In this paper, we consider an initial value problem for the 2-D compressible Navier-Stokes equations without heat conductivity. We prove the global existence of a strong solution when the initial perturbation is small in H² and its L¹ norm is bounded. Moreover, we derive some decay estimate for such a solution.

1. Introduction

The 2-D compressible Navier-Stokes equations for (x, t) ∈ ℝ² × ℝ⁺ are rewritten as

()

which govern the motion of gases, where ρ, u, P, θ stand for the density, velocity, pressure, and absolute temperature functions, respectively.

is the specific total energy with E as the specific internal energy,

is the stress tensor, k is the coefficient of heat conduction, I is the identity matrix, and μ and λ are the coefficients of viscosity and second coefficient of viscosity, respectively, satisfying

()

As one of the most important systems in fluid dynamics, there are lots of results on the well-posedness, blow-up phenomenon, large time or asymptotic behavior, and optimal decay rates of solutions based on different assumptions in different cases and function spaces. Among them, for the case with a positive coefficient of heat conduction k > 0, Kazhikhov and Shelukhin studied the global existence in one dimension [1, 2]. The global existence of multidimensional case was established in [3–6]; more results on global existence for different kinds of solutions can be found in [3–10]. For the study of the large time behavior, asymptotic behavior, and optimal decay rates of solutions, one can refer to [4, 11–15]. The references [15–17] and [10, 18] restricted the systems under the case of k = 0 and k > 0, respectively. Danchin [8, 9] proved the existence and uniqueness of strong solutions to the compressible Navier-Stokes equations in hybrid Besov spaces, and Tan and Wang [6] studied the global existence of strong small solutions in H^l, l ≥ 4. For the case of k = 0, Tan and Wang [5] proved the global solvability in three-dimensional space for the less regular solutions to the compressible Navier-Stokes equations in the H²-framework; however, they needed to assume that the L¹-norm of the initial perturbation is bounded which is important in the proof of global existence. Later, ref. [3] removed this assumption by using some techniques with regard to the homogeneous Besov space and the hybrid Besov space.

Compared to the Cauchy problem, the equilibrium state of pressure increases with time. Xin et al. [16, 17, 19] investigated the blow-up phenomenon of the compressible Navier-Stokes equations in inhomogeneous Sobolev space; they proved that the smooth or strong solutions will blow up in any positive time if the initial data have an isolated mass group, no matter how small they are. On the other hand, we would like to introduce some research on the Serrin-type regularity (blow-up) criteria for the incompressible Navier-Stokes system. These criteria are obtained from [20, 21]; later, many authors successfully extended these blow-up criteria to the compressible flow (for example, [22–29] and references therein).

In this paper, we consider this problem in ℝ² with the case k = 0 and assume that the gas is ideal and polytropic, i.e.,

, where R > 0 and A > 0 are the universal gas constant, γ > 1 is the adiabatic exponent,

is the entropy, and c_v = R/(γ − 1) is a constant which represents the specific heat at a constant volume. Furthermore, we also assume that R = A = 1 without loss of generality. Then, we have

()

and system (1) in terms of variables

can be reformulated in terms of variables

()

where

is the classical dissipation function. We are concerned with the initial value problem to system (4) with initial data satisfying

()

where p_∞ > 0 and s_∞ are the given constants. For the global existence and the decay estimate in the case of two dimensions, we have the following theorem.

Theorem 1. Let p_∞ > 0, s_∞ be two constants. There exists a small constant ε₀ such that if , and are bounded, then there exists a unique global solution of the initial value problems (4) and (5) satisfying

()

Finally, there is a constant C₀ such that for any t ≥ 0, the solution

has the decay properties

()

Remark 2.

(1)
The L^q decay estimates of (7) for 2 ≤ q ≤ 4 are optimal which coincide with the L^q decay of the heat equation and are much slower than the decay rate in ℝ³. In [3, 5], they obtain that which ensures that is bounded. But in ℝ², we only have and therefore, is unbounded. This is the main difficulty about the proof of existence in ℝ² (see Section 3 below)
(2)
Due to k = 0, we cannot gain any diffusion of and the L^∞ decay estimate of (7) is slower than the decay of the heat equation

1.1. Notation

In this paper, we use L^p, H^m to denote the L^p and Sobolev spaces on ℝ² with norms and , respectively. We use C to denote the constants depending only on physical coefficients and C₀ to be constants depending additionally on the initial data.

This paper is organized as follows: In Section 2, we reformulate problems (4) and (5), introduce two main propositions, and illustrate that we only need to prove Proposition 4. The rest of the paper is devoted to proving Proposition 4. The proof of the energy estimate part is in Section 3, and the decay rate part is in Section 4.

2. Reformulated System

We reformulate system (4) by setting

()

where ρ_∞ = ρ(p_∞, s_∞).

Changing variables as

, initial value problems (4) and (5) are reformulated as

()

where the nonlinearities are given by

()

For any T > 0, we define the solution space by

()

and the solution norm by

()

Taking a standard contraction mapping argument, we have the following propositions for the local existence (see [30]).

Proposition 3 (see [3].)Suppose that the initial data satisfy (p₀, v₀, s₀) ∈ H²(ℝ²) and . Then, there exists a positive constant T₀ depending on such that Cauchy problem (9) has a unique solution (p, v, s) ∈ X(0, T₀) satisfying

()

This, together with the proposition below, is sufficient to derive Theorem 1; the proof is based on the standard continuity argument.

Proposition 4. Let (p₀, v₀, s₀) ∈ H²(ℝ²) and (p₀, v₀) ∈ L¹(ℝ²); for any T > 0, there exists ε > 0 such that the solution (p, v, s) of the initial value problem (9) in [0, T] satisfies

()

Then, this solution is unique with the energy estimates

()

and the decay properties

()

The rest of paper is used to prove Proposition 4.

3. Energy Estimate

For later use, we introduce some useful analytic results.

Lemma 5. Let f ∈ H²(ℝ²); then, we have the following Sobolev inequalities:

()

Lemma 6 (lower-order energy estimate for (p, v)). Under the assumption of Proposition 4, there exists a δ₁ > 0 arbitrarily small and independent of ε, such that

()

Proof. Multiplying (2.1)₁, (2.1)₂ by p, v, respectively, integrating them over ℝ² and then adding them together, we obtain

()

〈p, f〉 and 〈v, g〉 can be estimated as follows. For the first term, Lemma 5 together with (14) and the Hölder inequality implies

()

For the second term,

()

Similar to the proof of (24), by Lemma 5, (14), the Hölder inequality, and the fact that

()

which is derived from (3) and (14), we have

()

Therefore,

()

Hence, combining (23), (24), and (28) yields

()

Next, we shall estimate . Multiplying (2.1)₂ by ∇p and integrating them over ℝ², we get

()

where

()

which are based on the similar calculation and Young’s inequality. In brief, we obtain

()

Finally, multiplying (32) by δ₁ that is small but fixed and adding it to (29), we can derive (22). This completes the proof of the lemma.

Next, we turn to estimate the higher-order energy for (p, v).

Lemma 7 (higher-order energy estimate for (p, v)). Under the assumption of Lemma 6, there is a small enough but fixed δ₂ > 0, which is independent of ε, such that

()

Proof. Applying ∇ to (2.1)₁, (2.1)₂ and multiplying by ∇p, ∇v, respectively, integrating them over ℝ² and using the same calculation technique as before, we get

()

Next, applying ∇² to (2.1)₁, (2.1)₂ and multiplying by ∇²p, ∇²v, respectively, we can deduce

()

where

()

The terms on the right-hand side are estimated as below:

()

Thus,

()

Now applying ∇ to (2.1)₂ and multiplying by ∇²p, we have

()

Similar to the estimate of , we transform the formula 〈−∇v_t, ∇²p〉 as below:

()

and derive

()

Multiplying (42) by δ₂ that is small but fixed, and combining it with (34) and (39), we finally obtain (33). This completes the proof of the lemma.

The lemma below gives the energy estimate for the entropy s.

Lemma 8. For 0 ≤ t ≤ T, it holds that

()

Proof. For each multi-index α with 0 ≤ |α| ≤ 2, we apply to (2.1)₃, multiply it by , integrate it on ℝ², and then sum them up to deduce

()

For W₁, one has

()

where, for each α with 0 ≤ |α| ≤ 2, one can infer that

()

and for each α, β with 0 ≤ |β| ≤ |α| − 1, 1 ≤ |α| ≤ 2, and β ≤ α, it holds that

()

Therefore, we have

()

Now, we turn to estimate W₂; using (14), we obtain

()

Combining (44) with (48) and (49) yields (43).

Now, we are in the position to prove the energy estimate in Proposition 4.

Proof of (15) and (16). The first step is to prove (15), energy estimate for (p, v). By defining

()

and adding (22) and (33) together, one has

()

Integrating the above equation directly in time gives

()

Hence,

()

which implies

()

when ε is small enough. In other words, (15) is valid.

The second step is to prove (16), the energy estimate for s. Summing up (22), (33), and (43) gives

()

where

()

Therefore,

()

By Grönwall’s inequality and (15), we obtain

()

which implies (16).

4. Decay Rates

In this section, we prove the decay rates in Proposition 4.

Proof of (17)–(20). Letting

()

we rewrite (33) as follows:

()

Next, by adding to both sides of the above inequality, we deduce that for some constant α > 0,

()

Therefore,

()

To deal with , we rewrite the solution of system (9) as

()

where H(t) = (p(t), v(t)) and is a matrix-valued differential operator given by

()

and the solution semigroup has the following property (see [31, 32]).

Lemma 9. Let k > 0 be an integer; then, ∀t ≥ 0,

()

Then, gathering (63) and (65), one has

()

where and the nonlinear source can be estimated as follows:

()

Hence,

()

Now, if we define

, then

()

which implies that

()

Combining (68) and (70) and letting ε be small enough, we can deduce that

()

Substituting (71) into (62), one has

()

Now by the definition of M(t), we obtain

()

Letting ε > 0 be small enough, the inequality above gives us

()

Thus, (70) gives

()

Meanwhile, applying (65) with k = 0 to (14), by (15) and (17), one can infer that

()

Then, by Sobolev’s inequality, (75), and (76), we get

()

Thus, by interpolation inequality, one has

()

for any 2 ≤ q ≤ 4, 0 ≤ t ≤ T, and

()

for any 4 < q ≤ ∞, 0 ≤ t ≤ T. Therefore, the optimal decay rates (17)–(19) have been proven.

We use the estimates above and (9) to deduce that

()

Therefore, (20) is finally proven.

5. Conclusion

The motivation of this paper is to refine the previous works of [3, 5]. As mentioned above, to prove the global existence of the compressible Navier-Stokes equations, ref. [3] needs to use some complicated techniques based on the notions of the homogeneous Besov space and the hybrid Besov space to remove a condition in [5]. However, in this paper, we use a much simpler method to achieve this, to complete a prior estimate on entropy which is important to prove the global existence of the compressible Navier-Stokes equations by some simple analysis.

The results are in the H²-framework; it is possible to consider similar problems in functional space with lower regularity.

Disclosure

The manuscript has been submitted as a preprint according to the following link: “https://www.authorea.com/users/488979/articles/572918-global-existence-and-decay-estimate-for-the-2-d-compressible-navier-stokes-equations-without-heat-conductivity.”

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 11901117).

Open Research

Data Availability

All data generated or analysed during this study are included in this published article.

References

1 Kazhikhov A. and Shelukhin V., Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas: PMM vol. 41, n≗ 2, 1977, pp. 282-291, Journal of Applied Mathematics and Mechanics. (1977) 41, no. 2, 273–282, https://doi.org/10.1016/0021-8928(77)90011-9, 2-s2.0-0001258059.
10.1016/0021-8928(77)90011-9
Web of Science® Google Scholar
2 Kazhikhov A., Cauchy problem for viscous gas equations, Siberian Mathematical Journal. (1982) 23, no. 1, 44–49, https://doi.org/10.1007/BF00971419, 2-s2.0-0001862785.
10.1007/BF00971419
Web of Science® Google Scholar
3 Chen Q., Tan Z., Wu G., and Zou W., The initial value problem for the compressible Navier-Stokes equations without heat conductivity, Journal of Differential Equations. (2020) 268, no. 9, 5469–5490, https://doi.org/10.1016/j.jde.2019.11.025.
10.1016/j.jde.2019.11.025
Google Scholar
4 Duan R. and Ma H., Global existence and convergence rates for the 3-D compressible Navier-Stokes equations without heat conductivity, Indiana University Mathematics Journal. (2008) 57, no. 5, 2299–2320, https://doi.org/10.1512/iumj.2008.57.3326, 2-s2.0-57449122104.
10.1512/iumj.2008.57.3326
Web of Science® Google Scholar
5 Tan Z. and Wang H., Global existence and optimal decay rate for the strong solutions in H² to the 3-D compressible Navier-Stokes equations without heat conductivity, Journal of Mathematical Analysis and Applications. (2012) 394, no. 2, 571–580, https://doi.org/10.1016/j.jmaa.2012.05.001, 2-s2.0-84862011715.
10.1016/j.jmaa.2012.05.001
Google Scholar
6 Tan Z. and Wang Y., On hyperbolic-dissipative systems of composite type, Journal of Differential Equations. (2016) 260, no. 2, 1091–1125, https://doi.org/10.1016/j.jde.2015.09.025, 2-s2.0-84947865549.
10.1016/j.jde.2015.09.025
Web of Science® Google Scholar
7 Charve F. and Danchin R., A global existence result for the compressible Navier–Stokes equations in the critical L^p framework, Archive for Rational Mechanics and Analysis. (2010) 198, no. 1, 233–271, https://doi.org/10.1007/s00205-010-0306-x, 2-s2.0-77956063840.
10.1007/s00205-010-0306-x
Web of Science® Google Scholar
8 Danchin R., Global existence in critical spaces for compressible Navier-Stokes equations, Inventiones Mathematicae. (2000) 141, no. 3, 579–614, https://doi.org/10.1007/s002220000078, 2-s2.0-0034349433.
10.1007/s002220000078
Web of Science® Google Scholar
9 Danchin R., Global existence in critical spaces for flows of compressible viscous and heat-conductive gases, Archive for Rational Mechanics and Analysis. (2001) 160, no. 1, 1–39, https://doi.org/10.1007/s002050100155, 2-s2.0-0012062948.
10.1007/s002050100155
Web of Science® Google Scholar
10 Ponce G., Global existence of small solutions to a class of nonlinear evolution equations, Nonlinear Analysis. (1985) 9, no. 5, 399–418, https://doi.org/10.1016/0362-546X(85)90001-X, 2-s2.0-46549102029.
10.1016/0362-546X(85)90001-X
Web of Science® Google Scholar
11 Duan R., Ukai S., Yang T., and Zhao H., Optimal convergence rates for the compressible Navier–Stokes equations with potential forces, Mathematical Models and Methods in Applied Sciences. (2007) 17, no. 5, 737–758, https://doi.org/10.1142/S021820250700208X, 2-s2.0-34248393109.
10.1142/S021820250700208X
Web of Science® Google Scholar
12 He L., Huang J., and Wang C., Large-time behavior for compressible Navier-Stokes-Fourier system in the whole space, Journal of Mathematical Fluid Mechanics. (2022) 24, no. 2, https://doi.org/10.1007/s00021-022-00673-8.
10.1007/s00021-022-00673-8
PubMed Google Scholar
13 Liu T. and Zeng Y., Compressible Navier-Stokes equations with zero heat conductivity, Journal of Differential Equations. (1999) 153, no. 2, 225–291, https://doi.org/10.1006/jdeq.1998.3554, 2-s2.0-0033541349.
10.1006/jdeq.1998.3554
Google Scholar
14 Ukai S., Yang T., and Zhao H., Convergence rate for the compressible Navier–Stokes equations with external force, Journal of Hyperbolic Differential Equations. (2006) 3, no. 3, 561–574, https://doi.org/10.1142/S0219891606000902, 2-s2.0-34247346836.
10.1142/S0219891606000902
Google Scholar
15 Wu G., Global existence and asymptotic behavior for the 3D compressible Navier-Stokes equations without heat conductivity in a bounded domain, Journal of Differential Equations. (2017) 262, no. 2, 844–861, https://doi.org/10.1016/j.jde.2016.09.046, 2-s2.0-84994745382.
10.1016/j.jde.2016.09.046
Google Scholar
16 Xin Z., Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density, Communications on Pure and Applied Mathematics. (1998) 51, no. 3, 229–240, https://doi.org/10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C.
10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C
Web of Science® Google Scholar
17 Xin Z. and Yan W., On blowup of classical solutions to the compressible Navier-Stokes equations, Communications in Mathematical Physics. (2013) 321, no. 2, 529–541, https://doi.org/10.1007/s00220-012-1610-0, 2-s2.0-84878775597.
10.1007/s00220-012-1610-0
Web of Science® Google Scholar
18 Deckelnick K., L² decay for the compressible Navier-Stokes equations in unbounded domains, Communications in Partial Differential Equations. (1993) 18, no. 9-10, 1445–1476, https://doi.org/10.1080/03605309308820981, 2-s2.0-0001373276.
10.1080/03605309308820981
Web of Science® Google Scholar
19 Li H., Wang Y., and Xin Z., Non-existence of classical solutions with finite energy to the Cauchy problem of the compressible Navier-Stokes equations, Archive for Rational Mechanics and Analysis. (2019) 232, no. 2, 557–590, https://doi.org/10.1007/s00205-018-1328-z, 2-s2.0-85055574455.
10.1007/s00205-018-1328-z
Web of Science® Google Scholar
20 Serrin J., On the interior regularity of weak solutions of the Navier-Stokes equations, Archive for Rational Mechanics and Analysis. (1962) 9, no. 1, 187–195, https://doi.org/10.1007/BF00253344, 2-s2.0-0000169262.
10.1007/BF00253344
Web of Science® Google Scholar
21 Struwe M., On partial regularity results for the Navier-Stokes equations, Communications on Pure and Applied Mathematics. (1988) 41, no. 4, 437–458, https://doi.org/10.1002/cpa.3160410404, 2-s2.0-84990553438.
10.1002/cpa.3160410404
Web of Science® Google Scholar
22 Choe H. and Yang M., Blow up criteria for the compressible Navier-Stokes equations, Contemporary Mathematics. (2018) 710, 65–84, https://doi.org/10.1090/conm/710/14364, 2-s2.0-85050250683.
10.1090/conm/710/14364
Google Scholar
23 Huang X., Li J., and Wang Y., Serrin-type blowup criterion for full compressible Navier-Stokes system, Archive for Rational Mechanics and Analysis. (2013) 207, no. 1, 303–316, https://doi.org/10.1007/s00205-012-0577-5, 2-s2.0-84871969951.
10.1007/s00205-012-0577-5
Web of Science® Google Scholar
24 Huang X., Li J., and Xin Z., Serrin-type criterion for the three-dimensional viscous compressible flows, SIAM Journal on Mathematical Analysis. (2011) 43, no. 4, 1872–1886, https://doi.org/10.1137/100814639, 2-s2.0-80052779279.
10.1137/100814639
Web of Science® Google Scholar
25 Jiu Q., Wang Y., and Ye Y., Refined blow-up criteria for the full compressible Navier-Stokes equations involving temperature, Journal of Evolution Equations. (2021) 21, no. 2, 1895–1916, https://doi.org/10.1007/s00028-020-00660-4.
10.1007/s00028-020-00660-4
Web of Science® Google Scholar
26 Lei Z. and Xin Z., On scaling invariance and type-I singularities for the compressible Navier-Stokes equations, Science China. Mathematics. (2019) 62, no. 11, 2271–2286, https://doi.org/10.1007/s11425-018-9363-1, 2-s2.0-85062641167.
10.1007/s11425-018-9363-1
Google Scholar
27 Sun Y., Wang C., and Zhang Z., A Beale-Kato-Majda blow-up criterion for the 3-D compressible Navier-Stokes equations, Journal de Mathématiques Pures et Appliquées. (2011) 95, no. 1, 36–47, https://doi.org/10.1016/j.matpur.2010.08.001, 2-s2.0-78650245997.
10.1016/j.matpur.2010.08.001
Web of Science® Google Scholar
28 Sun Y., Wang C., and Zhang Z., A Beale-Kato-Majda criterion for three dimensional compressible viscous heat-conductive flows, Archive for Rational Mechanics and Analysis. (2011) 201, no. 2, 727–742, https://doi.org/10.1007/s00205-011-0407-1, 2-s2.0-79960576445.
10.1007/s00205-011-0407-1
Web of Science® Google Scholar
29 Wen H. and Zhu C., Blow-up criterions of strong solutions to 3D compressible Navier-Stokes equations with vacuum, Advances in Mathematics. (2013) 248, 534–572, https://doi.org/10.1016/j.aim.2013.07.018, 2-s2.0-84883502164.
10.1016/j.aim.2013.07.018
Web of Science® Google Scholar
30 Matsumura A. and Nishida T., The initial value problem for the equations of motion of viscous and heat-conductive gases, Journal of Mathematics of Kyoto University. (1980) 20, 67–104.
10.1215/kjm/1250522322
Google Scholar
31 Kobayashi T., Some estimates of solutions for the equations of motion of compressible viscous fluid in the three-dimensional exterior domain, Journal of Differential Equations. (2002) 184, no. 2, 587–619, https://doi.org/10.1006/jdeq.2002.4158, 2-s2.0-0037144874.
10.1006/jdeq.2002.4158
Web of Science® Google Scholar
32 Kobayashi T. and Shibata Y., Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in ℝ³, Communications in Mathematical Physics. (1999) 200, no. 3, 621–659, https://doi.org/10.1007/s002200050543, 2-s2.0-0033472764.
10.1007/s002200050543
Web of Science® Google Scholar

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Global Existence and Large Time Behavior for the 2-D Compressible Navier-Stokes Equations without Heat Conductivity

Abstract

1. Introduction

1.1. Notation

2. Reformulated System

3. Energy Estimate

4. Decay Rates

5. Conclusion

Disclosure

Conflicts of Interest

Acknowledgments

Open Research

Data Availability

References

References

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Global Existence and Large Time Behavior for the 2-D Compressible Navier-Stokes Equations without Heat Conductivity

Abstract

1. Introduction

1.1. Notation

2. Reformulated System

3. Energy Estimate

4. Decay Rates

5. Conclusion

Disclosure

Conflicts of Interest

Acknowledgments

Open Research

Data Availability

References

References

Related

Information