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Some results for the large time behavior of Hamilton-Jacobi equations with Caputo time derivative
On the compressible Navier-Stokes equations in the whole space: From non-isentropic flow to isentropic flow
1. | Department of Mathematical Sciences, Tsinghua University, Beijing, China |
2. | School of Mathematical Sciences, Beihang University, 100191 Beijing, China |
$ \varepsilon $ |
$ S^{N}_ \varepsilon $ |
$ c_S $ |
$ S^{N}_ \varepsilon |_{t = 0} = c_S+O( \varepsilon ) $ |
$ ( \rho^{N}_ \varepsilon , \, \boldsymbol{u}^{N}_ \varepsilon , \, S^{N}_ \varepsilon ) $ |
$ ( \rho_0, \, \boldsymbol{u}_0, \, c_S+ \varepsilon S_0) $ |
$ (1, \boldsymbol{0}, c_S) $ |
$ ( \rho^{N}_ \varepsilon , u^{N}_ \varepsilon ) $ |
$ ( \rho^{I}, \, u^{I}) $ |
$ ( \rho_0, \, \boldsymbol{u}_0) $ |
$ \begin{eqnarray*} ( \rho^{N}_ \varepsilon (t), \, \boldsymbol{u}^{N}_ \varepsilon (t)) = ( \rho^{I}(t), \, \boldsymbol{u}^{I}(t))+O(\epsilon), \end{eqnarray*} $ |
References:
[1] |
H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, , Grundlehren der Mathematischen Wissenschaften, Springer 2011.
doi: 10.1007/978-3-642-16830-7. |
[2] |
J.-M. Bony, Calcul symbolique et propagation des singularités pour les $\mathop {\rm{q}}\limits^{\rm{'}} $uations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup., 14 (1981), 209–246.
doi: 10.24033/asens.1404. |
[3] |
F. Charve and R. Danchin,
A global existence result for the compressible Navier-Stokes equations in the critical $L^p$ framework, Arch. Ration. Mech. Anal., 198 (2010), 233-271.
doi: 10.1007/s00205-010-0306-x. |
[4] |
Q. Chen, C. Miao and Z. Zhang,
Well-posedness in critical spaces for the compressible Navier-Stokes equations with density dependent viscosities, Rev. Mat. Iberoam., 26 (2010), 915-946.
doi: 10.4171/RMI/621. |
[5] |
Q. Chen, C. Miao and Z. Zhang,
Global well-posedness for compressible Navier-Stokes equations with highly oscillating initial velocity, Comm. Pure Appl. Math., 63 (2010), 1173-1224.
doi: 10.1002/cpa.20325. |
[6] |
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), 1375-1402.
doi: 10.4171/RMI/872. |
[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), 3435-3467.
doi: 10.1016/j.jde.2015.01.012. |
[8] |
R. Danchin,
Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614.
doi: 10.1007/s002220000078. |
[9] |
R. Danchin,
Local theory in critical spaces for compressible viscous and heat-conductive gases, Comm. Partial Differential Equations, 26 (2001), 1183-1233.
doi: 10.1081/PDE-100106132. |
[10] |
R. Danchin,
Global existence in critical spaces for flows of compressible viscous and heat-conductive gases, Arch. Ration. Mech. Anal., 160 (2001), 1-39.
doi: 10.1007/s002050100155. |
[11] |
R. Danchin,
Well-posedness in critical spaces for barotropic viscous fluids with truly not constant density, Comm. Partial Differential Equations, 32 (2007), 1373-1397.
doi: 10.1080/03605300600910399. |
[12] |
R. Danchin and L. He,
The incompressible limit in $L^p$ type critical spaces, Math. Ann., 366 (2016), 1365-1402.
doi: 10.1007/s00208-016-1361-x. |
[13] |
R. Danchin and P. B. Mucha,
Compressible Navier-Stokes system: large solutions and incompressible limit, Adv. Math., 320 (2017), 904-925.
doi: 10.1016/j.aim.2017.09.025. |
[14] |
R. Danchin and J. Xu,
Optimal Time-decay Estimates for the Compressible Navier-Stokes Equations in the Critical $L^p$Framework, Arch. Ration. Mech. Anal., 224 (2017), 53-90.
doi: 10.1007/s00205-016-1067-y. |
[15] |
D. Fang, T. Zhang and R. Zi, Global solutions to the isentropic compressible Navier-Stokes equations with a class of large initial data, SIAM J. Math. Anal., 50 (2018), 4983–5026.
doi: 10.1137/17M1122062. |
[16] |
E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford Lecture Series in Mathematics and its Applications, 26. Oxford University Press, Oxford, 2004. |
[17] |
E. Feireisl and A. Novotný,
Weak-strong uniqueness property for the full Navier-Stokes-Fourier system., Arch. Ration. Mech. Anal., 204 (2012), 683-706.
doi: 10.1007/s00205-011-0490-3. |
[18] |
B. Haspot,
Existence of global strong solutions in critical spaces for barotropic viscous fluids, Arch. Ration. Mech. Anal., 202 (2011), 427-460.
doi: 10.1007/s00205-011-0430-2. |
[19] |
B. Haspot,
Global existence of strong solutions for viscous shallow water system with large initial data on the rotational part, J. Differential Equations, 262 (2017), 4931-4978.
doi: 10.1016/j.jde.2017.01.010. |
[20] |
X. Huang, J. Li and Z. Xin,
Global well-posedness of classical solutions with large oscillations and vacuum to the three dimensional isentropic compressible Navier-Stokes equations, Comm. Pure Appl. Math., 65 (2012), 549-585.
doi: 10.1002/cpa.21382. |
[21] |
X. Huang and J. Li,
Global classical and weak solutions to the three-dimensional full compressible Navier-Stokes system with vacuum and large oscillations, Arch. Ration. Mech. Anal., 227 (2018), 995-1059.
doi: 10.1007/s00205-017-1188-y. |
[22] |
A. Matsumura and T. Nishida,
The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids, Proc. Japan Acad. Ser. A Math. Sci., 55 (1979), 337-342.
doi: 10.3792/pjaa.55.337. |
[23] |
A. Matsumura and T. Nishida,
The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.
doi: 10.1215/kjm/1250522322. |
[24] |
J. Nash,
Le problème de Cauchy pour les équations différentielles d'un fluide général, Bulletin de la Soc. Math. de France, 90 (1962), 487-497.
doi: 10.24033/bsmf.1586. |
show all references
References:
[1] |
H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, , Grundlehren der Mathematischen Wissenschaften, Springer 2011.
doi: 10.1007/978-3-642-16830-7. |
[2] |
J.-M. Bony, Calcul symbolique et propagation des singularités pour les $\mathop {\rm{q}}\limits^{\rm{'}} $uations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup., 14 (1981), 209–246.
doi: 10.24033/asens.1404. |
[3] |
F. Charve and R. Danchin,
A global existence result for the compressible Navier-Stokes equations in the critical $L^p$ framework, Arch. Ration. Mech. Anal., 198 (2010), 233-271.
doi: 10.1007/s00205-010-0306-x. |
[4] |
Q. Chen, C. Miao and Z. Zhang,
Well-posedness in critical spaces for the compressible Navier-Stokes equations with density dependent viscosities, Rev. Mat. Iberoam., 26 (2010), 915-946.
doi: 10.4171/RMI/621. |
[5] |
Q. Chen, C. Miao and Z. Zhang,
Global well-posedness for compressible Navier-Stokes equations with highly oscillating initial velocity, Comm. Pure Appl. Math., 63 (2010), 1173-1224.
doi: 10.1002/cpa.20325. |
[6] |
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), 1375-1402.
doi: 10.4171/RMI/872. |
[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), 3435-3467.
doi: 10.1016/j.jde.2015.01.012. |
[8] |
R. Danchin,
Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614.
doi: 10.1007/s002220000078. |
[9] |
R. Danchin,
Local theory in critical spaces for compressible viscous and heat-conductive gases, Comm. Partial Differential Equations, 26 (2001), 1183-1233.
doi: 10.1081/PDE-100106132. |
[10] |
R. Danchin,
Global existence in critical spaces for flows of compressible viscous and heat-conductive gases, Arch. Ration. Mech. Anal., 160 (2001), 1-39.
doi: 10.1007/s002050100155. |
[11] |
R. Danchin,
Well-posedness in critical spaces for barotropic viscous fluids with truly not constant density, Comm. Partial Differential Equations, 32 (2007), 1373-1397.
doi: 10.1080/03605300600910399. |
[12] |
R. Danchin and L. He,
The incompressible limit in $L^p$ type critical spaces, Math. Ann., 366 (2016), 1365-1402.
doi: 10.1007/s00208-016-1361-x. |
[13] |
R. Danchin and P. B. Mucha,
Compressible Navier-Stokes system: large solutions and incompressible limit, Adv. Math., 320 (2017), 904-925.
doi: 10.1016/j.aim.2017.09.025. |
[14] |
R. Danchin and J. Xu,
Optimal Time-decay Estimates for the Compressible Navier-Stokes Equations in the Critical $L^p$Framework, Arch. Ration. Mech. Anal., 224 (2017), 53-90.
doi: 10.1007/s00205-016-1067-y. |
[15] |
D. Fang, T. Zhang and R. Zi, Global solutions to the isentropic compressible Navier-Stokes equations with a class of large initial data, SIAM J. Math. Anal., 50 (2018), 4983–5026.
doi: 10.1137/17M1122062. |
[16] |
E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford Lecture Series in Mathematics and its Applications, 26. Oxford University Press, Oxford, 2004. |
[17] |
E. Feireisl and A. Novotný,
Weak-strong uniqueness property for the full Navier-Stokes-Fourier system., Arch. Ration. Mech. Anal., 204 (2012), 683-706.
doi: 10.1007/s00205-011-0490-3. |
[18] |
B. Haspot,
Existence of global strong solutions in critical spaces for barotropic viscous fluids, Arch. Ration. Mech. Anal., 202 (2011), 427-460.
doi: 10.1007/s00205-011-0430-2. |
[19] |
B. Haspot,
Global existence of strong solutions for viscous shallow water system with large initial data on the rotational part, J. Differential Equations, 262 (2017), 4931-4978.
doi: 10.1016/j.jde.2017.01.010. |
[20] |
X. Huang, J. Li and Z. Xin,
Global well-posedness of classical solutions with large oscillations and vacuum to the three dimensional isentropic compressible Navier-Stokes equations, Comm. Pure Appl. Math., 65 (2012), 549-585.
doi: 10.1002/cpa.21382. |
[21] |
X. Huang and J. Li,
Global classical and weak solutions to the three-dimensional full compressible Navier-Stokes system with vacuum and large oscillations, Arch. Ration. Mech. Anal., 227 (2018), 995-1059.
doi: 10.1007/s00205-017-1188-y. |
[22] |
A. Matsumura and T. Nishida,
The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids, Proc. Japan Acad. Ser. A Math. Sci., 55 (1979), 337-342.
doi: 10.3792/pjaa.55.337. |
[23] |
A. Matsumura and T. Nishida,
The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.
doi: 10.1215/kjm/1250522322. |
[24] |
J. Nash,
Le problème de Cauchy pour les équations différentielles d'un fluide général, Bulletin de la Soc. Math. de France, 90 (1962), 487-497.
doi: 10.24033/bsmf.1586. |
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