Remarks on the a priori bound for the vorticity of the axisymmetric Navier-Stokes equations

Zujin Zhang; Chenxuan Tong

Applications of Mathematics (2022)

  • Volume: 67, Issue: 4, page 485-507
  • ISSN: 0862-7940

Abstract

top
We study the axisymmetric Navier-Stokes equations. In 2010, Loftus-Zhang used a refined test function and re-scaling scheme, and showed that | ω r ( x , t ) | + | ω z ( r , t ) | ≤ C r 10 , 0 < r ≤ 1 2 . By employing the dimension reduction technique by Lei-Navas-Zhang, and analyzing ω r , ω z and ω θ / r on different hollow cylinders, we are able to improve it and obtain | ω r ( x , t ) | + | ω z ( r , t ) | ≤ C | ln r | r 17 / 2 , 0 < r ≤ 1 2 .

How to cite

top

Zhang, Zujin, and Tong, Chenxuan. "Remarks on the a priori bound for the vorticity of the axisymmetric Navier-Stokes equations." Applications of Mathematics 67.4 (2022): 485-507. <http://eudml.org/doc/298306>.

@article{Zhang2022,
abstract = {We study the axisymmetric Navier-Stokes equations. In 2010, Loftus-Zhang used a refined test function and re-scaling scheme, and showed that \[ |\omega ^r(x,t)|+|\omega ^z(r,t)|\le \frac\{C\}\{r^\{10\}\},\quad 0<r\le \frac\{1\}\{2\}. \] By employing the dimension reduction technique by Lei-Navas-Zhang, and analyzing $\omega ^r$, $\omega ^z$ and $\omega ^\theta /r$ on different hollow cylinders, we are able to improve it and obtain \[ |\omega ^r(x,t)|+|\omega ^z(r,t)|\le \frac\{C|\{\rm ln\} r|\}\{r^\{17/2\}\},\quad 0<r\le \frac\{1\}\{2\}. \]},
author = {Zhang, Zujin, Tong, Chenxuan},
journal = {Applications of Mathematics},
keywords = {axisymmetric Navier-Stokes equations; weighted a priori bounds},
language = {eng},
number = {4},
pages = {485-507},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Remarks on the a priori bound for the vorticity of the axisymmetric Navier-Stokes equations},
url = {http://eudml.org/doc/298306},
volume = {67},
year = {2022},
}

TY - JOUR
AU - Zhang, Zujin
AU - Tong, Chenxuan
TI - Remarks on the a priori bound for the vorticity of the axisymmetric Navier-Stokes equations
JO - Applications of Mathematics
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 4
SP - 485
EP - 507
AB - We study the axisymmetric Navier-Stokes equations. In 2010, Loftus-Zhang used a refined test function and re-scaling scheme, and showed that \[ |\omega ^r(x,t)|+|\omega ^z(r,t)|\le \frac{C}{r^{10}},\quad 0<r\le \frac{1}{2}. \] By employing the dimension reduction technique by Lei-Navas-Zhang, and analyzing $\omega ^r$, $\omega ^z$ and $\omega ^\theta /r$ on different hollow cylinders, we are able to improve it and obtain \[ |\omega ^r(x,t)|+|\omega ^z(r,t)|\le \frac{C|{\rm ln} r|}{r^{17/2}},\quad 0<r\le \frac{1}{2}. \]
LA - eng
KW - axisymmetric Navier-Stokes equations; weighted a priori bounds
UR - http://eudml.org/doc/298306
ER -

References

top
  1. Chae, D., Lee, J., 10.1007/s002090100317, Math. Z. 239 (2002), 645-671. (2002) Zbl0992.35068MR1902055DOI10.1007/s002090100317
  2. Chen, H., Fang, D., Zhang, T., 10.3934/dcds.2017081, Discrete Contin. Dyn. Syst. 37 (2017), 1923-1939. (2017) Zbl1364.35114MR3640581DOI10.3934/dcds.2017081
  3. Chen, Q., Zhang, Z., 10.1016/j.jmaa.2006.09.069, J. Math. Anal. Appl. 331 (2007), 1384-1395. (2007) Zbl1151.35067MR2313720DOI10.1016/j.jmaa.2006.09.069
  4. Gala, S., 10.1016/j.na.2010.09.026, Nonlinear Anal., Theory Methods Appl., Ser. A 74 (2011), 775-782. (2011) Zbl1205.35194MR2738629DOI10.1016/j.na.2010.09.026
  5. Gallay, T., Šverák, V., 10.5802/cml.25, Confluentes Math. 7 (2015), 67-92. (2015) Zbl1356.35159MR4059266DOI10.5802/cml.25
  6. Hou, T. Y., Li, C., 10.3934/dcds.2005.12.1, Discrete Contin. Dyn. Syst. 12 (2005), 1-12. (2005) Zbl1274.76185MR2121245DOI10.3934/dcds.2005.12.1
  7. Kreml, O., Pokorný, M., A regularity criterion for the angular velocity component in axisymmetric Navier-Stokes equations, Electron J. Differ. Equ. 2007 (2007), Article ID 08, 10 pages. (2007) Zbl1387.35456MR2278422
  8. Kubica, A., Pokorný, M., Zajączkowski, W., 10.1002/mma.1586, Math. Methods Appl. Sci. 35 (2012), 360-371. (2012) Zbl1236.35107MR2876817DOI10.1002/mma.1586
  9. Ladyzhenskaya, O. A., Unique global solvability of the three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry, Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 7 (1968), 155-177 Russian. (1968) Zbl0195.10603MR0241833
  10. Lei, Z., Navas, E. A., Zhang, Q. S., 10.1007/s00220-015-2496-4, Commun. Math. Phys. 341 (2016), 289-307. (2016) Zbl1341.35102MR3439228DOI10.1007/s00220-015-2496-4
  11. Lei, Z., Zhang, Q. S., 10.1016/j.jfa.2011.06.016, J. Funct. Anal. 261 (2011), 2323-2345. (2011) Zbl1244.35105MR2824580DOI10.1016/j.jfa.2011.06.016
  12. Lei, Z., Zhang, Q. S., 10.2140/pjm.2017.289.169, Pac. J. Math. 289 (2017), 169-187. (2017) Zbl06722850MR3652459DOI10.2140/pjm.2017.289.169
  13. Leonardi, S., Málek, J., Nečas, J., Pokorný, M., 10.4171/ZAA/903, Z. Anal. Anwend. 18 (1999), 639-649. (1999) Zbl0943.35066MR1718156DOI10.4171/ZAA/903
  14. Loftus, J. B., Zhang, Q. S., A priori bounds for the vorticity of axially symmetric solutions to the Navier-Stokes equations, Adv. Differ. Equ. 15 (2010), 531-560. (2010) Zbl1195.35240MR2643234
  15. Neustupa, J., Pokorný, M., 10.21136/MB.2001.134015, Math. Bohem. 126 (2001), 469-481. (2001) Zbl0981.35046MR1844284DOI10.21136/MB.2001.134015
  16. Pokorný, M., 10.1142/9789812777201_0022, Elliptic and Parabolic Problems World Scientific, Singapore (2002), 233-242. (2002) Zbl1033.35085MR1937543DOI10.1142/9789812777201_0022
  17. Renc{ł}awowicz, J., Zajączkowski, W. M., 10.1007/s00021-019-0447-0, J. Math. Fluid Mech. 21 (2019), Article ID 51, 14 pages. (2019) Zbl1448.76055MR4010661DOI10.1007/s00021-019-0447-0
  18. Uhkovskii, M. R., Iudovich, V. I., 10.1016/0021-8928(68)90147-0, J. Appl. Math. Mech. 32 (1968), 52-62. (1968) Zbl0172.53405MR0239293DOI10.1016/0021-8928(68)90147-0
  19. Wei, D., 10.1016/j.jmaa.2015.09.088, J. Math. Anal. Appl. 435 (2016), 402-413. (2016) Zbl1330.35298MR3423404DOI10.1016/j.jmaa.2015.09.088
  20. Zhang, P., Zhang, T., 10.1093/imrn/rns232, Int. Math. Res. Not. 2014 (2014), 610-642. (2014) Zbl07357636MR3163561DOI10.1093/imrn/rns232
  21. Zhang, Z., 10.4064/ap3856-3-2016, Ann. Pol. Math. 117 (2016), 181-196. (2016) Zbl1359.35163MR3539076DOI10.4064/ap3856-3-2016
  22. Zhang, Z., 10.1016/j.amc.2016.10.001, Appl. Math. Comput. 296 (2017), 18-22. (2017) Zbl1411.35222MR3572774DOI10.1016/j.amc.2016.10.001
  23. Zhang, Z., 10.1016/j.jmaa.2017.12.069, J. Math. Anal. Appl. 461 (2018), 1-6. (2018) Zbl1390.35239MR3759525DOI10.1016/j.jmaa.2017.12.069
  24. Zhang, Z., 10.1016/j.camwa.2018.06.035, Comput. Math. Appl. 76 (2018), 1420-1426. (2018) Zbl1434.35072MR3850659DOI10.1016/j.camwa.2018.06.035
  25. Zhang, Z., 10.1016/j.aml.2019.01.022, Appl. Math. Lett. 92 (2019), 139-143. (2019) Zbl1417.35098MR3903956DOI10.1016/j.aml.2019.01.022
  26. Zhang, Z., Ouyang, X., Yang, X., 10.11948/2017034, J. Appl. Anal. Comput. 7 (2017), 554-558. (2017) Zbl07309521MR3602437DOI10.11948/2017034

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.