Abstract
In this paper, we investigate the initial-boundary value problem to the heat conductive compressible Navier–Stokes equations. Local existence and uniqueness of strong solutions is established with any such initial data that the initial density \(\rho _0\), velocity \(u_0\), and temperature \(\theta _0\) satisfy \(\rho _0\in W^{1,q}\), with \(q\in (3,6)\), \(u_0\in H^1_0\), and \(\sqrt{\rho _0}\theta _0\in L^2\). The initial density is assumed to be only nonnegative and thus the initial vacuum is allowed. In addition to the necessary regularity assumptions, we do not require any initial compatibility conditions such as those proposed in Cho and Kim (J Differ Equ 228(2):377–411, 2006), which although are widely used in many previous works, put some inconvenient constrains on the initial data. Due to the weaker regularities of the initial data and the absence of the initial compatibility conditions, leading to weaker regularities of the solutions compared with those in the previous works, the uniqueness of solutions obtained in the current paper does not follow from the arguments used in the existing literatures. Our proof of the uniqueness of solutions is based on the following new idea of two-stages argument: (1) showing that the difference of two solutions (or part of their components) with the same initial data is controlled by some power functions of the time variable; (2) carrying out some singular-in-time weighted energy differential inequalities fulfilling the structure of the Grönwall inequality. The existence is established in the Euler coordinates, while the uniqueness is proved in the Lagrangian coordinates first and then transformed back to the Euler coordinates.
Similar content being viewed by others
References
Bresch, D., Jabin, P.-E.: Global existence of weak solutions for compressible Navier–Stokes equations: thermodynamically unstable pressure and anisotropic viscous stress tensor. Ann. Math. (2) 188(2), 577–684 (2018)
Chen, G.-Q., Hoff, D., Trivisa, K.: Global solutions of the compressible Navier–Stokes equations with large discontinuous initial data. Commun. Partial Differ. Equ. 25(11–12), 2233–2257 (2000)
Chen, Q., Miao, C., Zhang, Z.: Global well-posedness for compressible Navier–Stokes equations with highly oscillating initial velocity. Commun. Pure Appl. Math. 63(9), 1173–1224 (2010)
Chikami, N., Danchin, R.: On the well-posedness of the full compressible Navier–Stokes system in critical Besov spaces. J. Differ. Equ. 258(10), 3435–3467 (2015)
Cho, Y., Choe, H.J., Kim, H.: Unique solvability of the initial boundary value problems for compressible viscous fluids. J. Math. Pures Appl. (9) 83(2), 243–275 (2004)
Cho, Y., Kim, H.: Existence results for viscous polytropic fluids with vacuum. J. Differ. Equ. 228(2), 377–411 (2006)
Choe, H.J., Kim, H.: Strong solutions of the Navier–Stokes equations for isentropic compressible fluids. J. Differ. Equ. 190(2), 504–523 (2003)
Danchin, R.: Global existence in critical spaces for compressible Navier–Stokes equations. Invent. Math. 141(3), 579–614 (2000)
Danchin, R.: Local theory in critical spaces for compressible viscous and heat-conductive gases. Commun. Partial Differ. Equ. 26(7–8), 1183–1233 (2001)
Danchin, R., Mucha, P.B.: The incompressible Navier–Stokes equations in vacuum. Comm. Pure Appl. Math. 72(7), 1351–1385 (2019)
Danchin, R., Xu, J.: Optimal decay estimates in the critical \(L^p\) framework for flows of compressible viscous and heat-conductive gases. J. Math. Fluid Mech. 20(4), 1641–1665 (2018)
Fang, D., Zhang, T., Zi, R.: Global solutions to the isentropic compressible Navier–Stokes equations with a class of large initial data. SIAM J. Math. Anal. 50(5), 4983–5026 (2018)
Feireisl, E.: On the motion of a viscous, compressible, and heat conducting fluid. Indiana Univ. Math. J. 53(6), 1705–1738 (2004)
Feireisl, E., Novotný, A., Petzeltová, H.: On the existence of globally defined weak solutions to the Navier–Stokes equations. J. Math. Fluid Mech. 3(4), 358–392 (2001)
Gong, H., Li, J., Liu, X., Zhang, X.: Local well-posedness of isentropic compressible Navier–Stokes equations with vacuum. Commun. Math. Sci. 18(7), 1891–1909 (2020)
Graffi, D.: Il teorema di unicità nella dinamica dei fluidi compressibili. J. Ration. Mech. Anal. 2, 99–106 (1953)
Hoff, D.: Discontinuous solutions of the Navier–Stokes equations for multidimensional flows of heat-conducting fluids. Arch. Ration. Mech. Anal. 139(4), 303–354 (1997)
Huang, X.D.: On local strong and classical solutions to the three-dimensional barotropic compressible Navier–Stokes equations with vacuum. Sci. China Math. 64(8), 1771–1788 (2021)
Huang, X., Li, J.: Global classical and weak solutions to the three-dimensional full compressible Navier–Stokes system with vacuum and large oscillations. Arch. Ration. Mech. Anal. 227(3), 995–1059 (2018)
Huang, X., Li, J., Xin, Z.: Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier–Stokes equations. Commun. Pure Appl. Math. 65(4), 549–585 (2012)
Itaya, N.: On the Cauchy problem for the system of fundamental equations describing the movement of compressible viscous fluid. Kodai Math. Sem. Rep. 23, 60–120 (1971)
Jiang, S., Zhang, P.: On spherically symmetric solutions of the compressible isentropic Navier–Stokes equations. Commun. Math. Phys. 215(3), 559–581 (2001)
Jiang, S., Zhang, P.: Axisymmetric solutions of the 3D Navier–Stokes equations for compressible isentropic fluids. J. Math. Pures Appl. (9) 82(8), 949–973 (2003)
Jiang, S., Zlotnik, A.: Global well-posedness of the Cauchy problem for the equations of a one-dimensional viscous heat-conducting gas with Lebesgue initial data. Proc. R. Soc. Edinb. Sect. A 134(5), 939–960 (2004)
Kanel, J.I.: A model system of equations for the one-dimensional motion of a gas. Differencial’nye Uravnenija 4, 721–734 (1968)
Kazhikhov, A.V.: On the Cauchy problem for the equations of a viscous gas. Sibirsk. Mat. Zh. 23(1), 60–64 (1982)
Kazhikhov, A.V., Shelukhin, V.V.: Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas. J. Appl. Math. Mech. 41(2), 273–282 (1977). translated from Prikl. Mat. Meh. 41 (1977), no. 2, 282–291
Lai, S., Xu, H., Zhang, J.: Well-posedness and exponential decay for the Navier–Stokes equations of viscous compressible heat-conductive fluids with vacuum. Math. Models Methods Appl. Sci. 32(9), 1725–1784 (2022)
Li, J.: Local existence and uniqueness of strong solutions to the Navier–Stokes equations with nonnegative density. J. Differ. Equ. 263(10), 6512–6536 (2017)
Li, J.: Global well-posedness of the one-dimensional compressible Navier–Stokes equations with constant heat conductivity and nonnegative density. SIAM J. Math. Anal. 51(5), 3666–3693 (2019)
Li, J.: Global well-posedness of non-heat conductive compressible Navier–Stokes equations in 1D. Nonlinearity 33(5), 2181–2210 (2020)
Li, J.: Global small solutions of heat conductive compressible Navier–Stokes equations with vacuum: smallness on scaling invariant quantity. Arch. Ration. Mech. Anal. 237(2), 899–919 (2020)
Li, J., Liang, Z.: Some uniform estimates and large-time behavior of solutions to one-dimensional compressible Navier–Stokes system in unbounded domains with large data. Arch. Ration. Mech. Anal. 220(3), 1195–1208 (2016)
Li, J., Xin, Z.: Global well-posedness and large time asymptotic behavior of classical solutions to the compressible Navier–Stokes equations with vacuum. Ann. PDE 5(1), 37 pp (2019)
Li, J., Xin, Z.: Entropy bounded solutions to the one-dimensional compressible Navier–Stokes equations with zero heat conduction and far field vacuum. Adv. Math. 361, 106923, 50 pp (2020)
Li, J., Xin, Z.: Entropy-bounded solutions to the one-dimensional heat conductive compressible Navier–Stokes equations with far field vacuum. Commun. Pure Appl. Math. 75(11), 2393–2445 (2022)
Li, J., Xin, Z.: Local and global well-posedness of entropy-bounded solutions to the compressible Navier–Stokes equations in multi-dimensions. Sci. China Math. (2022). https://doi.org/10.1007/s11425-022-2047-0
Lions, P.-L.: Mathematical topics in fluid mechanics. Vol. 2, Oxford Lecture Series in Mathematics and its Applications, 10, The Clarendon Press, Oxford University Press, New York (1998)
Matsumura, A., Nishida, T.: The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids. Proc. Jpn. Acad. Ser. A Math. Sci. 55(9), 337–342 (1979)
Matsumura, A., Nishida, T.: The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ. 20(1), 67–104 (1980)
Matsumura, A., Nishida, T.: Initial-boundary value problems for the equations of motion of general fluids, in Computing methods in applied sciences and engineering, V (Versailles,: 389–406. North-Holland, Amsterdam (1981)
Merle, F., Rapha’el, P., Rodnianski, I., Szeftel, J.: On the implosion of a compressible fluid I: smooth self-similar inviscid profiles. Ann. Math. (2) 196, 567–778 (2022)
Merle, F., Rapha’el, P., Rodnianski, I., Szeftel, J.: On the implosion of a compressible fluid II: singularity formation. Ann. Math. (2) 196, 779–889 (2022)
Nash, J.: Le problème de Cauchy pour les équations différentielles d’un fluide général. Bull. Soc. Math. France 90, 487–497 (1962)
Salvi, R., Straškraba, I.: Global existence for viscous compressible fluids and their behavior as \(t\rightarrow \infty \). J. Fac. Sci. Univ. Tokyo Sect. IA Math. 40(1), 17–51 (1993)
Serrin, J.: Mathematical principles of classical fluid mechanics, in Handbuch der Physik (herausgegeben von S. Flügge), Bd. 8/1, Strömungsmechanik I (Mitherausgeber C. Truesdell), 125–263, Springer-Verlag, Berlin
Solonnikov, V.A.: Solvability of the problem of the motion of a viscous incompressible fluid that is bounded by a free surface. Izv. Akad. Nauk SSSR Ser. Mat. 41(6), 1388–1424 (1977)
Vol’pert, A.I., Hudjaev, S.I.: The Cauchy problem for composite systems of nonlinear differential equations. Mat. Sb. (N.S.) 87(129), 504–528 (1972)
Wen, H., Zhu, C.: Global solutions to the three-dimensional full compressible Navier–Stokes equations with vacuum at infinity in some classes of large data. SIAM J. Math. Anal. 49(1), 162–221 (2017)
Zlotnik, A.A., Amosov, A.A.: Stability of generalized solutions of the one-dimensional motion of a viscous heat-conducting gas. (Russian), Math. Notes 63(5–6), 736–746 (1998); translated from Mat. Zametki 63(6), 835–846 (1998)
Acknowledgements
This work was supported in part by the National Natural Science Foundation of China (11971009 and 11871005), by the Key Project of National Natural Science Foundation of China (12131010), and by the Guangdong Basic and Applied Basic Research Foundation (2020B1515310005, 2020B1515310002, and 2021A1515010247).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work.
Additional information
Communicated by D. Bresch.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
5. Appendix
5. Appendix
In this “Appendix”, we give the proof of Proposition 4.1 and Proposition 4.2, as well as a lemma used in (4.34).
Proof of Proposition 4.1
Solving (4.8) yields
and, thus,
where \(C_*:=e^{\int _0^{{T_0}}\Vert \text {div} u\Vert _\infty dt}\). Since \(\text {det}\nabla \psi (x,t)=J(\psi (x,t),t)\), it holds that
from which, by (5.1), one gets
Letting \(\alpha \rightarrow \infty \) in the above leads to the estimate for \(\alpha =\infty \). Therefore,
Applying the above to \(g(\psi (x,t))\) leads to
Combining (5.2) with (5.3) leads to (4.12).
It follows from (4.7) that
and, thus,
Therefore
Similarly, one gets from (4.6) that
Thanks to (5.1), (5.4), and (5.5), it follows from (4.6)–(4.8) and (5.2) that
One gets from (4.7) that
and, thus, by (5.4), it follows that
and further that
Integrating the above over \(\Omega \), it follows from the Hölder inequality and (5.2) that
and, thus,
Applying the Grönwall inequality to the above yields
Similarly, one derives from (4.6) and (4.8) that
Conclusion (4.9) follows from (5.1) and (5.4)–(5.8).
Fix \(t_0\in [0,{T_0}]\) and denote
Then, it is clear that
By direct calculations and recalling the definitions of \(a_{ij}(y,t)\) and \(b_{ij}(y,t)\), one has
Therefore, it follows from (5.4) and (5.5) that
Thanks to (5.9), it follows from (5.2) and (5.3) that
As a result, recalling the definition of G, one gets
which applied to \(g(\psi (x,t_0))\) yields
Therefore (4.11) holds.
By direct calculations and recalling the definitions of \(a_{ij}(y,t)\) and \(b_{ij}(y,t)\), one has
Then, it follows from (5.4) and (5.5) that
Thanks to these, it follows from the Hölder and Sobolev inequalities, (5.7)–(5.8), and (5.2)–(5.3) that: for \(1\le \alpha <3\),
for \(\alpha =3\),
and for \(3<\alpha \le q\),
Therefore, for any \(1\le \alpha \le q\), it holds that
Thanks to these, by (5.10)–(5.11), and recalling the definition of G, it follows that
for any \(1\le \alpha \le q\). This proves
which, applied to \(g(\psi (x,t))\), yields further
Therefore, (4.10) holds. \(\square \)
Proof of Proposition 4.2
By Proposition 4.1, it holds that \(\Vert h(\varphi (\cdot ,t),t)\Vert \le C\Vert h\Vert \) for any \(t\in [0,{T_0}]\) and, thus,
Fix \(t_0\in [0,{T_0}]\) and take arbitrary \(\varepsilon >0\). Choose \(\xi \in C_c^\infty (\Omega )\) such that
(i) By the Hölder inequality and by Proposition 4.1, one deduces
for any \(t\in [0,{T_0}]\). Recalling (4.1) and by Proposition 4.1, it follows from the Gagliardo-Nirenberg and Hölder inequalities that
Plugging this estimate into (5.12) leads to
which implies \(\Vert h(\varphi (\cdot ,t),t)-h(\varphi (\cdot ,t_0),t_0)\Vert \rightarrow 0\) as \(t\rightarrow t_0\), for any \(t_0\in [0,{T_0}]\). Therefore, \(h(\varphi (\cdot ,t),t)\in C([0,{T_0}]; L^2)\).
(ii) Note that
Since \(\text {det}\nabla \psi (x,t)=J(\psi (x,t),t)>0\), one deduces
Since \(\chi (\psi (x,t_0))J(\psi (x,t_0),t_0)\in L^2\), guaranteed by Proposition 4.1, one has
Recalling (4.2) and (4.8), one has
and, thus, by Proposition 4.1, it follows
Thanks to this, one has
Using the Hölder inequality and by Proposition 4.1, it follows
For \(R_3\), it follows from the Hölder inequality and (5.13) that
Combining (5.16)–(5.18), one gets
With the aid of this and recalling (5.15), one derives
which implies that \(h(\varphi (\cdot ,t),t)\) is weakly continuous in \(L^2(\Omega )\) at any \(t_0\in [0,{T_0}]\). Therefore, \(h(\varphi (\cdot ,t),t)\in C_w([0,{T_0}]; L^2)\). \(\square \)
Finally, we prove the following lemma which is used in (4.34) during the proof of the uniqueness in the previous section.
Lemma 5.1
Given a bounded domain \(\Omega \) in \({\mathbb {R}}^3\). Let \(\Phi \in W^{2,q}\) with \(q\in (3,6)\) be a bijective mapping on \(\Omega \). Denote \(x=\Phi (y)\) and \(y=\Psi (x)\). Then, it holds that
Proof
We only give the proof of the first identity while the second one can be proved in the same way. In the proof of this lemma, for a matrix \(A=(a_{ij})_{3\times 3}\), we use \(R_i(A)\), \(M_{ij}(A)\), and \(A_{adj}\), respectively, to denote the i-th row of A, the minor of the entry \(a_{ij}\), and the classical adjoint of A that is, \(A_{adj}=((-1)^{i+j}M_{ij})^T\). Denote \(\nabla _xy=(\partial _{x_i}y_j)_{3\times 3}\) and \(\nabla _yx=(\partial _{y_i}x_j)_{3\times 3}\). Then, the chain rule gives \(\nabla _xy\nabla _yx=I\). Thanks to this, one deduces
Therefore,
It remains to show that \(\text {div}_y\Big (R_i((\nabla _yx)_{adj})\Big )=0\) for \(i=1,2,3.\) We only prove the case \(i=1\), the proofs for \(i=2,3\) are the same. By definition, one has
where the determinant is understood by expanding along the first column. By direct calculations, one can verify that the above determinant is identically zero and, thus, the conclusion holds. \(\square \)
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Li, J., Zheng, Y. Local Existence and Uniqueness of Heat Conductive Compressible Navier–Stokes Equations in the Presence of Vacuum Without Initial Compatibility Conditions. J. Math. Fluid Mech. 25, 14 (2023). https://doi.org/10.1007/s00021-022-00761-9
Accepted:
Published:
DOI: https://doi.org/10.1007/s00021-022-00761-9
Keywords
- Compressible Navier–Stokes equations
- Existence and uniqueness
- Vacuum
- Singular-in-time weighted estimates
- Lagrangian coordinates