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Local Existence and Uniqueness of Heat Conductive Compressible Navier–Stokes Equations in the Presence of Vacuum Without Initial Compatibility Conditions

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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.

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References

  1. 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)

  2. 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)

    Article  MathSciNet  MATH  Google Scholar 

  3. 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)

    MathSciNet  MATH  Google Scholar 

  4. 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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. 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)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cho, Y., Kim, H.: Existence results for viscous polytropic fluids with vacuum. J. Differ. Equ. 228(2), 377–411 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Choe, H.J., Kim, H.: Strong solutions of the Navier–Stokes equations for isentropic compressible fluids. J. Differ. Equ. 190(2), 504–523 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Danchin, R.: Global existence in critical spaces for compressible Navier–Stokes equations. Invent. Math. 141(3), 579–614 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Danchin, R.: Local theory in critical spaces for compressible viscous and heat-conductive gases. Commun. Partial Differ. Equ. 26(7–8), 1183–1233 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Danchin, R., Mucha, P.B.: The incompressible Navier–Stokes equations in vacuum. Comm. Pure Appl. Math. 72(7), 1351–1385 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  11. 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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. 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)

    Article  MathSciNet  MATH  Google Scholar 

  13. Feireisl, E.: On the motion of a viscous, compressible, and heat conducting fluid. Indiana Univ. Math. J. 53(6), 1705–1738 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. 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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. 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)

    Article  MathSciNet  MATH  Google Scholar 

  16. Graffi, D.: Il teorema di unicità nella dinamica dei fluidi compressibili. J. Ration. Mech. Anal. 2, 99–106 (1953)

    MATH  Google Scholar 

  17. 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)

    Article  MathSciNet  MATH  Google Scholar 

  18. 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)

    Article  MathSciNet  MATH  Google Scholar 

  19. 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)

    Article  MathSciNet  MATH  Google Scholar 

  20. 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)

    Article  MathSciNet  MATH  Google Scholar 

  21. 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)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jiang, S., Zhang, P.: On spherically symmetric solutions of the compressible isentropic Navier–Stokes equations. Commun. Math. Phys. 215(3), 559–581 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. 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)

    Article  MathSciNet  MATH  Google Scholar 

  24. 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)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kanel, J.I.: A model system of equations for the one-dimensional motion of a gas. Differencial’nye Uravnenija 4, 721–734 (1968)

    MathSciNet  Google Scholar 

  26. Kazhikhov, A.V.: On the Cauchy problem for the equations of a viscous gas. Sibirsk. Mat. Zh. 23(1), 60–64 (1982)

  27. 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

  28. 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)

    Article  MathSciNet  MATH  Google Scholar 

  29. 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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. 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)

    Article  MathSciNet  MATH  Google Scholar 

  31. Li, J.: Global well-posedness of non-heat conductive compressible Navier–Stokes equations in 1D. Nonlinearity 33(5), 2181–2210 (2020)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. 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)

    Article  MathSciNet  MATH  Google Scholar 

  33. 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)

    Article  MathSciNet  MATH  Google Scholar 

  34. 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)

  35. 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)

  36. 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)

    Article  MathSciNet  Google Scholar 

  37. 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

  38. 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)

  39. 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)

    Article  MathSciNet  MATH  Google Scholar 

  40. 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)

    MathSciNet  MATH  Google Scholar 

  41. 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)

  42. 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)

    Article  MathSciNet  MATH  Google Scholar 

  43. 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)

    Article  MathSciNet  MATH  Google Scholar 

  44. Nash, J.: Le problème de Cauchy pour les équations différentielles d’un fluide général. Bull. Soc. Math. France 90, 487–497 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  45. 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)

    MathSciNet  MATH  Google Scholar 

  46. 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

  47. 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)

  48. 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)

    MathSciNet  Google Scholar 

  49. 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)

    Article  MathSciNet  MATH  Google Scholar 

  50. 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)

Download references

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).

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  1. South China Research Center for Applied Mathematics and Interdisciplinary Studies, School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China

    Jinkai Li & Yasi Zheng

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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

$$\begin{aligned} J(y,t)=e^{-\int _0^t\text {div} u(\varphi (y,s),s)ds} \end{aligned}$$

and, thus,

$$\begin{aligned} \frac{1}{C_*}\le J(y,t)\le C_*,\quad \forall (y,t)\in \Omega \times [0,{T_0}], \end{aligned}$$
(5.1)

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

$$\begin{aligned} \Vert g(\varphi (\cdot ,t))\Vert _\alpha= & {} \left( \int _\Omega |g(\varphi (y,t))|^\alpha \textrm{d}y\right) ^\frac{1}{\alpha }=\left( \int _\Omega |g(x)|^\alpha |\text {det}\nabla \psi (x,t)| \textrm{d}x\right) ^\frac{1}{\alpha }\\= & {} \left( \int _\Omega |g(x)|^\alpha J(\psi (x,t),t) \textrm{d}x\right) ^\frac{1}{\alpha }, \end{aligned}$$

from which, by (5.1), one gets

$$\begin{aligned} C_*^{-\frac{1}{\alpha }}\Vert g\Vert _\alpha \le \Vert g(\varphi (\cdot ,t))\Vert _\alpha \le C_*^\frac{1}{\alpha }\Vert g\Vert _\alpha ,\quad \alpha \in [1,\infty ). \end{aligned}$$

Letting \(\alpha \rightarrow \infty \) in the above leads to the estimate for \(\alpha =\infty \). Therefore,

$$\begin{aligned} \Vert g\Vert _\alpha \simeq \Vert g(\varphi (\cdot ,t))\Vert _\alpha ,\quad \alpha \in [1,\infty ]. \end{aligned}$$
(5.2)

Applying the above to \(g(\psi (x,t))\) leads to

$$\begin{aligned} \Vert g\Vert _\alpha \simeq \Vert g(\psi (\cdot ,t))\Vert _\alpha ,\quad \alpha \in [1,\infty ]. \end{aligned}$$
(5.3)

Combining (5.2) with (5.3) leads to (4.12).

It follows from (4.7) that

$$\begin{aligned} \partial _t|B|^2=2B:(B\nabla u(\varphi ,t))\le C\Vert \nabla u\Vert _\infty |B|^2 \end{aligned}$$

and, thus,

$$\begin{aligned} |B|^2(y,t)\le 3e^{C\int _0^t\Vert \nabla u\Vert _\infty \textrm{d}s},\quad \forall (y,t)\in \Omega \times [0,{T_0}]. \end{aligned}$$

Therefore

$$\begin{aligned} \sup _{0\le t\le {T_0}}\Vert B\Vert _\infty (t)\le Ce^{C\int _0^{T_0}\Vert \nabla u\Vert _\infty \textrm{d}t}\le C. \end{aligned}$$
(5.4)

Similarly, one gets from (4.6) that

$$\begin{aligned} \sup _{0\le t\le T_0}\Vert A\Vert _\infty \le C. \end{aligned}$$
(5.5)

Thanks to (5.1), (5.4), and (5.5), it follows from (4.6)–(4.8) and (5.2) that

$$\begin{aligned} \sup _{0\le t\le {T_0}}\Vert (J_t, A_t, B_t)\Vert \le C\sup _{0\le t\le {T_0}}\Vert \nabla u(\varphi (\cdot ,t),t)\Vert \le C\sup _{0\le t\le {T_0}}\Vert \nabla u\Vert \le C. \end{aligned}$$
(5.6)

One gets from (4.7) that

$$\begin{aligned} \partial _t\partial _iB=\partial _iB\nabla u(\varphi ,t)+B\nabla \partial _lu(\varphi ,t)\partial _i\varphi _l =\partial _iB\nabla u(\varphi ,t)+B\nabla \partial _lu(\varphi ,t)b_{il} \end{aligned}$$

and, thus, by (5.4), it follows that

$$\begin{aligned} \partial _t|\nabla B|^2= & {} 2\partial _iB:(\partial _iB\nabla u(\varphi ,t)+B\nabla \partial _lu(\varphi ,t)b_{il})\\\le & {} C(\Vert \nabla u\Vert _\infty |\nabla B|^2+|\nabla B||B|^2|\nabla ^2u(\varphi ,t)|)\\\le & {} C(\Vert \nabla u\Vert _\infty |\nabla B|^2+|\nabla ^2u(\varphi ,t)||\nabla B|), \end{aligned}$$

and further that

$$\begin{aligned} \partial _t|\nabla B|^q\le C(\Vert \nabla u\Vert _\infty |\nabla B|^q+|\nabla ^2u(\varphi ,t)|\nabla B|^{q-1}). \end{aligned}$$

Integrating the above over \(\Omega \), it follows from the Hölder inequality and (5.2) that

$$\begin{aligned} \frac{d}{dt}\Vert \nabla B\Vert _q^q\le & {} C\left( \Vert \nabla u\Vert _\infty \Vert \nabla B\Vert _q^q+\Vert \nabla ^2u(\varphi ,t)\Vert _q\Vert \nabla B\Vert _q^{q-1}\right) \\\le & {} C\left( \Vert \nabla u\Vert _\infty \Vert \nabla B\Vert _q^q+\Vert \nabla ^2u\Vert _q\Vert \nabla B\Vert _q^{q-1}\right) \\ \end{aligned}$$

and, thus,

$$\begin{aligned} \frac{d}{dt}\Vert \nabla B\Vert _q\le C\left( \Vert \nabla u\Vert _\infty \Vert \nabla B\Vert _q+\Vert \nabla ^2u\Vert _q\right) . \end{aligned}$$

Applying the Grönwall inequality to the above yields

$$\begin{aligned} \sup _{0\le t\le {T_0}}\Vert \nabla B\Vert _q\le Ce^{C\int _0^{T_0}\Vert \nabla u\Vert _\infty \textrm{d}t}\int _0^{T_0}\Vert \nabla ^2u\Vert _q \textrm{d}t\le C. \end{aligned}$$
(5.7)

Similarly, one derives from (4.6) and (4.8) that

$$\begin{aligned} \sup _{0\le t\le {T_0}}(\Vert \nabla A\Vert _q+\Vert \nabla J\Vert _q)\le C. \end{aligned}$$
(5.8)

Conclusion (4.9) follows from (5.1) and (5.4)–(5.8).

Fix \(t_0\in [0,{T_0}]\) and denote

$$\begin{aligned} G(y):=g(\varphi (y,t_0)),\quad \forall y\in \Omega . \end{aligned}$$

Then, it is clear that

$$\begin{aligned} g(x)=G(\psi (x,t_0)),\quad \forall x\in \Omega . \end{aligned}$$

By direct calculations and recalling the definitions of \(a_{ij}(y,t)\) and \(b_{ij}(y,t)\), one has

$$\begin{aligned}{} & {} \partial _iG(y)=b_{il}(y,t_0)\partial _lg(\varphi (y,t_0)),\quad \partial _ig(x) =a_{il}(\psi (x,t_0),t_0)\partial _lG(\psi (x,t_0)). \end{aligned}$$

Therefore, it follows from (5.4) and (5.5) that

$$\begin{aligned}{} & {} |\nabla G(y)|\le C|\nabla g(\varphi (y,t_0))|,\quad |\nabla g(x)|\le C|\nabla G(\psi (x,t_0))|. \end{aligned}$$
(5.9)

Thanks to (5.9), it follows from (5.2) and (5.3) that

$$\begin{aligned} \Vert \nabla G\Vert _\alpha \le C\Vert \nabla g(\varphi (\cdot ,t_0))\Vert _\alpha \le C\Vert \nabla g\Vert _\alpha ,\quad 1\le \alpha \le \infty , \end{aligned}$$
(5.10)
$$\begin{aligned} \Vert \nabla g\Vert _\alpha \le C\Vert \nabla G(\psi (\cdot ,t_0))\Vert _\alpha \le C\Vert \nabla G\Vert _\alpha ,\quad 1\le \alpha \le \infty . \end{aligned}$$
(5.11)

As a result, recalling the definition of G, one gets

$$\begin{aligned} \Vert \nabla [g(\varphi (\cdot ,t_0))]\Vert _\alpha \simeq \Vert \nabla g\Vert _\alpha ,\quad 1\le \alpha \le \infty , \end{aligned}$$

which applied to \(g(\psi (x,t_0))\) yields

$$\begin{aligned} \Vert \nabla g\Vert _\alpha \simeq \Vert \nabla [ g(\psi (\cdot ,t_0))]\Vert _\alpha ,\quad 1\le \alpha \le \infty . \end{aligned}$$

Therefore (4.11) holds.

By direct calculations and recalling the definitions of \(a_{ij}(y,t)\) and \(b_{ij}(y,t)\), one has

$$\begin{aligned} \partial _{ij}^2G(y)= & {} \partial _ib_{jl}(y,t_0)\partial _lg(\varphi (y,t_0))+b_{il}(y,t_0)b_{jm}(y,t_0) \partial _{lm}^2g(\varphi (y,t_0)),\\ \partial _{ij}^2g(x)= & {} a_{jm}(\psi (x,t_0),t_0)\partial _ma_{il}(\psi (x,t_0),t_0)\partial _lG(\psi (x,t_0))\\{} & {} +a_{il}(\psi (x,t_0),t_0)a_{jm}(\psi (x,t_0),t_0) \partial _{lm}^2G(\psi (x,t_0)). \end{aligned}$$

Then, it follows from (5.4) and (5.5) that

$$\begin{aligned}{} & {} |\nabla ^2G(y)|\le C|\nabla B(y,t_0)||\nabla g(\varphi (y,t_0))|+C|\nabla ^2g(\varphi (y,t_0))|,\\{} & {} |\nabla ^2g(x)|\le C|\nabla A(\psi (x,t_0),t_0)||\nabla G(\psi (x,t_0))|+C|\nabla ^2G(\psi (x,t_0))|. \end{aligned}$$

Thanks to these, it follows from the Hölder and Sobolev inequalities, (5.7)–(5.8), and (5.2)–(5.3) that: for \(1\le \alpha <3\),

$$\begin{aligned} \Vert \nabla ^2G\Vert _\alpha\le & {} C\left( \Vert \nabla B\Vert _3\Vert \nabla g(\varphi (\cdot ,t_0))\Vert _{\frac{3\alpha }{3-\alpha }}+\Vert \nabla ^2g(\varphi ( \cdot , t_0))\Vert _\alpha \right) \\\le & {} C\left( \Vert \nabla B\Vert _q\Vert \nabla g\Vert _{\frac{3\alpha }{3-\alpha }}+\Vert \nabla ^2g\Vert _\alpha \right) \le C\Vert \nabla g\Vert _{W^{1,\alpha }}, \\ \Vert \nabla ^2g\Vert _\alpha\le & {} C\left( \Vert \nabla A(\psi (\cdot ,t_0),t_0)\Vert _3\Vert \nabla G(\psi (\cdot ,t_0))\Vert _{\frac{3\alpha }{3-\alpha }}+\Vert \nabla ^2G(\psi (\cdot ,t_0))\Vert _\alpha \right) \\\le & {} C\left( \Vert \nabla A\Vert _3\Vert \nabla G\Vert _{\frac{3\alpha }{3-\alpha }}+\Vert \nabla ^2G\Vert _\alpha \right) \le C\Vert \nabla G\Vert _{W^{1,\alpha }}; \end{aligned}$$

for \(\alpha =3\),

$$\begin{aligned} \Vert \nabla ^2G\Vert _3\le & {} C\left( \Vert \nabla B\Vert _q\Vert \nabla g(\varphi (\cdot ,t_0))\Vert _{\frac{3q}{q-3}}+\Vert \nabla ^2g(\varphi ( \cdot , t_0))\Vert _3\right) \\\le & {} C\left( \Vert \nabla g\Vert _{\frac{3q}{q-3}}+\Vert \nabla ^2g\Vert _3\right) \le C\Vert \nabla g\Vert _{W^{1,3}},\\ \Vert \nabla ^2g\Vert _3\le & {} C\left( \Vert \nabla A(\psi (\cdot ,t_0),t_0)\Vert _q\Vert \nabla G(\psi (\cdot ,t_0))\Vert _{\frac{3q}{q-3}}+\Vert \nabla ^2G(\psi (\cdot ,t_0))\Vert _3\right) \\\le & {} C\left( \Vert \nabla A \Vert _q\Vert \nabla G\Vert _{\frac{3q}{q-3}}+\Vert \nabla ^2G\Vert _3\right) \le C\Vert \nabla G\Vert _{W^{1,3}}; \end{aligned}$$

and for \(3<\alpha \le q\),

$$\begin{aligned} \Vert \nabla ^2G\Vert _\alpha\le & {} C\left( \Vert \nabla B\Vert _q\Vert \nabla g(\varphi (\cdot ,t_0))\Vert _\infty +\Vert \nabla ^2g(\varphi ( \cdot , t_0))\Vert _\alpha \right) \\\le & {} C\left( \Vert \nabla g\Vert _\infty +\Vert \nabla ^2g\Vert _\alpha \right) \le C\Vert \nabla g\Vert _{W^{1,\alpha }},\\ \Vert \nabla ^2g\Vert _\alpha\le & {} C\left( \Vert \nabla A(\psi (\cdot ,t_0),t_0)\Vert _q\Vert \nabla G(\psi (\cdot ,t_0))\Vert _\infty +\Vert \nabla ^2G(\psi (\cdot ,t_0))\Vert _\alpha \right) \\\le & {} C\left( \Vert \nabla A\Vert _q\Vert \nabla G\Vert _\infty +\Vert \nabla ^2G\Vert _\alpha \right) \le C\Vert \nabla G\Vert _{W^{1,\alpha }}. \end{aligned}$$

Therefore, for any \(1\le \alpha \le q\), it holds that

$$\begin{aligned} \Vert \nabla ^2G\Vert _\alpha \le C\Vert \nabla g\Vert _{W^{1,\alpha }},\quad \Vert \nabla ^2g\Vert _\alpha \le C\Vert \nabla G\Vert _{W^{1,\alpha }}. \end{aligned}$$

Thanks to these, by (5.10)–(5.11), and recalling the definition of G, it follows that

$$\begin{aligned} \Vert \nabla [g(\varphi (\cdot ,t_0))]\Vert _{W^{1,\alpha }}\le C\Vert \nabla g\Vert _{W^{1,\alpha }},\quad \Vert \nabla g\Vert _{W^{1,\alpha }}\le C \Vert \nabla [g(\varphi (\cdot ,t_0))]\Vert _{W^{1,\alpha }}, \end{aligned}$$

for any \(1\le \alpha \le q\). This proves

$$\begin{aligned} \Vert \nabla [g(\varphi (\cdot ,t_0))]\Vert _{W^{1,\alpha }}\simeq \Vert \nabla g\Vert _{W^{1,\alpha }},\quad \forall \ 1\le \alpha \le q, \end{aligned}$$

which, applied to \(g(\psi (x,t))\), yields further

$$\begin{aligned} \Vert \nabla g\Vert _{W^{1,\alpha }}\simeq \Vert \nabla [g(\psi (x,t_0))]\Vert _{W^{1,\alpha }},\quad \forall \ 1\le \alpha \le q. \end{aligned}$$

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,

$$\begin{aligned} \Vert h(\varphi (\cdot ,t),t)\Vert _{L^\infty (0,{T_0}; L^2)}\le C\Vert h\Vert _{L^\infty (0,{T_0}; L^2)}. \end{aligned}$$

Fix \(t_0\in [0,{T_0}]\) and take arbitrary \(\varepsilon >0\). Choose \(\xi \in C_c^\infty (\Omega )\) such that

$$\begin{aligned} \Vert \xi -h(\cdot ,t_0)\Vert _{L^2}\le \varepsilon . \end{aligned}$$

(i) By the Hölder inequality and by Proposition 4.1, one deduces

$$\begin{aligned}{} & {} \Vert h(\varphi (\cdot ,t),t)-h(\varphi (\cdot ,t_0),t_0)\Vert \nonumber \\{} & {} \quad \le \Vert h(\varphi (\cdot ,t),t)-h(\varphi (\cdot ,t),t_0)\Vert + \Vert h(\varphi (\cdot ,t),t_0)-\xi (\varphi (\cdot ,t))\Vert \nonumber \\{} & {} \quad \quad +\Vert \xi (\varphi (y,t))-\xi (\varphi (y,t_0))\Vert +\Vert \xi (\varphi (y,t_0))-h(\varphi (\cdot ,t_0),t_0)\Vert \nonumber \\{} & {} \quad \le C (\Vert h(\cdot ,t)-h(\cdot ,t_0)\Vert + \Vert h(\cdot ,t_0)-\xi \Vert +\Vert \xi (\varphi (\cdot ,t)) -\xi (\varphi (\cdot ,t_0))\Vert ) \nonumber \\{} & {} \quad \le C(\Vert h(\cdot ,t)-h(\cdot ,t_0)\Vert +\varepsilon +\Vert \xi (\varphi (\cdot ,t))-\xi (\varphi (\cdot ,t_0))\Vert ) \end{aligned}$$
(5.12)

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

$$\begin{aligned}{} & {} \left\| \xi (\varphi (\cdot ,t))-\xi (\varphi (\cdot ,t_0))\right\| = \left\| \int _{t_0}^t\nabla \xi (\varphi (\cdot ,s))\cdot \partial _t\varphi (\cdot ,s) \textrm{d}s\right\| \nonumber \\{} & {} \quad =\left\| \int _{t_0}^t \nabla \xi (\varphi (\cdot ,s))\cdot u(\varphi (\cdot ,s),s) \textrm{d}s\right\| \le \left| \int _{t_0}^t\Vert \nabla \xi (\varphi (\cdot ,s))\Vert \Vert u\Vert _\infty \textrm{d}s\right| \nonumber \\{} & {} \quad \le C\left| \int _{t_0}^t\Vert \nabla \xi \Vert \Vert \nabla u\Vert ^\frac{1}{2}\Vert \nabla ^2u\Vert ^\frac{1}{2} \textrm{d}s\right| \le C |t-t_0|^\frac{3}{4},\quad \forall t\in [0,{T_0}]. \end{aligned}$$
(5.13)

Plugging this estimate into (5.12) leads to

$$\begin{aligned} \Vert h(\varphi (\cdot ,t),t)-h(\varphi (\cdot ,t_0),t_0)\Vert \le C(\Vert h(\cdot ,t)-h(\cdot ,t_0)\Vert +\varepsilon + |t-t_0|^\frac{3}{4}), \end{aligned}$$
(5.14)

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

$$\begin{aligned}{} & {} \int _\Omega [h(\varphi (y,t),t)-h(\varphi (y,t_0),t_0)]\chi (y) \textrm{d}y\\{} & {} \quad =\int _\Omega [h(\varphi (y,t),t)-h(\varphi (y,t),t_0)]\chi (y) \textrm{d}y+\int _\Omega [h(\varphi (y,t),t_0)-\xi (\varphi (y,t))]\chi (y) \textrm{d}y\\{} & {} \quad \quad +\int _\Omega [\xi (\varphi (y,t))-\xi (\varphi (y,t_0))]\chi (y) \textrm{d}y+\int _\Omega [\xi (\varphi (y,t_0))-h(\varphi (y,t_0),t_0)]\chi (y) \textrm{d}y\\{} & {} \quad =:R_1+R_2+R_3+R_4. \end{aligned}$$

Since \(\text {det}\nabla \psi (x,t)=J(\psi (x,t),t)>0\), one deduces

$$\begin{aligned} R_1= & {} \int _\Omega [h(\varphi (y,t),t)-h(\varphi (y,t),t_0)]\chi (y) \textrm{d}y\\= & {} \int _\Omega [h(x,t)-h(x,t_0)]\chi (\psi (x,t))|\text {det}\nabla \psi (x,t)| \textrm{d}x\\= & {} \int _\Omega [h(x,t)-h(x,t_0)]\chi (\psi (x,t))J(\psi (x,t),t) \textrm{d}x\\= & {} \int _\Omega [h(x,t)-h(x,t_0)][\chi (\psi (x,t))J(\psi (x,t),t)-\chi (\psi (x,t_0))J(\psi (x,t_0),t_0)] \textrm{d}x\\{} & {} +\int _\Omega [h(x,t)-h(x,t_0)]\chi (\psi (x,t_0))J(\psi (x,t_0),t_0) \textrm{d}x\\=: & {} R_{11}+R_{12}. \end{aligned}$$

Since \(\chi (\psi (x,t_0))J(\psi (x,t_0),t_0)\in L^2\), guaranteed by Proposition 4.1, one has

$$\begin{aligned} R_{12}\rightarrow 0,\quad \text{ as } t\rightarrow t_0. \end{aligned}$$
(5.15)

Recalling (4.2) and (4.8), one has

$$\begin{aligned}{} & {} \chi (\psi (x,t))J(\psi (x,t),t)-\chi (\psi (x,t_0))J(\psi (x,t_0),t_0)\\{} & {} \quad =\int _{t_0}^t[\chi (\psi (x,s))(\partial _tJ(\psi (x,s),s)+\nabla J(\psi (x,s),s)\cdot \partial _t\psi (x,s))\\{} & {} \quad \quad +\nabla \chi (\psi (x,s))\cdot \partial _t\psi (x,s)J(\psi (x,s),s)] \textrm{d}s \\{} & {} \quad =-\int _{t_0}^t(u(x,s)\cdot \nabla )\psi (x,s)\cdot [\chi (\psi (x,s)) \nabla J(\psi (x,s),s)\\{} & {} \quad \quad +\nabla \chi (\psi (x,s))J(\psi (x,s),s)] \textrm{d}s-\int _{t_0}^t\chi (\psi (x,s))\text {div} u(x,s)J(\psi (x,s),s) \textrm{d}s\\{} & {} \quad =-\int _{t_0}^tu(x,s)\cdot A(\psi (x,s),s)\cdot [\chi (\psi (x,s)) \nabla J(\psi (x,s),s)\\{} & {} \quad \quad +\nabla \chi (\psi (x,s))J(\psi (x,s),s)] \textrm{d}s\\ {}{} & {} \quad -\int _{t_0}^t\chi (\psi (x,s))\text {div} u(x,s)J(\psi (x,s),s) \textrm{d}s \end{aligned}$$

and, thus, by Proposition 4.1, it follows

$$\begin{aligned}{} & {} \Vert \chi (\psi (\cdot ,t))J(\psi (\cdot ,t),t)-\chi (\psi (\cdot ,t_0))J(\psi (\cdot ,t_0),t_0)\Vert \\{} & {} \quad \le \left| \int _{t_0}^t\Vert u\Vert _\infty \Vert A(\psi (\cdot ,s),s)\Vert \Vert \nabla \chi \Vert _\infty \Vert J\Vert _\infty + \Vert u\Vert _\infty \Vert A\Vert _\infty \Vert \chi \Vert _\infty \Vert \nabla J(\psi (\cdot ,s),s)\Vert \textrm{d}s\right| \\{} & {} \quad \quad +\left| \int _{t_0}^t\Vert \chi \Vert _\infty \Vert \text {div} u\Vert \Vert J\Vert _\infty \textrm{d}s\right| \le C\left| \int _{t_0}^t\left( \Vert \nabla u\Vert ^\frac{1}{2}\Vert \nabla ^2u\Vert ^\frac{1}{2}\right) \textrm{d}s\right| +C|t-t_0|\le C|t-t_0|^\frac{3}{4}. \end{aligned}$$

Thanks to this, one has

$$\begin{aligned} |R_{11}|\le & {} \Vert h(\cdot ,t)-h(\cdot ,t_0)\Vert \Vert \chi (\psi (\cdot ,t))J(\psi (\cdot ,t),t)-\chi (\psi (\cdot ,t_0))J(\psi (\cdot ,t_0),t_0)\Vert \nonumber \\\le & {} C|t-t_0|^\frac{3}{4}. \end{aligned}$$
(5.16)

Using the Hölder inequality and by Proposition 4.1, it follows

$$\begin{aligned} |R_2+R_4|\le & {} C\Vert \chi \Vert (\Vert h(\varphi (\cdot ,t),t_0)-\xi (\varphi (\cdot ,t))\Vert +\Vert h(\varphi (\cdot ,t_0),t_0)-\xi (\varphi (\cdot ,t_0))\Vert )\nonumber \\\le & {} C\Vert \chi \Vert \Vert h(\cdot , t_0)-\xi \Vert \le C \varepsilon . \end{aligned}$$
(5.17)

For \(R_3\), it follows from the Hölder inequality and (5.13) that

$$\begin{aligned} |R_3|\le \Vert \chi \Vert \Vert \xi (\varphi (y,t))-\xi (\varphi (y,t_0))\Vert \le C |t-t_0|^\frac{3}{4}. \end{aligned}$$
(5.18)

Combining (5.16)–(5.18), one gets

$$\begin{aligned} \left| \int _\Omega [h(\varphi (y,t),t)-h(\varphi (y,t_0),t_0)]\chi (y) \textrm{d}y\right| \le C |t-t_0|^\frac{3}{4}+C\varepsilon +|R_{12}|. \end{aligned}$$

With the aid of this and recalling (5.15), one derives

$$\begin{aligned} \int _\Omega [h(\varphi (y,t),t)-h(\varphi (y,t_0),t_0)]\chi (y) \textrm{d}y\rightarrow 0,\quad \text{ as } t\rightarrow t_0, \end{aligned}$$

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

$$\begin{aligned} \text {div}_y\left( \frac{\partial _{x_i}y}{\text {det}\nabla _xy}\right) =0 \quad \text{ and }\quad \text {div}_x\left( \frac{\partial _{y_i}x}{\text {det}\nabla _yx}\right) =0. \end{aligned}$$

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

$$\begin{aligned} \frac{\nabla _xy}{\text {det} \nabla _xy}=\frac{(\nabla _yx)^{-1}}{{\text {det} \nabla _xy}}=\frac{(\nabla _yx)_{adj}}{\text {det}\nabla _yx{\text {det} \nabla _xy}}=(\nabla _yx)_{adj}. \end{aligned}$$

Therefore,

$$\begin{aligned} \text {div}_y\left( \frac{\partial _{x_i}y}{\text {det}\nabla _xy}\right) =\text {div}_y\Big (R_i((\nabla _yx)_{adj})\Big ). \end{aligned}$$

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

$$\begin{aligned} \text {div}_y\Big (R_1((\nabla _yx)_{adj})\Big )=\left| \begin{array}{ccc} \partial _{y_1} &{} \partial _{y_1}x_2 &{} \partial _{y_1}x_3 \\ \partial _{y_2} &{} \partial _{y_2}x_2 &{} \partial _{y_2}x_3 \\ \partial _{y_3} &{} \partial _{y_3}x_2 &{} \partial _{y_3}x_3 \end{array} \right| , \end{aligned}$$

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. \(\square \)

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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

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