On the energy conservation law for the compressible Navier-Stokes equations

[1] I. Akramov, T. De, biec, J. Skipper, and E. Wiedemann, Energy conservation for the compressible Euler and

Navier–Stokes equations with vacuum, Anal. PDE 13 (2020), no. 3, 789–811.

[2] M. Aoki and T. Iwabuchi, Energy conservation law for weak solutions of the full compressible Navier–Stokes

equations, J. Differential Equations 341 (2022), 481–503.

[3]

, On the ill-posedness for the full system of compressible Navier–Stokes equations, preprint.

[4] M. Cannone, A generalization of a theorem by Kato on Navier–Stokes equations, Rev. Mat. Iberoamericana

13 (1997), no. 3, 515–541.

[5] Q. Chen, C. Miao, and Z. Zhang, On the ill-posedness of the compressible Navier–Stokes equations in the

critical Besov spaces, Rev. Mat. Iberoam. 31 (2015), no. 4, 1375–1402.

[6] A. Cheskidov and X. Luo, Energy equality for the Navier–Stokes equations in weak-in-time Onsager spaces,

Nonlinearity 33 (2020), no. 4, 1388–1403.

[7] N. Chikami and R. Danchin, On the well-posedness of the full compressible Navier–Stokes system in critical

Besov spaces, J. Differential Equations 258 (2015), no. 10, 3435–3467.

[8] R. Danchin, Global existence in critical spaces for flows of compressible viscous and heat-conductive gases,

Arch. Ration. Mech. Anal. 160 (2001), no. 1, 1–39.

[9]

, Local theory in critical spaces for compressible viscous and heat-conductive gases, Comm. Partial

Differential Equations 26 (2001), no. 7-8, 1183–1233.

[10] E. Feireisl, Dynamics of viscous compressible fluids, Oxford Lecture Series in Mathematics and its Applications, vol. 26, Oxford University Press, Oxford, 2004.

[11] H. Fujita and T. Kato, On the Navier–Stokes initial value problem. I, Arch. Rational Mech. Anal. 16 (1964),

269–315.

[12] Y. Giga and T. Miyakawa, Solutions in Lr of the Navier–Stokes initial value problem, Arch. Rational Mech.

Anal. 89 (1985), no. 3, 267–281.

[13] Y. Giga, Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of the Navier–

Stokes system, J. Differential Equations 62 (1986), no. 2, 186–212.

[14] B. Haspot, Well-posedness in critical spaces for the system of compressible Navier–Stokes in larger spaces, J.

Differential Equations 251 (2011), no. 8, 2262–2295.

[15] E. Hopf, Uber

die Anfangswertaufgabe f¨

ur die hydrodynamischen Grundgleichungen, Math. Nachr. 4 (1951),

213–231 (German).

[16] T. Iwabuchi and M. Nakamura, Small solutions for nonlinear heat equations, the Navier–Stokes equation, and

the Keller-Segel system in Besov and Triebel-Lizorkin spaces, Adv. Differential Equations 18 (2013), no. 7-8,

687–736.

[17] T. Iwabuchi and T. Ogawa, Ill-posedness for the Cauchy problem of the two-dimensional compressible Navier–

Stokes equations for an ideal gas, J. Elliptic Parabol. Equ. 7 (2021), no. 2, 571–587.

[18]

, Ill-posedness for the compressible Navier–Stokes equations under barotropic condition in limiting

Besov spaces, J. Math. Soc. Japan 74 (2022), no. 2, 353–394.

[19] T. Kato, Strong Lp -solutions of the Navier–Stokes equation in Rm , with applications to weak solutions, Math.

Z. 187 (1984), no. 4, 471–480.

[20] T. Kato and H. Fujita, On the nonstationary Navier–Stokes system, Rend. Sem. Mat. Univ. Padova 32

(1962), 243–260.

[21] H. Koch and D. Tataru, Well-posedness for the Navier–Stokes equations, Adv. Math. 157 (2001), no. 1,

22–35.

[22] H. Kozono and H. Sohr, Remark on uniqueness of weak solutions to the Navier–Stokes equations, Analysis

16 (1996), no. 3, 255–271.

11

[23] O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Mathematics and its Applications, Vol. 2, Gordon and Breach Science Publishers, New York-London-Paris, 1969. Second English edition,

revised and enlarged; Translated from the Russian by Richard A. Silverman and John Chu.

[24] J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math. 63 (1934), no. 1, 193–248

(French).

[25] P.-L. Lions, Mathematical topics in fluid mechanics. Vol. 2, Oxford Lecture Series in Mathematics and its

Applications, vol. 10, The Clarendon Press, Oxford University Press, New York, 1998. Compressible models;

Oxford Science Publications.

[26] K. Masuda, Weak solutions of Navier–Stokes equations, Tohoku Math. J. (2) 36 (1984), no. 4, 623–646.

[27] Q.-H. Nguyen, P.-T. Nguyen, and B. Q. Tang, Energy equalities for compressible Navier–Stokes equations,

Nonlinearity 32 (2019), no. 11, 4206–4231.

[28] F. Planchon, Asymptotic behavior of global solutions to the Navier–Stokes equations in R3 , Rev. Mat.

Iberoamericana 14 (1998), no. 1, 71–93, DOI 10.4171/RMI/235. MR1639283

[29] G. Prodi, Un teorema di unicit`

a per le equazioni di Navier–Stokes, Ann. Mat. Pura Appl. (4) 48 (1959),

173–182 (Italian).

[30] J. Serrin, On the interior regularity of weak solutions of the Navier–Stokes equations, Arch. Rational Mech.

Anal. 9 (1962), 187–195.

[31]

, The initial value problem for the Navier–Stokes equations, Nonlinear Problems (Proc. Sympos.,

Madison, Wis., 1962), Univ. Wisconsin Press, Madison, Wis., 1963, pp. 69–98. MR0150444

[32] M. Shinbrot, The energy equation for the Navier–Stokes system, SIAM J. Math. Anal. 5 (1974), 948–954.

[33] H. Sohr, Zur Regularit¨

atstheorie der instation¨

aren Gleichungen von Navier–Stokes, Math. Z. 184 (1983),

no. 3, 359–375 (German).

[34] Y. Taniuchi, On generalized energy equality of the Navier–Stokes equations, Manuscripta Math. 94 (1997),

no. 3, 365–384.

[35] C. Yu, Energy conservation for the weak solutions of the compressible Navier–Stokes equations, Arch. Ration.

Mech. Anal. 225 (2017), no. 3, 1073–1087.

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