Abstract
In this note, we study the blowup of classical solutions to the three-dimensional compressible Navier–Stokes equations with revised Maxwell’s law. First, we improve the previous blowup result with initial density away from vacuum by removing three restrictions. Next, we present a blowup result for the classical solutions with decay at far fields when the shear relaxation time is zero by introducing a new averaged quantity.
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1 Introduction
In this note, we consider the following three-dimensional compressible Navier–Stokes equations with revised Maxwell’s law (see [10] for instance):
where the fluid density \(\rho =\rho (x,t)\), the fluid velocity \({\textbf{u}}={\textbf{u}}(x,t)=(u_{1}(x,t),u_{2}(x,t),u_{3}(x,t))\) and the stress tensor \({\mathbb {S}}={\mathbb {S}}(x,t)\) are the unknown functions with \((x,t)\in {\mathbb {R}}^{3}\times {\mathbb {R}}^{+}\). The function \(p(\rho )=a\rho ^{\gamma }\) represents the pressure with \(a>0\) and \(\gamma >1\) being two constants, \(\gamma \) is called the adiabatic exponent. The stress tensor \({\mathbb {S}}\) is assumed to satisfy the revised Maxwell’s law: \({\mathbb {S}}={\mathbb {S}}_{1}+S_{2}{\mathbb {I}}\), where \({\mathbb {S}}_{1}\) and \(S_{2}\) are given by the following equations
where \(\mu \) and \(\lambda \) are positive constants, \(\tau _{1}\) and \(\tau _{2}\) denote the shear relaxation time and the compressible relaxation time, respectively, \({\mathbb {I}}\) represents \(3\times 3\) identity matrix. Equations (1.2) and (1.3) were first proposed by Yong [14] from some mathematical point of view, later Chakraborty and Sader [2] showed the fact that the division of \({\mathbb {S}}\) into two parts has its physical meanings, where the importance of this model for describing high frequency limits is underlined together with the presentation of numerical experiments.
There were some works about the mathematical analysis of the Navier–Stokes equations with revised Maxwell’s law in the literature. Yong proved that system (1.1)–(1.3) is symmetric hyperbolic system, which implies local well-posedness immediately. Again due to the symmetric hyperbolic property of system (1.1)–(1.3), the important property of finite propagation speed is available, which allows one to define some averaged quantities as in [8] and shows finite time blowup of solutions by establishing a Riccati-type inequality, see [10]. Precisely, Wang and Hu [10] considered system (1.1)–(1.3) with initial data
they showed that if the initial data \((\rho _{0}-{\overline{\rho }},{\textbf{u}}_{0},{\mathbb {S}}_{10},S_{20})\) are compactly supported in \(B_{R}:=\{x\in {\mathbb {R}}^{3}||x|\le R\}\) for some \(R>0\) with \({\overline{\rho }}\) being any positive constant, then there exists a constant \(\sigma \) such that
on \(D=\{x\in {\mathbb {R}}^{3}||x|>R+\sigma t\}\) for \(C^{1}\) solutions \((\rho (\cdot ,t),{\textbf{u}}(\cdot ,t),{\mathbb {S}}_{1}(\cdot ,t),S_{2}(\cdot ,t))\) to the Cauchy problem (1.1)–(1.4) on [0, T], see Proposition 2.1 of [10]. This guarantees that the following averaged quantities are finite:
By establishing a Riccati-type inequality of F(t), the authors of Wang and Hu [10] proved that the life span of any \(C^{1}\) solution to (1.1)–(1.4) must be finite for some special large initial data. Unfortunately, the following three restrictions are required in [10]:
In this note, our first aim is to give a blowup result for the \(C^{1}\) solutions to (1.1)–(1.4) without the conditions (1.9).
When \(\tau _{1}=\tau _{2}=0\), the system (1.1)–(1.3) is reduced to the classical compressible Navier–Stokes system. Many authors have studied the blowup of smooth solutions to the classical compressible Navier–Stokes system, see [1, 3, 5,6,7, 9, 12, 13] and the references therein. Particularly, the authors of Jiu [5] and Wang et al. [9] investigated the blowup of classical solutions to the classical compressible Navier–Stokes system when the density and the velocity decay at far fields. Motivated by Jiu [5] and Wang et al. [9], we can present a blowup result of classical solutions to (1.1)–(1.4) when \(\tau _{1}=0\) and the density and the velocity decay at far fields. This is our second aim of this note. Different from Jiu [5] and Wang et al. [9], for the problem (1.1)–(1.4) we cannot obtain the conversation of the total energy or the decrease with time of the total energy. This makes us difficult to get the upper bound of the momentum of inertia, which is crucial to establish the Riccati-type inequality of a weighted momentum. To overcome this difficulty, we introduce a new averaged quantity J(t) [see (3.3) below], which decreases with time and is related to the total energy.
Remark 1.1
For the results about the non-isentropic Navier–Stokes equations with revised Maxwell’s law, we can refer to [4, 11].
2 Improvement on the Blowup Result of Wang and Hu [10]
Our result in this section is stated as follows.
Theorem 2.1
Let \((\rho ,u,{\mathbb {S}}_{1},S_{2})\) be a \(C^{1}\) solution to the Cauchy problem (1.1)–(1.4) for \(0\le t\le T_{1}\) with the initial data \((\rho _{0}-{\overline{\rho }},{\textbf{u}}_{0},{\mathbb {S}}_{10},S_{20})\) being compactly supported in \(B_{R}:=\{x\in {\mathbb {R}}^{3}||x|\le R\}\). For any fixed \(t_{1}^{*}>0\), if
and
then \(T_{1}<t_{1}^{*}\), where
Proof
By (1.6), (1.1)\(_{1}\) and (1.5), we get
which implies that
In view of (1.7), (1.3) and (1.5), one has
so we obtain
By (1.2), we know that
due to
Define
then by (2.8), we have
It follows from (1.8), (1.1)\(_{2}\), (1.5), (2.7) and (2.11) that
where
By the Schwarz inequality, it holds
By (1.6) and (2.5), we know that
Combining (2.12), (2.14) and (2.15), we obtain
For any fixed \(t_{1}^{*}>0\), when \(0\le t\le t_{1}^{*}\), it follows from (2.16) that
By (2.1), we know that
which together with (2.17) imply that \(F'(0)>0\), so \(F(t)>F(0)>0\) holds at least for a small time \(t>0\), then by (2.17) we derive that
and
hold whenever F(t) exists and \(0<t\le t_{1}^{*}\). We divide (2.19) by \(F(t)^{2}\) and integrate the resultant inequality over \([0,t_{1}^{*}]\) to have
On the other hand, by (2.2) we know that
which together with (2.21) imply that \(T_{1}<t_{1}^{*}\). We complete the proof of Theorem 2.1. \(\square \)
Remark 2.1
In Theorem 2.1 above, we show the blowup of classical solutions to the problem (1.1)–(1.4) without the three restrictions \(m(0)\ge 0\), \(\textrm{Trace}\{{\mathbb {S}}_{10}(x)\}=0\) and \(A(0)\le 0\), so we have improved the blowup result of [10].
3 Blowup for Solutions Decay at Far Fields
In this section, we need the following averaged quantities:
where \(I(t),E(t),E_{k}(t)\) and \(E_{i}(t)\) represent the momentum of inertia, the total energy, the kinetic energy and the internal energy, respectively, which have been used in [5, 9]. We remark that the averaged quantity J(t) is new in the literature. In this section, we only consider the classical solutions with decay at far fields. Precisely, for any \(T>0\), we require that the solutions \((\rho ,{\textbf{u}},{\mathbb {S}}_{1},S_{2})\) satisfy that
We should remark that the condition (3.4) guarantees that the integration by parts in our calculations makes sense (see also [5, 9]).
The main result of this section is stated as follows.
Theorem 3.1
Let \((\rho ,{\textbf{u}},{\mathbb {S}}_{1},S_{2})\) be a \(C^{1}\) solution to the Cauchy problem (1.1)–(1.4) with \(\tau _{1}=0\) satisfying (3.4) for \(0\le t\le T_{2}\) with the initial data satisfying \(0<|A(0)|,I(0),J(0)<+\infty \). For any fixed \(t_{2}^{*}>0\), if
and
then \(T_{2}<t_{2}^{*}\).
Proof
We multiply (1.1)\(_{2}\) by \({\textbf{u}}\) and integrate it over \({\mathbb {R}}^{3}\) to have
where we have used \({\mathbb {S}}={\mathbb {S}}_{1}+S_{2}{\mathbb {I}}\) and (1.2) with \(\tau _{1}=0\). We use integration by part and (1.1)\(_{1}\) to obtain
Similarly,
We estimate the integral \(\mu \int _{{\mathbb {R}}^{3}}\textrm{div}[(\nabla {\textbf{u}}+\nabla {\textbf{u}}^{T})-\frac{2}{3}\textrm{div}{\textbf{u}}{\mathbb {I}}]\cdot {\textbf{u}}dx\) as follows. Noticing that
one has
Consequently,
Combining (3.7)–(3.10), (3.14) and (3.2), we obtain
We multiply (1.3) by \(\frac{S_{2}}{\lambda }\) and integrate the resultant equation over \({\mathbb {R}}^{3}\) to have
In view of (3.15), (3.16) and (3.3), it holds
which implies that
By (3.1), (1.1)\(_{1}\) and (1.8), we know that
In view of (1.8) and (1.1)\(_{2}\), we get
where we have used \({\mathbb {S}}={\mathbb {S}}_{1}+S_{2}{\mathbb {I}}\) and (1.2) with \(\tau _{1}=0\). Using (3.4) and integration by part, we obtain
It follows from (3.20) to (3.24) that
which together with (3.2) lead to
We integrate (1.3) over \({\mathbb {R}}^{3}\) and use the condition \(|{\textbf{u}}|\in L^{\infty }(0,T;L^{1}({\mathbb {R}}^{3}))\) in (3.4) to have
which together with (1.7) and (3.26) imply that
By (3.19), (3.28), (3.2), (3.3) and (3.18), we know that
We integrate (3.29) over \({\mathbb {R}}^{3}\) twice and use (3.19) to obtain
By the Schwarz inequality, (3.1) and (3.2), it holds
which together with (3.28) and (3.30) imply that
This corresponds to (2.16), so the rest proof is similar to the one of Theorem 2.1, we omit the details. \(\square \)
In fact, if we assume \(A(0)\le 0\), we can obtain a refined blowup result for the Cauchy problem (1.1)–(1.4) with solutions decay at far fields.
Corollary 3.1
Denote
Under the assumptions of Theorem 3.1, if we further assume that \(A(0)\le 0\), \(F(0)>0\) and
then the life span \(T_{2}\) of the classical solutions to (1.1)–(1.4) satisfies that
Proof
By the condition \(A(0)\le 0\) and (3.28)–(3.30), we know that
and
Then (3.32) becomes
The inequality (3.36) implies that F(t) increases with time, which together with \(F(0)>0\) lead to the fact that \(F(t)>0\) for \(t\in [0,T_{2}]\). We use (3.33) to rewrite (3.38) as
Noticing that
we have \(c_{2}>0\). So we divide (3.39) by \(F(t)^{2}\) and integrate the resultant inequality over \([0,T_{2}]\) to obtain
which can be solved as (3.35). The proof of Corollary 3.1 is finished. \(\square \)
Remark 3.1
The condition (3.34) in Corollary 3.1 is used to ensure that the upper bound in (3.35) is positive. In fact, by using (3.34) and the monotonicity of the function \(\tan x\) in \((0,\frac{\pi }{2})\), we have
Remark 3.2
If \(\tau _{1}>0\), we do not know how to treat the integral \(\int _{{\mathbb {R}}^{3}}\textrm{div}{\mathbb {S}}_{1}\cdot {\textbf{u}}dx\), which is crucial to obtain the inequality like (3.18). Perhaps we need to construct some new averaged quantities for the case of \(\tau _{1}>0\).
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Acknowledgements
This work is supported by the Project of Youth Backbone Teachers of Colleges and Universities in Henan Province (2019GGJS176) and the Vital Science Research Foundation of Henan Province Education Department (22A110024).
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Dong, J. Remarks on Blowup of Solutions for Compressible Navier–Stokes Equations with Revised Maxwell’s Law. Bull. Malays. Math. Sci. Soc. 46, 38 (2023). https://doi.org/10.1007/s40840-022-01437-3
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DOI: https://doi.org/10.1007/s40840-022-01437-3