Existence and regularity results for semi‐linearized compressible 2D fluids with generalized diffusion

We consider semi‐linearized compressible Navier–Stokes equations, in a two‐dimensional periodic domain, introducing in this scheme generalized diffusive terms. After rewriting the equations in terms of a potential flow for the velocity, we prove existence and regularity for a suitable class of weak solutions.

The basic choice behind the introduction ofF(u) in the above model, is related to the idea of Prouse 1 who suggested the possibility that the standard viscous stress S for the incompressible Navier-Stokes equations is suitable in the case of relatively low flow velocities, while for rather high velocities a modification of the type introduced above is reasonable. Here, we make similar assumptions, although for moderately high values of the flow velocity (also assuming moderate mean flow velocity values), and not necessarily in the presence of turbulence: Due to these properties of the fluid flow we consider an intermediate scheme between the Navier-Stokes equations and system (1.1), that is, (u t + (u · ∇)u) + ∇p = Δu + ( + )∇(divu) + 1 ∇(divF(u)), in Q T , t + div( u) = 0, in Q T . (1.2) In this specific circumstance, instead of taking the viscous stress S of the usual type S = 2 D(u) + divu I, for compressible flows, with D(u) = sym∇u = ( ∇u + ∇u ⊤ ) ∕2 and I ∶= I 2×2 = I the second-rank unit tensor, we assume for S an augmented structure given by S = 2 D(u) + divu I + 1 divF(u) I, that, in components, can be rewritten as follows with i, , k = 1, 2 and where ij denotes the Kronecker delta function. We emphasize that with respect to (1.1) 1 , here ) , i, = 1, 2, has been replaced with Here and in what follows, all the fields involved in the considered problem are assumed to be periodic in x = (x 1 , x 2 ) with period 1.

PRELIMINARIES
Given q ≥ 1, by L q (D), we denote the customary Lebesgue spaces with norm || · || q . In particular, we have that || · || ∶= || · || L 2 (D) and (· , ·) denotes the L 2 -inner product. Moreover, by W k, q (D), k a non-negative integer and q as before, we denote the usual Sobolev spaces with norm || · || k, q . For k > 0, and q = 2, we use the notation H k = W k,2 (D). The dual of W 1, q (D) is denoted by W −1, q ′ with norm || · || −1, q ′ ; the dual of H q is denoted by H −q , and ⟨· , · ⟩ = ⟨· , · ⟩ H −1 ,H 1 indicates the duality pairing. Similarly we define Lebesgue and Sobolev spaces on Q T = (0, T) × D, T > 0, that is, L q (Q T ), q ≥ 1, with norm || · || L q (Q T ) . Let us recall that all the considered fields are periodic of period 1 in the variables x 1 and In what follows, in order to keep the notation compact, we use the same kind of symbols for scalar and vector fields (the same convention is used also for the related spaces, without specifying the space dimension or the number of the components), distinguishing the different cases only when it is strictly required by the context. Let X be a real Banach space with norm || · || X . We consider the usual Bochner spaces L q (0, T; X), with norm denoted by || · || L q (0,T;X) . Moreover, C([0, T]; X) denotes the space of the continuous functions taking values in X. We also use the Orlicz space L (D) associated to the convex function = (r), for r ≥ 0, given by (r) = (1 + r) ln(1 + r) − r (which is equal to ln − + 1, when = 1 + r), see, for example, Vaigant and Kazhikhov. 3 Again, in this case, we use the notation || · || ∶= || · || L (D) .
In most of the cases we omit the explicit dependence on D in the considered Lebesgue, Sobolev, Bochner, and Orlicz spaces.
Here and in the sequel, we denote by C or c positive constants that may assume different values, even in the same equation; when the constants depend on quantities of interest, these are explicitly placed in parentheses or as subscripts.

Basic estimates
Let us recall classical Ladyzhenskaya's inequality (see, e.g., Ladyzhenskaya 6 ). For v ∈ H 1 we have with C depending only on the domain D.
We will also use the following Gagliardo-Nirenberg inequalities on D (see, e.g., Nirenberg 7 ), that is, where C depends on Lebesgue space indexes, the exponents to the interpolating norms, the order of the involved derivatives, as well as on the domain D. Furthermore, we will also exploit Hölder's and Young's inequalities.

MAIN RESULTS
Introducing suitable assumptions onF, we state the main theorems of this paper.

System (1.6) in reduced form
Integrating the momentum equation (1.6) 1 one obtains the following coupled with t + div ( ∇ ) = 0 in Q T . As mentioned in Section 1, the structure ofF, in the principal part of this study, is as followsF with > 0. In the sequel we always assume = = 1∕4, and = 1∕2, to get For this system, which is supplied with periodic boundary conditions, we also require that In addition, we impose the following requirements on the solution, that is, and also that The termF(∇ ) mimics-at least in its form-the extra stress-tensor in the case of some types of generalized Newtonian fluids (see, e.g., previous works [8][9][10][11] ). We setF where > 0 and m = p − 2, with 1 < p ≤ 2.
Let us now introduce the notion of "regular weak solution" for the considered system. The expression regular weak solutions has been introduced in Berselli and Lewandowski 12 and, although the context is different, the basic idea is the same: The term "weak" refers to the fact that the equations are satisfied in distributional sense, while the solutions are called "regular" because of the properties of the spaces to which they belong. However, in order to be concise, in the following we will always refer to weak solutions rather than regular weak solutions. For the problem (3.2)-to-(3.6), we state the following existence theorem which will be proved in Section 5.

Preliminary problem
In this case, we actually substituteF =F(∇ ), in relation (3.1) (and consequently also in the system (3.2)), with F ∶= F( ), and soF = F( ) + ∇ , > 0, (3.8) where F ∈ C 2 (R) and, following the same scheme as in Temam,5 we require that for i = 1, 2 and m ≥ 1/2. Here, c 1 and c 2 are positive constants. This model will be considered in Section 4. The notion of regular weak solution can be directly derived by adapting that one given in Definition 3.1, taking into account relation (3.9) in place of (3.6).
We have the following existence criterion.  Moreover, if it holds that inf x∈D 0 (x) > 0 and that sup x∈D 0 (x) < ∞, then it follows that inf Q T (t, x) > 0, and that sup Q T (t, x) < ∞.
The calculations we perform to prove this result will then be used, in Section 5, also to prove Theorem 3.1.
Remark 3.2. In the case of Theorem 3.1, assuming that inf x∈D 0 (x) > 0 and that sup x∈D 0 (x) < ∞, in order to get the bounds inf Q T (t, x) > 0 and sup Q T (t, x) < ∞, more regularity on the considered solutions seems to be needed (see Appendix A for more details).

SYSTEM (3.2) UNDER THE HYPOTHESES (3.8)-(3.9)
Let us consider the system For this problem, we prove existence and regularity results for the weak solutions by using suitable energy estimates.

A priori estimates
Let us start with the following result based on formal estimates that, however, can be made more rigorous by introducing a suitable approximating Galerkin scheme (see, e.g., Vaigant and Kazhikhov 3 ). Lemma 4.1. If the initial data ( 0 , y 0 ) are such that 0 ∈ L (D) and y 0 ∈ H 1 then there exists a time T 0 = T 0 (|| 0 || , || 0 || H 1 ) such that, for any 0 < t < T 0 , the following inequality holds true, that is, where C and c are two positive constants. Moreover, we also have that for any 0 < T < T 0 and 0 ≤ t ≤ T.

then there exists a constant C depending on p and D such that the inequality
holds for any t ∈ [0, T], T < T 0 .

then there exists a constant C depending on p such that the inequality
holds for any T < T 0 .
Proof. Estimate (4.9) is a consequence of (4.8), (3.9) and equation (4.1) 1 : The latter is multiplied by − |Δy| p − 2 Δy and subsequently integrated over Q T . Indeed, in such a case we have that where C = C(|v|). The conclusion follows by using calculations similar to those present in the proof of Lemma 4.1 and Lemma 4.2.
In order, to get improved a priori estimates, we differentiate (4.1) 1 with respect to x 1 , x 2 , and with respect to t to get, respectively, the estimates and In the following we omit the sums on the indices to keep the notation as concise as possible. For arbitrary q ≥ 2, we multiply (4.10) by q|∇y| q − 2 ∇ y and integrate on D to get Then, substituting − divF( ) + (v · ∇) = 1 + Δ − t in such a relation, we obtain (4.12) Let s ≥ 2. Multiplying (4.11) by s|y t | s − 2 y t in L 2 , and substituting = 1 (4.13) Thus, the sum of the above two relations is as follows d dt (4.14) To bound the terms on the right-hand side of the above relation, we observe that where we used Ladyzhenskaya's inequality. Moreover, we have and, for q≥4, we also have , y 0 ∈ H 2 , then there exists a constant C depending on T 0 such that the inequalities (4.20) and hold true for any t ∈ [0, T].
Proof. By taking q = 4 and s = 2 in (4.14), making derivatives explicit and rearranging the terms in such a relation (especially those coming from the first addendum on the right-hand side), we have d dt where, to keep the notation concise, we omit the summations made on the indices for the derivatives and the map components.
Proceeding as in Bessaih, 2 for the first ten terms from I 1 to I 10 , we have the following inequalities that we list for the sake of completeness (see also Vaigant and Kazhikhov, 3 for more details on the other terms I i , i = 1, … , 10), that is, and Then, in particular, we have that (4.24) Let us take into account the terms coming from (v · ∇) , that is, and (4.26) Now, consider the new terms coming from divF (y). Let us start with I 11 to get where, to control || || 4m 8m , we used (2.2). Finally, observe that where, in particular, we used (2.1) and (4.2).
To close the differential estimate (4.22), we integrate in time (in the time-interval [0, t]) such an estimate and we use the previous relations (4.23)-to-(4.28). Let us take into account the worst term coming from (4.24), that is, (4.29) To control this last term we use, up to integrate in t, relation (4.9): In the present case, that is, p = 4 and s = 2, relation (4.9) gives (4.31) Therefore, using (4.29) along with (4.30) and (4.31), we get The remaining terms in the right-hand side of (4.22) are easier to deal with: In essence, the same type of estimates used above allows us to bound them (see also Bessaih 2 ). Therefore, from (4.22) along with (4.23)-to-(4.31), we infer where y 0 ∈ H 2 → W 1, 4 (D). Using Gronwall's lemma we obtain Also, exploiting this inequality along with (4.8), (4.30) and (4.31), it follows that In fact, in relation (4.8), for p = 4, the worst term on the right-hand side, besides ∫ t 0 ||Δ (s)|| 4 4 ds, still remain ∫ t 0 ||∇ (s)|| 8 8 ds. Thus, the same calculations as above can be used to close this last integral inequality. As a very last point, we observe that (4.32) requires initial data ( 0 , y 0 ) ∈ L 3 (D) × H 2 , actually 0 is taken in L 3 (D) in order to use (4.8), and subsequently get (4.33). Indeed, if (y k , k ) is the family of Galerkin approximating functions used to make rigorous the previous calculations (see the end of this subsection for more details), then to bound ( t y k )(0) = y k, t (0), for ∈ L 2 (D) with || || ≤ 1, consider  Proof. As a consequence of the previous estimates, we also have that for any ∈ H 1 , with || || H 1 = 1, the following relation holds true, that is, and so As a direct consequence, the conclusion follows.

Higher order estimates
Now, let us consider the Equation (4.12) for q ≥ 4 and Equation (4.13) for s = 2. Adding these relations, without using  Proof. Due to the hypotheses on the initial data, we have that y 0 ∈ H 2 → W 1, q (D), q ≥ 1. Let us consider the case q > 4 for (4.36) (the cases 1 ≤ q ≤ 4 are consequence of the previous estimates), and make use of (4.35). When q = 4 the calculations are as in the proof of Theorem 4.1. For the first four terms J i , i = 1, 2, 3, 4, on the right-hand side of (4.35), we use the same estimates as in Bessaih; 2 we report in details only the case of J 1 , that is, Now, using Gagliardo-Nirenberg's inequality (2.1), with m = (q − 2)∕q, and exponent = (q − 4)∕(2(q − 2)), along with (4.18) and (4.19), we obtain , and, hence, we can conclude that where we used Young's inequality with exponents = 2q∕(q − 4) and ′ = 2q∕(q + 4). For the second term on the right-hand side of the above inequality, we use again (4.9) to bound ||Δ || 4 4 . As a consequence, we actually have to control ||∇ || 8 8 . Hence, where we used Gagliardo-Nirenberg's inequality (2.3). The remaining terms J 2 , J 3 and J 4 are simpler and can be treated similarly: We only list here, for the sake of completeness, the related estimates Let us consider the remaining terms. Observe that J 7 can be estimated exactly as in (4.28). We now have that and Consider J 5 and J 6 to get (4.38) Consequently, using (4.35) along with (4.37)-to-(4.38), (4.34) and (4.9), we have that where we used Ladyzhenskaya's inequality to get || t || 4q∕q+4 4 ≤ || t || 2q∕(q+4) ||∇ t || 2q∕(q+4) . The conclusion follows by an application of Gronwall's lemma.
The obtained estimates are sufficient for proving the existence of solutions. Indeed, we can use the method for constructing solutions given in Vaigant and Kazhikhov. 3 According to this scheme, approximating solutions {(y k , k )} are found by the Galerkin method (see, e.g., Galdi; 13 see also Bisconti 14 ). In particular {∇ y k } is compact in L 2 (Q T ). Thanks to the previous estimates, we can use classical compactness arguments (see, e.g., Temam; 15 see also the argument in previous works 2,3 ) to extract a convergent subsequence (still denoted by {(y k , k )}) Thus, the passage to the limit in the nonlinear terms in (4.1) 2 is justified. For the nonlinear term in (4.1) 1 , to pass to the limit, it is enough to observe that since F(y k ) is bounded, uniformly with respect to k, in L 2 (Q T ), then F( k ) ⇀ A in L 2 (Q T ). Therefore, using that F = F(s) is continuous along with the fact that {y k } converges a.e. on D tõ, due to the uniqueness of the limit, it follows that A = F(̃).

Upper and lower bounds for the density
The estimates previously obtained allow us to prove the density is actually bounded provided that the initial density 0 is bounded as well. To this end, we use the same approach provided in Vaigant and Kazhikhov 3 (see also Bessaih 2 ). We have the following two lemmas whose proofs, close to those of [ 2,Lemma 4.6,and Lemma 4.7], are provided in Appendix A.

CASE OFF(∇ ) = S(∇ )
Let us consider the following system with > 0 and 1 < p ≤ 2. In the sequel we use the notation S(∇ ) = ( + |∇ |) p−2 ∇ , introduced in (3.6) and we always assume ≥ 1. Further hypotheses on will be introduced later. This section devoted to the proof of Theorem 3.1.

Energy estimates
Also, in this case we proceed formally by providing a number of a priori estimates that, combined with a compactness criterion à la Aubin-Lions, allow us to prove the existence of a weak solution for the considered system supplied with appropriate initial data. As before, calculations can be made rigorous by using a suitable Galerkin scheme (see, e.g., previous works 3, 10,11,13 ). Let us recall some properties characterizing the non-linear term S(∇ y). It can be proved that S(v), v ∈ R 2 , satisfies the following relation: There exists a positive constant C 1 such that (see, e.g., Diening 16,17 ) for any vector w ∈ R 2 . Also, for any pair of vectors v, w ∈ R 2 , the following relations hold true, that is, and with C 2 and C 3 positive constants. The proofs of these estimates are given 16 in the case of second-rank tensors, but they can easily adapted to our simpler case.
Remark 5.1. It holds that where in the last step we used (5.2), with C 1 = p − 1, and the derivatives with respect to D l are evaluated at the point D = ∇ . Here D l ∕ x s = s l and D i ∕ x s = s i . For further details see, for example, previous works 10, 16,17 Lemma 5.1. Let T > 0. If the initial data ( 0 , y 0 ) are such that 0 ∈ L (D) and y 0 ∈ H 1 then there exists a positive constant C, such that for any 0 < t < T the following inequality holds true, that is, Proof. Multiplying equation (5.1) 1 by Δy, in L 2 , and using relation (5.5), we find Then, multiplying equation (5.1) 2 by ln , in L 2 , we get and summing them up, and integrating in time, we get ds, and the conclusion follows by an application of Gronwall's lemma and selecting = 1∕2.
Remark 5.2. Let us multiply (5.1) 1 by y t and integrate on D to get Now, we have that where ( ) ∶= ∫ D L(|∇ |)dx ≥ 0, with Moreover, we have that L(t) ≃ ( + t) p − 2 t 2 and also that (see, e.g., Berselli and Bisconti 10 ) In particular, this shows that Then, using (5.7) along with the above relations, we reach that, however, cannot be closed at this level due to the lack of a direct control on || ||. Therefore, to control ||y t || we resort to higher order estimates.

Estimates in higher-order norms
Let us start with the following result Proof. Multiply equation (5.1) 2 against 3 2 to get t 3 + 3 2 div ( ∇ ) = 0, that can be rewritten as , (5.9) where in the last step we used (5.1) 1 . By Young's inequality, we have that and also that Integrating (5.9) over D, and using the above controls, we obtain and relation (5.8) follows directly.
In particular, for the term Θ, we have that where we used Hölder's, Young's and Ladyzhenskaya's inequalities. Hence, we reach where we exploited Poincaré's inequality along with (5.6), andĈ =Ĉ(T) is a suitable positive constant. The first two terms in the right-hand side of (5.12) can be easily reabsorbed on the left-hand side of (5.11), provided that is taken sufficiently large.
Similarly, the worst terms coming from ∫ D |divS(∇ )| 4 dx can be reabsorbed on the left-hand side of (5.11) provided that is large enough. In fact, we have that ) ds where c = c(p), and in the last two steps we used Ladyzhenskaya's inequality and (5.6). As a consequence of the above control used along with (5.11) and (5.12) (here we set C ∶= max{Ĉ, C}), we have that relation (5.10) follows directly.
Differentiating (5.1) 1 with respect to x = (x 1 , x 2 ), and with respect to t, we get, respectively, the following controls and Now, multiplying (5.14) by y t in L 2 , we reach and so d dt Observe that where we used (5.2), with C 1 = p − 1, and the derivatives with respect to D k = k are evaluated at the point D = ∇ , and Therefore, using the above relation along with (5.13), we obtain (5.16) Multiplying (5.13) by q|∇y| q − 2 ∇ y, in L 2 , integrating by parts, and substituting − div (5.17) Summing up (5.16) and (5.17), we obtain d dt Let us now return to the Galerkin scheme. Take t 0 = 0. Defining for ∈ L 2 (0, T; H 1 ), which is a test having the same regularity of y k , the following quantity Now, using (5.23) along with (5.22), and recalling that k ⇀ and y k → y in L 2 (Q T ) (actually we also have that y k → y in L 2 (0, T; H 1 )), we infer In particular, in passing to the limit in the first term, we used the triangle inequality ds| along with the strong convergence of y k and the weak convergence of k , in L 2 (Q T ). Hence, by using (5.22) along with (5.24) and (5.25) (i.e., subtracting (5.22) from (5.25)), we get Choosing = y + for some smooth and letting → 0, we can conclude that  = S(∇ ).
Remark 5.3. In this case, due to the form of S(∇ ) = ( + |∇ |) p−2 ∇ , does not seem possible to reproduce-in an elementary way-the calculations in the proof of Lemma 4.6, which are used to provide upper and lower bounds for . In order to retrive such bounds, we resort to higher order estimates assuming more regularity on the initial data. A sketch of these additional calculations is provided in Appendix A.
As a consequence of the above results, Theorem 3.1 follows directly. Proof of Lemma 4.6. Assuming that (t, x) > 0, (t, x) ∈ Q T and T < T 0 , and arguing as in Galdi, 3 we can rewrite (4.1) 2 in the form t ln + ∇ · ∇ ln + Δ = 0.

Case ofF(∇ ) = S(∇ ): Upper and lower bounds for by using estimates for higher order derivatives
In order to use an approach similar to the one just exploited to bound the density in (5.1), we require more regularity on the initial data to get improved solutions. Also, the parameter > 0 is taken as large as needed. We ssume that t , D 3 , and ∇ , are sufficiently regular (here D 3 = i k , i, , k = 1, 2). Starting from (5.1) 1 , i.e.