Abstract
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 and its norm is bounded. Moreover, we derive some decay estimate for such a solution.
1. Introduction
The 2-D compressible Navier-Stokes equations for are rewritten as
which govern the motion of gases, where stand for the density, velocity, pressure, and absolute temperature functions, respectively. is the specific total energy with as the specific internal energy, is the stress tensor, is the coefficient of heat conduction, is the identity matrix, and and 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 , 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 and , 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 . For the case of 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 -framework; however, they needed to assume that the -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 with the case and assume that the gas is ideal and polytropic, i.e., , where and are the universal gas constant, is the adiabatic exponent, is the entropy, and is a constant which represents the specific heat at a constant volume. Furthermore, we also assume that 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 and 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 be two constants. There exists a small constant 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 such that for any , the solution has the decay properties
Remark 2. (1)The decay estimates of (7) for are optimal which coincide with the decay of the heat equation and are much slower than the decay rate in In [3, 5], they obtain that which ensures that is bounded. But in we only have and therefore, is unbounded. This is the main difficulty about the proof of existence in (see Section 3 below)(2)Due to we cannot gain any diffusion of and the decay estimate of (7) is slower than the decay of the heat equation
1.1. Notation
In this paper, we use to denote the and Sobolev spaces on with norms and , respectively. We use to denote the constants depending only on physical coefficients and 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 .
Changing variables as , initial value problems (4) and (5) are reformulated as where the nonlinearities are given by
For any , 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 and . Then, there exists a positive constant depending on such that Cauchy problem (9) has a unique solution 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 and ; for any , there exists such that the solution of the initial value problem (9) in 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 ; then, we have the following Sobolev inequalities:
Lemma 6 (lower-order energy estimate for ). Under the assumption of Proposition 4, there exists a arbitrarily small and independent of , such that
Proof. Multiplying by , respectively, integrating them over and then adding them together, we obtain and 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 by and integrating them over , we get where which are based on the similar calculation and Young’s inequality. In brief, we obtain Finally, multiplying (32) by 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 .
Lemma 7 (higher-order energy estimate for ). Under the assumption of Lemma 6, there is a small enough but fixed , which is independent of , such that
Proof. Applying to and multiplying by , , respectively, integrating them over and using the same calculation technique as before, we get Next, applying to and multiplying by , , respectively, we can deduce where The terms on the right-hand side are estimated as below: Thus, Now applying to and multiplying by , we have Similar to the estimate of , we transform the formula as below: and derive Multiplying (42) by 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 .
Lemma 8. For , it holds that
Proof. For each multi-index with , we apply to , multiply it by , integrate it on , and then sum them up to deduce For , one has where, for each with , one can infer that and for each with , , and , it holds that Therefore, we have Now, we turn to estimate ; 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 . By defining and adding (22) and (33) together, one has
Integrating the above equation directly in time gives Hence, which implies
when is small enough. In other words, (15) is valid.
The second step is to prove (16), the energy estimate for . 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 , Therefore, To deal with , we rewrite the solution of system (9) as where and is a matrix-valued differential operator given by and the solution semigroup has the following property (see [31, 32]).
Lemma 9. Let be an integer; then, , 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 be small enough, we can deduce that
Substituting (71) into (62), one has
Now by the definition of , we obtain
Letting be small enough, the inequality above gives us
Thus, (70) gives
Meanwhile, applying (65) with 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 , and
for any . 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 -framework; it is possible to consider similar problems in functional space with lower regularity.
Data Availability
All data generated or analysed during this study are included in this published article.
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).