φ ∂η = ∂θ ∂η = 0 on ∂Ω × (0, T ∗), as well as the initial conditions in (1.5). We have stated the Theorem with the minimal regularity required to obtain the existence of solutions to system (1.1)-(1.5). However, with some additional and long computations we can show that the solutions are more regular; indeed, u ∈ L∞(0, T ∗;V ) ∩ L2(0, T ∗;H2) and φ, θ ∈ L∞(0, T ∗;H2) ∩ L2(0, T ∗;H3). This can be done similarly as in the proof of [17, Theorem 4.3] (cf. Step 1), so we leave the details to interested readers. 3. Phase-field equation with thermo-induced interfacial thickness In this section, we investigate the existence of strong solutions to the phase- field equation with thermo-induced interfacial thickness. We are going to apply this existence result when treating the whole system. In this way, we can assume enough regularity for u and θ. In particular, as u will be coming from the Galerkin approximations, we assume that u is a smooth divergence-free vector field vanishing on ∂Ω and that θ ∈ L∞(0, T ;H1 ∩ L∞) ∩ L2(0, T ;H2) with θt ∈ L2(0, T ;L2). Proposition 3.1. Let T > 0 and φ0 ∈ H1 ∩L∞ such that ‖φ0‖L∞ ≤ 1. Then, the problem φt + u · ∇φ = ∇ · (ε(θ)∇φ)− 1 ε(θ) F ′(φ) in Ω× (0, T ), φ(0) = φ0 in Ω, ∂φ ∂η = 0 on ∂Ω× (0, T ), (3.1) has a unique solution φ ∈ L∞(0, T ;H1) ∩ L2(0, T ;H2) with φt ∈ L2(0, T ;L2) and |φ| ≤ 1 a.e. Ω× (0, T ). Proof. We start by introducing an approximate function F ′1(s) =  sF ′′(1) + F ′(1)− F ′′(1), s ≥ 1, F ′(s), −1 ≤ s ≤ 1, sF ′′(−1) + F ′(−1) + F ′′(−1), s ≤ −1. (3.2) Let F1(s) = ∫ s 0 F ′1(r) dr and observe that properties (2.5) are true for all s ∈ R and F ′′1 is bounded. See, e.g. [7, 13, 16]. Once the maximum principle for the unique solution φ1 of (3.1) with F1 instead of F is true, there will follow that F1(φ1) = F (φ1). Thenceforth, φ1 will be a solution of (3.1), which is unique. To not overburden the notation, we will omit the subscript on F1 and φ1. To show the existence of solutions we employ the Faedo-Galerkin method. Let αk and ψk, k ∈ N, be the eigenvalues and eigenfunctions of −∆ with homogeneous Neumann boundary conditions. For each m ∈ N, we look for a function φm(x, t) = ∑m k=1 b m k (t)ψk(x) such that, for k = 1, . . . ,m, (φ′m, ψk) + (u · ∇φm, ψk) + (ε(θ)∇φm,∇ψk) + ( 1 ε(θ) F ′(φm), ψk ) = 0, (3.3) φm(0) = φm0 , (3.4) 6 J. H. LOPES, G. PLANAS EJDE-2022/72 where φm0 ∈ span{w1, . . . , wm} and φm0 → φ0 in H1 as m → ∞. Then, (3.3)-(3.4) becomes a nonlinear system of ordinary differential equations for bm = (bm1 , . . . , b m m), d dt bmk (t) = − m∑ i=1 bmi (t)(u · ∇ψi, ψk)− m∑ i=1 bmi (t)(ε(θ)∇ψi,∇ψk) − ( 1 ε(θ) F ′(φm), ψk ) := fk(t, bm), (3.5) bmk (0) = kth component of φm0 . (3.6) As ε0 ≤ ε(·) ≤ ε1 and F ′′ is a bounded function, it is not difficult to see that f(t, bm) = (f1, . . . , fm) is locally Lipschitz continuous with respect to bm. There- fore, (3.5)-(3.6) has a unique maximal solution in [0, Tm) with Tm ≤ T such that bmk ∈ H1(0, Tm). To show that Tm = T , we multiply (3.3) by bmk and sum for k = 1, . . . ,m, to find 1 2 d dt ‖φm‖2 + ε0‖∇φm‖2 + ∫ Ω 1 ε(θ) F ′(φm)φmdx ≤ 0, where we have used that u is divergence-free and vanishes on the boundary. Integrating from 0 to t and using properties (2.5) of F give us ‖φm‖2 + ε0 ∫ T 0 ‖∇φm‖2dt ≤ CT |Ω|+ ‖φ0‖2, (3.7) where C is independent of m. Thus, we conclude that Tm = T and that {φm} is bounded in L∞(0, T ;L2) ∩ L2(0, T ;H1) independently of m. Next, we show that {φm} is bounded in L2(0, T ;H2). By multiplying (3.3) by αkb m k , and summing from k = 1, . . . ,m, we discover that (φ′m,−∆φm)− (u ·∇φm,∆φm)+(∇· (ε(θ)∇φm),∆φm)− ( 1 ε(θ) F ′(φm),∆φm ) = 0, from which it follows that 1 2 d dt ‖∇φm‖2 + ε0‖∆φm‖2 ≤ (u · ∇φm,∆φm)− (ε′(θ)∇θ · ∇φm,∆φm)− ( 1 ε(θ) F ′(φm),∆φm ) . (3.8) We proceed to estimate the right-hand side of the above inequality. First, note that |(u · ∇φm,∆φm)| ≤ ‖u‖L∞‖∇φm‖‖∆φm‖ ≤ δε0‖∆φm‖2 + C‖∇φm‖2, for any δ > 0 that will be chosen later and C depends on ‖u‖L∞ . For the next term, we recall that ε′ is a bounded function and use Gagliardo- Nirenberg interpolation inequality (2.2) together with elliptic estimates to arrive at |(ε′(θ)∇θ · ∇φm,∆φm)| ≤ C‖∇θ‖L4‖∇φm‖L4‖∆φm‖ ≤ δε0‖∆φm‖2 + C‖∇θ‖H1‖∇θ‖‖∇φm‖H1‖∇φm‖ ≤ 2δε0‖∆φm‖2 + C(‖θ‖2H2 + 1)‖∇φm‖2 + C‖θ‖2H2 , where C depends on ‖θ‖L∞(0,T ;H1) and we have used that {φm} is bounded in L∞(0, T ;L2). EJDE-2022/72 NON-ISOTHERMAL NAVIER-STOKES-ALLEN-CAHN SYSTEM 7 We define β1 = max{|F ′′(1)|, |F ′′(−1)|} and β2 = max { max −1≤s≤1 {|F ′(s)|}, |F ′(1)− F ′′(1)|, |F ′(−1) + F ′′(−1)| } . Since ε0 ≤ ε(·), by using (3.2), we have that∣∣( 1 ε(θ) F ′(φm),∆φm )∣∣ ≤ C ∫ Ω (β1|φm|+ β2)|∆φm| dx ≤ C‖φm‖‖∆φm‖+ C‖∆φm‖ ≤ δε0‖∆φm‖2 + C, (3.9) where we have used (3.7). By plugging the previous estimates into (3.8) and taking δ small enough, we arrive at d dt ‖∇φm‖2 + ε0‖∆φm‖2 ≤ C(‖θ‖2H2 + 1)‖∇φm‖2 + C‖θ‖2H2 + C. Since (‖θ‖2H2 + 1) is integrable on [0, T ], by Gronwall Lemma, we conclude that ‖∇φm‖2 ≤ CeTC(‖∇φ0‖2 + T + 1), and so ‖∇φm‖2 + ∫ T 0 ‖∆φm‖2dt ≤ C(T, θ, φ0, u). (3.10) We note that the constant C is an increasing function on T , so it can be chosen independent of T for T in any finite interval. This estimate and (3.7) imply that {φm} is bounded independently of m in L∞(0, T ;H1) ∩ L2(0, T ;H2). To guarantee some strong convergence for {φm}, we estimate {(φm)t}. For this, we multiply (3.3) by (bmk )′ and sum from k = 1, . . . ,m, to obtain ‖(φm)t‖2 + (u · ∇φm, (φm)t) + (ε(θ)∇φm,∇(φm)t) + ( 1 ε(θ) F ′(φm), (φm)t ) = 0. (3.11) Observe that (ε(θ)∇φm,∇(φm)t) = ∫ Ω ε(θ) 1 2 ∂ ∂t |∇φm|2dx = 1 2 ∫ Ω ∂ ∂t (ε(θ)|∇φm|2)dx− 1 2 ∫ Ω ε′(θ)θt|∇φm|2dx, so we can rewrite (3.11) as ‖(φm)t‖2 + 1 2 d dt ‖(ε(θ))1/2∇φm‖2 = −(u · ∇φm, (φm)t) + 1 2 ∫ Ω ε′(θ)θt|∇φm|2dx− ( 1 ε(θ) F ′(φm), (φm)t ) . (3.12) We estimate the right-hand side of (3.12) term by term. For the first term, |(u · ∇φm, (φm)t)| ≤ ‖u‖L∞‖∇φm‖‖(φm)t‖ ≤ δ‖(φm)t‖2 + C, for any δ > 0 to be chosen later and we have used (3.10). Next, as ε′ is bounded, Gagliardo-Nirenberg inequality (2.2), and elliptic estimates yield 1 2 ∫ Ω ε′(θ)θt|∇φm|2dx ≤ C‖θt‖‖∇φm‖2L4 8 J. H. LOPES, G. PLANAS EJDE-2022/72 ≤ C‖θt‖‖φm‖H2‖∇φm ≤ C‖∆φm‖2 + C‖θt‖2 + C, where we have used that {φm} is bounded in L∞(0, T ;H1). In a similar way to (3.9), it follows ∣∣∣( 1 ε(θ) F ′(φm), (φm)t )∣∣∣ ≤ δ‖(φm)t‖2 + C. Therefore, taking δ small enough, (3.12) can be estimated by ‖(φm)t‖2 + d dt ‖(ε(θ))1/2∇φm‖2 ≤ C‖∆φm‖2 + C‖θt‖2 + C. By integrating in time, we infer that {(φm)t} is bounded in L2(0, T ;L2) indepen- dently of m and, by the compactness lemma [22, Cor. 4], that there exists a subsequence (relabeled the same) of {φm} that converges strongly to a function φ in L2(0, T ;H1). Since F ′′ is bounded, F ′(φm) → F ′(φ) strongly in L2(0, T ;L2). Thus, we can pass to the limit as m → ∞ in (3.3) and infer the existence of a strong solution φ of (3.1). The proof of the maximum principle, i.e., if the initial datum satisfies ‖φ0‖L∞ ≤ 1 then |φ| ≤ 1 a.e. Ω × (0, T ), can be done as usual. We multiply equation (3.1) by the positive part (φ− 1)+ and integrate in Ω to obtain 1 2 d dt ‖(φ− 1)+‖2 + ∫ Ω ε(θ)|∇(φ− 1)+|2dx+ ∫ Ω 1 ε(θ) (F ′(φ)− F ′(1))(φ− 1)+dx = − ∫ Ω 1 ε(θ) F ′(1)(φ− 1)+dx ≤ 0 because F ′(1) ≥ 0 by (2.4). Next, we use the Mean Value Theorem to arrive at 1 2 d dt ‖(φ− 1)+‖2 + ∫ Ω 1 ε(θ) F ′′(γ)(φ− 1)(φ− 1)+dx ≤ 0. Finally, by (2.5) and the fact that ε0 ≤ ε(·) we deduce that 1 2 d dt ‖(φ− 1)+‖2 ≤ C1 ε0 ‖(φ− 1)+‖2. Gronwall Lemma and the fact that (φ0 − 1)+ = 0 imply that (φ − 1)+ = 0 and therefore φ ≤ 1. We proceed in analogous way by multiplying by the negative part (φ+ 1)− to obtain that φ ≥ −1. To finish, let φ1, φ2 ∈ L∞(0, T ;H1∩L∞)∩L2(0, T ;H2) be two solutions of (3.1). Then φ1 − φ2 solves (φ1 − φ2)t + u · ∇(φ1 − φ2) = ∇ · (ε(θ)∇(φ1 − φ2))− 1 ε(θ) (F ′(φ1)− F ′(φ2)) together with ∂ ∂η (φ1 − φ2) = 0 on ∂Ω× (0, T ) and (φ1 − φ2)(0) = 0 in Ω. Multiplying the above equation by φ1 − φ2 and integrating in Ω, we see 1 2 d dt ‖φ1 − φ2‖2 + ε0‖∇(φ1 − φ2)‖2 ≤ − ( 1 ε(θ) (F ′(φ1)− F ′(φ2)), φ1 − φ2 ) ≤ C‖F ′(φ1)− F ′(φ2)‖‖φ1 − φ2‖ ≤ C‖φ1 − φ2‖2, EJDE-2022/72 NON-ISOTHERMAL NAVIER-STOKES-ALLEN-CAHN SYSTEM 9 where we used that ∇ · u = 0, the Mean Value Theorem for F ′ and the fact that F ′′ is bounded. Gronwall Lemma gives the uniqueness. This completes the proof. � If we assume more regularity on the initial datum and that θ ∈ L∞(0, T ;H2) ∩ L2(0, T ;H3), the solution φ to (3.1) is more regular. Proposition 3.2. Under assumptions of Proposition 3.1 and, moreover, φ0 ∈ H2 satisfying ∂φ0 ∂η = 0 on ∂Ω, the solution φ of (3.1) satisfies φ ∈ L∞(0, T ;H2) ∩ L2(0, T ;H3). Proof. We derive further a priori estimates for the approximate solution. Multiply- ing (3.3) by α2 kb m k and adding from k = 1, . . . ,m, it follows that ((φm)t,∆ 2φm)+(u·∇φm,∆2φm)+(ε(θ)∇φm,∇∆2φm)+ ( 1 ε(θ) F ′(φm),∆2φm ) = 0. Note that ∂(∆φm) ∂η = 0 and ∂(∂tφm) ∂η = 0 on ∂Ω. Hence, we have that ((φm)t,∆ 2φm) = 1 2 d dt ‖∆φm‖2, (u · ∇φm,∆2φm) = −(∇u∇φm,∇∆φm)− (D2φmu,∇∆φm), (ε(θ)∇φm,∇∆2φm) = −(ε′(θ)∇θ · ∇φm,∆2φm)− (ε(θ)∆φm,∆ 2φm) = (ε′′(θ)∇θ∇θ · ∇φm,∇∆φm) + (ε′(θ)D2θ∇φm,∇∆φm) + (ε′(θ)D2φm∇θ,∇∆φm) + (ε′(θ)∇θ∆φm,∇∆φm) + (ε(θ)∇∆φm,∇∆φm),( 1 ε(θ) F ′(φm),∆2φm ) = ( ε′(θ) ε2(θ) ∇θF ′(φm),∇∆φm ) − ( 1 ε(θ) F ′′(φm)∇φm,∇∆φm ) . Therefore, as ε0 ≤ ε(·), one has 1 2 d dt ‖∆φm‖2 + ε0‖∇∆φm‖2 ≤ (∇u∇φm,∇∆φm) + (D2φmu,∇∆φm)− (ε′′(θ)∇θ∇θ · ∇φm,∇∆φm) − (ε′(θ)D2θ∇φm,∇∆φm)− (ε′(θ)D2φm∇θ,∇∆φm) − (ε′(θ)∇θ∆φm,∇∆φm)− ( ε′(θ) ε2(θ) ∇θF ′(φm),∇∆φm ) + ( 1 ε(θ) F ′′(φm)∇φm,∇∆φm ) := 8∑ i=1 Ii. (3.13) We estimate the right-hand side of (3.13) term by term by using Hölder, Young, Gagliardo-Nirenberg interpolation (2.2), and elliptic inequalities together with the fact that {φm} is bounded independently of m in L∞(0, T ;H1)∩L2(0, T ;H2) which was already proved in the previous Proposition. We denote by δ a positive constant that will be chosen later. Then, we have I1 ≤ δε0‖∇∆φm‖2 + C, I2 ≤ δε0‖∇∆φm‖2 + C‖∆φm‖2 + C, 10 J. H. LOPES, G. PLANAS EJDE-2022/72 where C depends on u. Since ε′′ is bounded, I3 ≤ δε0‖∇∆φm‖2 + C‖∇θ‖4L8‖φm‖H2‖∇φm‖ ≤ δε0‖∇∆φm‖2 + C‖θ‖4H2 (‖∆φm‖+ ‖φm‖) ≤ δε0‖∇∆φm‖2 + C‖∆φm‖2 + C, where C depends on ‖θ‖L∞(0,T ;H2). Next, since ε′ is bounded, I4 ≤ δε0‖∇∆φm‖2 + C‖θ‖H3‖θ‖H2‖∇φm‖H1‖∇φm‖ ≤ δε0‖∇∆φm‖2 + C‖θ‖H3 (‖∆φm‖+ ‖φm‖) ≤ δε0‖∇∆φm‖2 + C‖∆φm‖2 + C‖θ‖2H3 + C. Similarly, I5 ≤ δε0‖∇∆φm‖2 + C‖θ‖2H2‖φm‖H3‖φm‖H2 ≤ δε0‖∇∆φm‖2 + C (‖∇∆φm‖+ ‖∆φm‖+ ‖φm‖) (‖∆φm‖+ ‖φm‖) ≤ 3δε0‖∇∆φm‖2 + C‖∆φm‖2 + C and I6 ≤ 2δε0‖∇∆φm‖2 + C‖∆φm‖2 + C. As in (3.9), using that ε′ is bounded, I7 ≤ C ∫ Ω |∇θ| (β1|φm|+ β2) |∇∆φm|dx ≤ C‖∇θ‖L4‖φm‖L4‖∇∆φm‖+ C‖∇θ‖‖∇∆φm‖ ≤ 2δε0‖∇∆φm‖2 + C. Finally, as F ′′ is bounded, I8 ≤ δε0‖∇∆φm‖2 + C. By taking δ small enough and plugging the previous estimates into (3.13) we arrive at d dt ‖∆φm‖2 + ε0‖∇∆φm‖2 ≤ C‖∆φm‖2 + C‖θ‖2H3 + C. As θ ∈ L2(0, T ;H3), it follows that ‖∆φm‖2 + ε0 ∫ T 0 ‖∇∆φm‖2 dt ≤ C(T, θ, φ0, u). (3.14) We note again that the constant C is an increasing function on T , so it can be chosen independent of T for T in any finite interval. Hence, φm ∈ L∞(0, T ;H2) ∩ L2(0, T ;H3). This completes the proof. � 4. Proof of Theorem 2.1 To show the existence of a local solution to (1.1)-(1.5), we will use the semi- Galerkin method. We split the proof into four steps. In the first step, we introduce the approximate problem. In the second step, we show the existence of a local in time solution for the approximate problem. However, the time of existence depends on the approximate parameter. So, in the third step, we obtain a new differential inequality for some norms of the approximate solution which allows concluding the existence of small-time independent of the parameter. Finally, in the last step, EJDE-2022/72 NON-ISOTHERMAL NAVIER-STOKES-ALLEN-CAHN SYSTEM 11 we pass to the limit in the approximate problem showing the existence of a local solution to (1.1)-(1.5). Step 1: Approximate problem. Let wi and λi, i ∈ N, be the eigenfunctions and eigenvalues of the Stokes operator A. For m ∈ N, we denote by Pm the orthogonal projection from H onto Hm = span{w1, . . . , wm}. Fixed T > 0 and for m ∈ N, we consider the following approximate problem of finding um(x, t) = m∑ i=1 gmi (t)wi(x), φm and θm, satisfying (u′m, v) + (um · ∇um, v) + (ν(θm)Dum, Dv) = (( −∇ · (ε(θm)∇φm) + 1 ε(θm) F ′(φm) ) ∇φm, v ) − (∆θm∇θm, v), ∀ v ∈ Hm, (4.1) (φm)t + um · ∇φm = ∇ · (ε(θm)∇φm)− 1 ε(θm) F ′(φm) in Ω× (0, T ), (4.2) (θm)t + um · ∇θm = ∇ · ( k(θm)∇θm ) in Ω× (0, T ), (4.3) um(0) = Pmu0, φm(0) = φ0, θm(0) = θ0 in Ω, (4.4) ∂φm ∂η = ∂θm ∂η = 0 on ∂Ω× (0, T ). (4.5) Step 2: Existence of local solutions to the approximate problem. We have the following result. Lemma 4.1. There exist 0 < Tm ≤ T and um ∈ H1(0, Tm;Hm), φm, θm ∈ L∞(0, Tm;H2) ∩ L2(0, Tm;H3) such that (um, φm, θm) is the unique solution to (4.1)-(4.5). Proof. The existence of a unique local solution to this approximate problem can be done by using the Schauder Fixed Point Theorem. Indeed, given um ∈ C([0, T ];Hm) with um(t) = ∑m i=1 g m i (t)wi and (∑m i=1 |gmi (t)|2 )1/2 ≤ M , where M is a large enough constant that will be chosen later, there exists a unique θm ∈ L∞(0, T ;H2)∩ L2(0, T ;H3) with (θm)t ∈ L2(0, T ;L2) and ‖θm‖L∞ ≤ ‖θ0‖L∞ solution to (4.3). The well-posedness to (4.3) can be obtained by the Faedo-Galerkin method in the spirit of Section 3, see also [18, 28]. Moreover, it holds that ‖θm‖L∞(0,T ;H2)∩L2(0,T ;H3) ≤ Cm,M . (4.6) By Propositions 3.1 and 3.2, there exists a unique φm ∈ L∞(0, T ;H2)∩L2(0, T ;H3) with (φm)t ∈ L2(0, T ;L2) and ‖φm‖L∞ ≤ 1 solution to (4.2). Moreover, from (3.7), (3.10) and (3.14) it follows that ‖φm‖L∞(0,T ;H2)∩L2(0,T ;H3) ≤ Cm,M . (4.7) Once θm, φm are determined, we turn to look for functions ûm = ∑m i=1 ĝ m i (t)wi(x) satisfying (4.1), which is a system of m non-linear ordinary differential equations for {ĝmi (t)}mi=1. From the assumptions on the coefficients ν and ε, it is standard to show the local well-posedness of the initial value problem using the classical theory 12 J. H. LOPES, G. PLANAS EJDE-2022/72 of ordinary differential equations. To see that ûm is defined on the whole interval [0, T ], we take ûm as a test function in (4.1) to obtain 1 2 d dt ‖ûm‖2 + ν0‖Dûm‖2 ≤ −(∇ · (ε(θm)∇φm)∇φm, ûm) + ( 1 ε(θm) F ′(φm)∇φm, ûm ) − (∆θm∇θm, ûm). (4.8) We proceed to estimate the right-hand side of (4.8) term by term. For the first term, by using that ε and ε′ are bounded, Gagliardo-Nirenberg inequality (2.2), and elliptic estimates, we have − ( (ε′(θm)∇θm · ∇φm)∇φm, ûm ) − (ε(θm)∆φm∇φm, ûm) ≤ C‖ûm‖L∞‖∇θm‖‖∇φm‖2L4 + C‖ûm‖L∞‖∆φm‖‖∇φm‖ ≤ ν0 4 ‖Dûm‖2 + Cm‖∇θm‖2‖φm‖2H2 + Cm‖∆φm‖2‖∇φm‖2. Using the fact that ε0 ≤ ε(·) and that ‖φm‖L∞ ≤ 1 to estimate F ′, it follows( 1 ε(θm) F ′(φm)∇φm, ûm ) ≤ ν0 4 ‖Dûm‖2 + C‖∇φm‖2. Finally, rewriting the last term we have that −(∆θm∇θm, ûm) = (∇θm ⊗∇θm,∇ûm) ≤ ν0 4 ‖Dûm‖2 + C‖∇θm‖4. Therefore, (4.8) becomes d dt ‖ûm‖2 + ν0 2 ‖Dûm‖2 ≤ Cm ( ‖∇θm‖2‖φm‖2H2 + ‖∆φm‖2‖∇φm‖2 + ‖∇φm‖2 + ‖∇θm‖4 ) . Integration in time, (4.6) and (4.7) allow us to conclude that ‖ûm(t)‖2 + ν0 ∫ t 0 ‖Dûm‖2dt ≤ ‖u0‖2 + Cm,MT. (4.9) Hence, ûm ∈ L∞(0, T ;H) ∩ L2(0, T ;V ). In this way, we obtain a well-defined operator ΦmT : C([0, T ];Hm) → ( L∞(0, T ;H2) ∩ L2(0, T ;H3) )2 → H1(0, T ;Hm) um 7→ (φm, θm) 7→ ûm . We next show that the operator ΦmT is continuous. As it was shown in [28, pp. 432-433], we can assure that θm is continuous with respect to um. More precisely, let θm = θ1 m − θ2 m and um = u1 m − u2 m, then it holds ‖θm‖L∞(0,T ;L2)∩L2(0;T ;H1) ≤ Cm‖um‖L2(0,T ;H). To show that φm is continuous with respect to um, let φ1 m and φ2 m be two solutions to (4.2). So, φm = φ1 m − φ2 m satisfies ∂tφm + (u1 m − u2 m) · ∇φ1 m + u2 m · ∇φm = ∇ · ( ε(θ1 m)∇φm ) +∇ · ( (ε(θ1 m)− ε(θ2 m))∇φ2 m ) − 1 ε(θ1 m) F ′(φ1 m) + 1 ε(θ2 m) F ′(φ2 m). EJDE-2022/72 NON-ISOTHERMAL NAVIER-STOKES-ALLEN-CAHN SYSTEM 13 Multiplying by φm, integrating over Ω and denoting 1 ε(s) = G(s), we obtain 1 2 d dt ‖φm‖2 + ε0‖∇φm‖2 ≤ ( um · ∇φm, φ1 m ) − ( (ε(θ1 m)− ε(θ2 m))∇φ2 m,∇φm ) + ‖F ′′‖L∞ ∫ Ω |G(θ1 m)||φm|2dx+ ‖G′‖L∞ ∫ Ω |F ′(φ2 m)||θm||φm|dx, where we have used the Mean Value Theorem to estimate the last two terms as follows − ( 1 ε(θ1 m) F ′(φ1 m)− 1 ε(θ2 m) F ′(φ2 m), φm ) = − ( G(θ1 m)(F ′(φ1 m)− F ′(φ2 m)), φm ) − ( (G(θ1 m)−G(θ2 m))F ′(φ2 m), φm ) ≤ ‖F ′′‖L∞ ∫ Ω |G(θ1 m)||φm|2dx+ ‖G′‖L∞ ∫ Ω |F ′(φ2 m)||θm||φm|dx. Then, we have 1 2 d dt ‖φm‖2 + ε0‖∇φm‖2 ≤ ‖um‖‖∇φm‖‖φ1 m‖L∞ + C‖φm‖2 + C‖θm‖2 + ‖ε′‖L∞ ∫ Ω |θm||∇φ2 m||∇φm| dx ≤ ε0 4 ‖∇φm‖2 + C(‖um‖2 + ‖φm‖2 + ‖θm‖2 + ‖θm‖L4‖∇φ2 m‖L4‖∇φm‖) ≤ ε0 2 ‖∇φm‖2 + C(‖um‖2 + +‖φm‖2 + ‖θm‖2H1) and conclude that d dt ‖φm‖2 + ε0‖∇φm‖2 ≤ C(‖φm‖2 + ‖um‖2 + ‖θm‖2H1). Gronwall Lemma and the continuity of θm with respect to um yield ‖φm‖2 ≤ C(‖um‖2L2(0,T ;H) + ‖θm‖2L2(0,T ;H1)) ≤ C‖um‖ 2 L2(0,T ;H), which implies the continuity of φm with respect to um. It is not difficult to see the continuity of the solution ûm to the system of ordinary differential equations (4.1) with respect to φm and θm. Indeed, we can consider ûm = û1 m − û2 m, where ûim, i = 1, 2 are two solutions to (4.1). Writing the system that ûm satisfies and taking ûm as test function, we can deduce that ‖ûm‖2L∞(0,T ;H) ≤ Cm(‖φm‖2L2(0,T ;H1) + ‖θm‖2L2(0,T ;H1)). We only sketch how to deal with the higher-order nonlinear term since the other terms are easier. Notice that, by integrating by parts twice, we can rewrite − (∇ · (ε(θ1 m)∇φ1 m)∇φ1 m, ûm) = (ε(θ1 m)∇φ1 mD 2φ1 m, ûm) + (ε(θ1 m)∇φ1 m ⊗∇φ1 m,∇ûm) = (ε(θ1 m) 1 2 ∇|∇φ1 m|2, ûm) + (ε(θ1 m)∇φ1 m ⊗∇φ1 m,∇ûm) = −1 2 (ε′(θ1 m)∇θ1 m|∇φ1 m|2, ûm) + (ε(θ1 m)∇φ1 m ⊗∇φ1 m,∇ûm), 14 J. H. LOPES, G. PLANAS EJDE-2022/72 where we have used that ∇ · ûm = 0. We proceed similarly for i = 2. Hence, rearranging terms, − (∇ · (ε(θ1 m)∇φ1 m)∇φ1 m, ûm) + (∇ · (ε(θ2 m)∇φ2 m)∇φ2 m, ûm) = −1 2 ((ε′(θ1 m)− ε′(θ2 m))∇θ1 m|∇φ1 m|2, ûm)− 1 2 (ε′(θ2 m)∇θm|∇φ1 m|2, ûm) − 1 2 (ε′(θ2 m)∇θ2 m∇φm · ∇φ1 m, ûm)− 1 2 (ε′(θ2 m)∇θ2 m∇φ2 m · ∇φm, ûm) + ((ε(θ1 m)− ε(θ2 m))∇φ1 m ⊗∇φ1 m,∇ûm) + (ε(θ2 m)∇φm ⊗∇φ1 m,∇ûm) + (ε(θ2 m)∇φ2 m ⊗∇φm,∇ûm). Since ε′ and ε′′ are bounded and, thus, ε and ε′ are Lipschitz continuous functions, the terms on the right-hand side can be estimated by using Hölder, Young and Gagliardo-Nirenberg inequalities. For instance, for the first term |((ε′(θ1 m)− ε′(θ2 m))∇θ1 m|∇φ1 m|2, ûm)| ≤ C‖θm‖L4‖∇θ1 m‖L4‖∇φ1 m‖2L4‖ûm‖L∞ ≤ Cm‖θm‖H1‖θ1 m‖H2‖φ1 m‖H2‖ûm‖L2 ≤ Cm,M (‖θm‖2H1 + ‖ûm‖2L2), where we have used (4.6), (4.7) and the fact that all norms on a finite-dimensional vector space are equivalent. The remainder terms are treated similarly. Thenceforth, ΦmT is a continuous operator. Since Hm is a finite dimensional space, H1(0, T ;Hm) is compact in C([0, T ];Hm). Moreover, as ΦmT is a bounded operator from C([0, T ];Hm) into H1(0, T ;Hm), we conclude that ΦmT is a compact operator from C([0, T ];Hm) into itself. From (4.9), by taking M2 = ‖u0‖2/2, we can choose Tm small enough such that we are able to apply the Schauder Fixed Point Theorem and conclude that there exists (um, φm, θm) solution to the approximate problem (4.1)-(4.5) defined on [0, Tm]. The proof of the uniqueness of the solution is standard, so we omit the details. � Step 3: Local existence time independent of m. We will construct a new differ- ential inequality and combine it with a small data argument for the shifted approx- imate solution ũm(t) = um(t)− um(0), φ̃m(t) = φm(t)− φ0 and θ̃m(t) = θm(t)− θ0 to show the existence of small time T ∗ > 0 such the solution (um, φm, θm) to (4.1)- (4.5) is defined on [0, T ∗]. First, we rewrite the approximate problem in terms of the shifted approximate solution. Starting with the equation for the mean velocity, (ũ′m, v) + ((ũm + um(0)) · ∇ũm, v) + (ν(θ̃m + θ0)Dũm, Dv) = −((ũm + um(0)) · ∇um(0), v)− (ν(θ̃m + θ0)Dum(0), Dv) + (( −∇ · (ε(θ̃m + θ0)∇(φ̃m + φ0)) + 1 ε(θ̃m + θ0) F ′(φ̃m + φ0) ) ∇(φ̃m + φ0), v ) − (∆θ̃m∇(θ̃m + θ0), v)− (∆θ0∇(θ̃m + θ0), v),∀ v ∈ Hm, ũm(0) = 0. (4.10) EJDE-2022/72 NON-ISOTHERMAL NAVIER-STOKES-ALLEN-CAHN SYSTEM 15 For the phase-field, we have that ∂tφ̃m + (ũm + um(0)) · ∇φ̃m −∇ · (ε(θ̃m + θ0)∇φ̃m) = ∇ · (ε(θ̃m + θ0)∇φ0)− (ũm + um(0)) · ∇φ0 − 1 ε(θ̃m + θ0) F ′(φ̃m + φ0) (4.11) with φ̃m(0) = 0 and homogeneous Neumann boundary condition. For the temperature, it holds ∂tθ̃m + ũm · ∇(θ̃m + θ0)−∇ · (k(θ̃m + θ0)∇θ̃m) = ∇ · (k(θ̃m + θ0)∇θ0)− um(0) · ∇(θ̃m + θ0), (4.12) with θ̃m(0) = 0 and homogeneous Neumann boundary condition. Next, we take v = ũm in (4.10), 1 2 d dt ‖ũm‖2 + ν0‖Dũm‖2 ≤ − ((ũm + um(0)) · ∇um(0), ũm)− (ν(θ̃m + θ0)Dum(0), Dũm) − (∇ · (ε(θ̃m + θ0)∇φ̃m)∇φ̃m, ũm)− (∇ · (ε(θ̃m + θ0)∇φ0)∇φ̃m, ũm) − (∇ · (ε(θ̃m + θ0)∇φ̃m)∇φ0, ũm)− (∇ · (ε(θ̃m + θ0)∇φ0)∇φ0, ũm) + (F ′(φ̃m + φ0) ε(θ̃m + θ0) ∇φ̃m, ũm ) + (F ′(φ̃m + φ0) ε(θ̃m + θ0) ∇φ0, ũm ) − (∆θ̃m∇(θ̃m + θ0), ũm)− (∆θ0∇(θ̃m + θ0), ũm) := 10∑ i=1 Ii. (4.13) We estimate the right-hand side of (4.13) by using Hölder, Young, Ladyzhenskaya (2.1), Gagliardo-Nirenberg interpolation (2.2), and elliptic inequalities. Recall that φm and θm satisfy the maximum principle, i.e., ‖φm‖L∞ ≤ 1 and ‖θm‖L∞ ≤ ‖θ0‖L∞ , so ‖φ̃m‖ ≤ C‖φ0‖L∞ and ‖θ̃m‖ ≤ C‖θ0‖L∞ ; hence, ‖∇φ̃m‖L4 ≤ C‖φ̃m‖1/2H2 ‖φ̃m‖1/2L∞ ≤ C(‖∆φ̃m‖1/2 + 1). A similar estimate holds for θ̃m. Moreover, it holds ‖um(0)‖V ≤ ‖u0‖V . We denote by δ a positive constant that will be chosen later. Then we have I1 ≤ ‖u0‖V ‖ũm‖2L4 + C‖u0‖2V ‖ũm‖L4 ≤ C(‖∇ũm‖‖ũm‖+ ‖∇ũm‖) ≤ 2δν0‖Dũm‖2 + C‖ũm‖2 + C. For the next term, we use that ν is bounded, I2 ≤ C‖u0‖V ‖Dũm‖ ≤ δν0‖Dũm‖2 + C. Rewriting I3 and using that ε and ε′ are bounded, it follows that I3 = −((ε′(θ̃m + θ0)∇(θ̃m + θ0) · ∇φ̃m)∇φ̃m, ũm)− (ε(θ̃m + θ0)∆φ̃m∇φ̃m, ũm) ≤ C(‖∇θ̃m‖L4 + ‖∇θ0‖L4)‖∇φ̃m‖2L4‖ũm‖L4 + C‖∆φ̃m‖‖∇φ̃m‖L4‖ũm‖L4 ≤ C(‖∆θ̃m‖1/2 + 1)(‖∆φ̃m‖+ 1)‖∇ũm‖1/2‖ũm‖1/2 16 J. H. LOPES, G. PLANAS EJDE-2022/72 + C‖∆φ̃m‖(‖∆φ̃m‖1/2 + 1)‖∇ũm‖1/2‖ũm‖1/2 = C‖∆θ̃m‖1/2‖∆φ̃m‖‖∇ũm‖1/2‖ũm‖1/2 + C‖∆θ̃m‖1/2‖∇ũm‖1/2‖ũm‖1/2 + C‖∆φ̃m‖‖∇ũm‖1/2‖ũm‖1/2 + C‖∇ũm‖1/2‖ũm‖1/2 + C‖∆φ̃m‖3/2‖∇ũm‖1/2‖ũm‖1/2 ≤ 3δν0‖Dũm‖2 + 3δε0‖∆φ̃m‖2 + 2δk0‖∆θ̃m‖2 + C‖ũm‖2(‖Dũm‖2 + 1) + C. Similarly for the next three terms I4 = −((ε′(θ̃m + θ0)∇(θ̃m + θ0) · ∇φ0)∇φ̃m, ũm)− (ε(θ̃m + θ0)∆φ0∇φ̃m, ũm) ≤ C‖∇θ̃m‖L4‖∇φ̃m‖L4‖ũm‖L4 + C‖∇φ̃m‖L4‖ũm‖L4 ≤ C(‖∆θ̃m‖1/2 + 1)(‖∆φ̃m‖1/2 + 1)‖∇ũm‖1/2‖ũm‖1/2 + C(‖∆φ̃m‖1/2 + 1)‖∇ũm‖ ≤ 3δν0‖Dũm‖2 + 3δε0‖∆φ̃m‖2 + 2δk0‖∆θ̃m‖2 + C‖ũm‖2(‖Dũm‖2 + 1) + C, I5 = −((ε′(θ̃m + θ0)∇(θ̃m + θ0) · ∇φ̃m)∇φ0, ũm)− (ε(θ̃m + θ0)∆φ̃m∇φ0, ũm) ≤ C‖θ̃m‖1/2H2 ‖φ̃m‖1/2H2 ‖∇ũm‖1/2‖ũm‖1/2 + C‖φ̃m‖1/2H2 ‖∇ũm‖ + C‖∆φ̃m‖‖ũm‖1/2‖∇ũm‖1/2 ≤ 4δν0‖Dũm‖2 + 4δε0‖∆φ̃m‖2 + 2δk0‖∆θ̃m‖2 + C‖ũm‖2(‖Dũm‖2 + 1) + C, and I6 = −((ε′(θ̃m + θ0)∇(θ̃m + θ0) · ∇φ0)∇φ0, ũm)− (ε(θ̃m + θ0)∆φ0∇φ0, ũm) ≤ δk0‖∆θ̃m‖2 + 3δν0‖∇ũm‖2 + C‖ũm‖2 + C. Since ‖φ̃m‖ ≤ C‖φ0‖L∞ , ‖θ̃m‖ ≤ C‖θ0‖L∞ and ε0 ≤ ε(·) it follows that I7 ≤ C‖∇φ̃m‖‖ũm‖ ≤ C‖∇φ̃m‖2 + C‖ũm‖2, I8 ≤ C‖ũm‖2 + C. We observe that I9 will be canceled later. So, for the last term I10 ≤ ‖∆θ0‖(‖∇θ̃m‖L4 + ‖∇θ0‖L4)‖ũm‖L4 ≤ C(‖∆θ̃m‖1/2 + 1)‖∇ũm‖ ≤ δk0‖∆θ̃m‖2 + δν0‖Dũm‖2 + C. Now, we multiply (4.11) by −∆φ̃m, integrate over Ω and sum up, to arrive at 1 2 d dt ‖∇φ̃m‖2 + ε0‖∆φ̃m‖2 ≤ ((ũm + um(0)) · ∇φ̃m,∆φ̃m) + ((ũm + um(0)) · ∇φ0,∆φ̃m) − (ε′(θ̃m + θ0)∇(θ̃m + θ0) · ∇φ̃m,∆φ̃m) − (ε′(θ̃m + θ0)∇(θ̃m + θ0)∇φ0,∆φ̃m)− (ε(θ̃m + θ0)∆φ0,∆φ̃m) + (F ′(φ̃m + φ0) ε(θ̃m + θ0) ,∆φ̃m ) := 16∑ i=11 Ii. (4.14) EJDE-2022/72 NON-ISOTHERMAL NAVIER-STOKES-ALLEN-CAHN SYSTEM 17 We proceed to estimate (4.14) term by term in an analogous way as the previous estimates. We start with I11 ≤ (‖ũm‖L4 + ‖um(0)‖L4)‖∇φ̃m‖L4‖∆φ̃m‖ ≤ C(‖∇ũm‖1/2‖ũm‖1/2 + 1)(‖∆φ̃m‖1/2 + 1)‖∆φ̃m‖ ≤ C‖∆φ̃m‖3/2‖∇ũm‖1/2‖ũm‖1/2 + C‖∆φ̃m‖‖∇ũm‖1/2‖ũm‖1/2 + C‖∆φ̃m‖3/2 + C‖∆φ̃m‖ ≤ 4δε0‖∆φ̃m‖2 + δν0‖Dũm‖2 + C‖ũm‖2(‖Dũm‖2 + 1) + C, and similarly I12 ≤ C(‖∇ũm‖1/2‖ũm‖1/2 + 1)‖∆φ̃m‖ ≤ δε0‖∆φ̃m‖2 + δν0‖Dũm‖2 + C‖ũm‖2 + C. Since ε′ is bounded, it follows I13 ≤ (‖∇θ̃m‖L4 + ‖∇θ0‖L4)‖∇φ̃m‖L4‖∆φ̃m‖ ≤ C(‖∇θ̃m‖1/2H1 ‖∇θ̃m‖1/2 + 1)(‖∆φ̃m‖1/2 + 1)‖∆φ̃m‖ ≤ C(‖∆θ̃m‖1/2 + 1)‖∇θ̃m‖1/2(‖∆φ̃m‖3/2 + ‖∆φ̃m‖) + C‖∆φ̃m‖3/2 + C‖∆φ̃m‖ ≤ 6δε0‖∆φ̃m‖2 + δk0‖∆θ̃m‖2 + C‖∇θ̃m‖2(‖∆θ̃m‖2 + 1) + C. By using again that ε′ is bounded, one has I14 ≤ (‖∇θ̃m‖L4 + ‖∇θ0‖L4)‖∇φ0‖L4‖∆φ̃m‖ ≤ C(‖∆θ̃m‖1/2 + 1)‖∆φ̃m‖ ≤ δε0‖∆φ̃m‖2 + δk0‖∆θ̃m‖2 + C. As ε is bounded, I15 ≤ δε0‖∆φ̃m‖2 + C. For the last term, we use that ‖φ̃m‖ ≤ C and that ε0 ≤ ε(·) to obtain I16 ≤ δε0‖∆φ̃m‖2 + C. We multiply (4.12) by −∆θ̃m, integrate over Ω and sum up, to arrive at 1 2 d dt ‖∇θ̃m‖2 + k0‖∆θ̃m‖2 ≤ (ũm · ∇(θ̃m + θ0),∆θ̃m) + (um(0) · ∇(θ̃m + θ0),∆θ̃m) − (k′(θ̃m + θ0)∇(θ̃m + θ0) · ∇θ̃m,∆θ̃m) − (k′(θ̃m + θ0)∇(θ̃m + θ0) · ∇θ0,∆θ̃m)− (k(θ̃m + θ0)∆θ0,∆θ̃m) := 21∑ i=17 Ii. (4.15) We note that I17 = −I9, so it will be canceled later. I18 ≤ ‖um(0)‖L4(‖∇θ̃m‖L4 + ‖∇θ0‖L4)‖∆θ̃m‖ ≤ C(‖∆θ̃m‖1/2 + 1)‖∆θ̃m‖ 18 J. H. LOPES, G. PLANAS EJDE-2022/72 ≤ 2δk0‖∆θ̃m‖2 + C. Since k′ is bounded, in a similar way as for I13, we have that I19 ≤ (‖∇θ̃m‖L4 + ‖∇θ0‖L4)‖∇θ̃m‖L4‖∆θ̃m‖ ≤ C(‖∇θ̃m‖1/2H1 ‖∇θ̃m‖1/2 + 1)(‖∆θ̃m‖1/2 + 1)‖∆θ̃m‖ ≤ 5δk0‖∆θ̃m‖2 + C‖∇θ̃m‖2(‖∆θ̃m‖2 + 1) + C. Using again that k′ is bounded, it follows, analogously to I18, I20 ≤ δk0‖∆θ̃m‖2 + C. Finally, as k is bounded, I21 ≤ δk0‖∆θ̃m‖2 + C. By adding (4.13), (4.14) and (4.15), using estimates for I1 to I21, and taking δ small enough, we infer that d dt A(t) +B(t) ≤ CA(t)(B(t) + 1) + C (4.16) where A(t) = ‖ũm‖2 + ‖∇φ̃m‖2 + ‖∇θ̃m‖2, B(t) = ν0‖Dũm‖2 + ε0‖∆φ̃m‖2 + k0‖∆θ̃m‖2. We will prove that A(t) < 1 2C for t ∈ [0, T ∗], (4.17) where T ∗ < 1 4C( 1 2 + C) . (4.18) Since A(0) = 0, (4.17) holds for small times. Suppose by contradiction that there exists T ∗m ∈ [0, T ∗) such that A(t) < 1 2C in [0, T ∗m) and A(T ∗m) = 1 2C . (4.19) Using (4.19) into (4.16), it follows that d dt A(t) +B(t) ≤ C 1 2C (B(t) + 1) + C, which implies d dt A(t) + 1 2 B(t) ≤ 1 2 + C. Consequently, by (4.18), for t ∈ [0, T ∗m], A(t) ≤ ( 1 2 + C)T ∗ < 1 4C < 1 2C , leading to a contradiction with the definition of T ∗m given by (4.19). Therefore, (4.17) holds for any t ∈ [0, T ∗] and T ∗ is independent of m. Step 4: Passing to the limit as m → +∞. From the previous step, it follows that {um} is bounded in L∞(0, T ∗;H) ∩ L2(0, T ∗;V ), {φm}, {θm} are bounded in L∞(0, T ∗;H1 ∩ L∞) ∩ L2(0, T ∗;H2), independently of m. EJDE-2022/72 NON-ISOTHERMAL NAVIER-STOKES-ALLEN-CAHN SYSTEM 19 To pass to the limit on the nonlinear terms, it is necessary to obtain some strong convergences. To this end, observe that from (4.1) ‖(um)t‖V ′ ≤ C(‖um‖2L4 + ‖Dum‖+ ‖∇φm‖L4‖D2φm‖ + ‖∇φm‖2L4 + ‖∇φm‖+ ‖∇θm‖2L4). By Gagliardo-Nirenberg interpolation inequality (2.2) and the fact that ‖φm‖L∞ ≤ 1 and ‖θm‖L∞ ≤ ‖θ0‖L∞ , it follows that ‖∇φm‖2L4 ≤ C‖φm‖H2‖φm‖L∞ ≤ C‖φm‖H2 and ‖∇θm‖2L4 ≤ C‖θm‖H2 . Ladyzhenskaya inequality (2.1) together with the previous estimates, give us ‖(um)t‖V ′ ≤ C ( ‖um‖V + ‖∇um‖‖um‖+ ‖φm‖3/2H2 + ‖φm‖H2 + ‖θm‖H2 ) . Therefore, {(um)t} is bounded in L 4 3 (0, T ∗;V ′) independently of m. From (4.2), Ladyzhenskaya inequality (2.1), Gagliardo-Nirenberg interpolation inequality (2.2), ‖φm‖L∞ ≤ 1, and the fact that ε and ε′ are bounded, we see that ‖(φm)t‖ ≤ ‖um‖L4‖∇φm‖L4 + ‖∇φm‖L4‖∇θm‖L4 + ‖∆φm‖+ C ≤ C ( ‖um‖‖∇um‖+ ‖φm‖H2 + ‖θm‖H2 + 1 ) , hence, {(φm)t} is bounded in L2(0, T ∗;L2). Similarly, from (4.3), it follows that ‖(θm)t‖ ≤ C ( ‖um‖‖∇um‖+ ‖θm‖H2 + 1 ) . Thus, {(θm)t} is bounded in L2(0, T ∗;L2). By the uniform estimates in m and the compactness lemma, there exist functions u ∈ L∞(0, T ∗;H) ∩L2(0, T ∗;V ), φ, θ ∈ L∞(0, T ∗;H1 ∩ L∞) ∩ L2(0, T ∗;H2) such that, up subsequences, as m→∞, um ⇀ u in L2(0, T ∗;V ); um ∗ ⇀ u in L∞(0, T ∗;H); um → u in L2(0, T ∗;H); φm, θm ⇀ φ, θ in L2(0, T ∗;H2); φm, θm ∗ ⇀ φ, θ in L∞(0, T ∗;H1); φm, θm → φ, θ in L2(0, T ∗;H1) ∩ C([0, T ∗];Lp), 1 ≤ p <∞. All the previous convergences allow us to pass to the limit in (4.1)-(4.5), noting that ν(θm)→ ν(θ), ε(θm)→ ε(θ), k(θm)→ k(θ), and F ′(φm)→ F ′(φ) strongly in C([0, T ∗];Lp), 1 ≤ p <∞. We only show the convergence of the terms involving ε(θm) in the velocity equa- tion (4.1), since the convergence of the other terms is standard. We observe that − ( ∇ · (ε(θ)∇φ)∇φ,wj ) = ( ε(θ)∇φ⊗∇φ,∇wj) + ( ε(θ)∇φD2φ,wj ) . For any ψ ∈ C∞0 ([0, T ∗]), we can write∫ T∗ 0 ( − ( ∇ · (ε(θ)∇φ)∇φ,wj ) + ( ∇ · (ε(θm)∇φm)∇φm, wj )) ψ(t)dt = ∫ T∗ 0 ( ε(θ)∇φ⊗∇φ− ε(θm)∇φm ⊗∇φm,∇wj ) ψ(t)dt 20 J. H. LOPES, G. PLANAS EJDE-2022/72 + ∫ T∗ 0 ( ε(θ)∇φD2φ− ε(θm)∇φmD2φm, wj ) ψ(t)dt := J1 + J2. For the first term, as ε is bounded, we have that |J1| ≤ ∫ T∗ 0 ∣∣((ε(θ)− ε(θm))∇φ⊗∇φ,∇wj ) ψ(t) ∣∣dt + ∫ T∗ 0 ∣∣(ε(θm)(∇φ−∇φm)⊗∇φ,∇wj ) ψ(t) ∣∣dt + ∫ T∗ 0 ∣∣(ε(θm)∇φm ⊗ (∇φ−∇φm),∇wj ) ψ(t) ∣∣dt ≤ C‖ε(θ)− ε(θm)‖L∞(0,T∗;L2)‖φ‖2L2(0,T∗;H2) + C‖φm − φ‖L2(0,T∗;H1)‖φ‖L2(0,T∗;H1) + C‖φm‖L2(0,T∗;H1)‖φm − φ‖L2(0,T∗;H1) which converges to 0 as m→∞. For the second term, |J2| ≤ ∣∣∣∫ T∗ 0 ( ε(θ)∇φ(D2φ−D2φm), wj ) ψ(t)dt ∣∣∣ + ∫ T∗ 0 ∣∣(ε(θ)(∇φ−∇φm)D2φm, wj ) ψ(t) ∣∣dt + ∫ T∗ 0 ∣∣((ε(θ)− ε(θm))∇φmD2φm, wj ) ψ(t) ∣∣dt := J21 + J22 + J23. Since ε is bounded and ∇φ ∈ L2(0, T ∗;L2), it follows that J21 → 0 as m→∞. For the next term, J22 ≤ C‖φm − φ‖L2(0,T∗;H1)‖φm‖L2(0,T∗;H2) so, J22 → 0 as m→∞. Finally, J23 ≤ C‖ε(θ)− ε(θm)‖L∞(0,T∗;L4)‖φm‖2L2(0,T∗;H2), which implies that J23 → 0 as m→∞. We treat now the other term in the velocity equation concerning with ε. We have that∣∣∣∫ T∗ 0 ( 1 ε(θ) F ′(φ)∇φ,wj ) ψ(t)− ( 1 ε(θm) F ′(φm)∇φm, wj ) ψ(t)dt ∣∣∣ ≤ ∫ T∗ 0 ∣∣∣(( 1 ε(θ) − 1 ε(θm) ) F ′(φ)∇φ,wj ) ψ(t) ∣∣∣dt + ∫ T∗ 0 ∣∣∣( 1 ε(θm) (F ′(φ)− F ′(φm))∇φ,wj ) ψ(t) ∣∣∣dt + ∫ T∗ 0 ∣∣∣( 1 ε(θm) F ′(φm)(∇φ−∇φm), wj ) ψ(t) ∣∣∣dt. Since ε0 ≤ ε(·), ε and F ′ are bounded (because ‖φm‖L∞ ≤ 1), we can pass to the limit as m→∞ and show that the integral converges to zero. EJDE-2022/72 NON-ISOTHERMAL NAVIER-STOKES-ALLEN-CAHN SYSTEM 21 Thus, we conclude that (u, φ, θ) is a local solution to (1.1)-(1.5). The proof of Theorem 2.1 is complete. � 5. Conclusion We have considered a general system (1.1)-(1.5) involving temperature depen- dence on all main coefficients. We have proved the existence of local in time so- lutions in the two-dimensional case without any restriction on the size of initial conditions. This is due to the strong non-linear couplings between those equations due to the temperature dependence which brings new mathematical challenges. Since lower and higher order estimates cannot be obtained in a separate way, we have derived a novel higher order differential inequality for the shifted functions and combine it with a small time argument. It will be interesting to consider the three-dimensional case. However, our argu- ment fails in this case and other techniques should be developed. Acknowledgments. J. H. 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Juliana Honda Lopes Departamento de Matemática, Instituto de Matemática, Estat́ıstica e Computação, Cient́ıfica, Universidade Estadual de Campinas, Rua Sergio Buarque de Holanda, 651, 13083-859, Campinas - SP, Brazil Email address: juhonlopes@gmail.com Gabriela Planas Departamento de Matemática, Instituto de Matemática, Estat́ıstica e Computação Cient́ıfica, Universidade Estadual de Campinas, Rua Sergio Buarque de Holanda, 651, 13083-859, Campinas - SP, Brazil Email address: gplanas@unicamp.br 1. Introduction 2. Notation and main result 3. Phase-field equation with thermo-induced interfacial thickness 4. Proof of Theorem ?? 5. Conclusion Acknowledgments References