Gevrey class regularity and stability for the Debye-Huckel system in critical Fourier-Besov-Morrey spaces
DOI:
https://doi.org/10.5269/bspm.62517Resumen
In this paper, we study the analyticity of mild solutions to the Debye-Huckel system with small initial data in critical Fourier-Besov-Morrey spaces. Specifically, by using the Fourier localization argument, the Littlewood-Paley theory and bilinear-type fixed point theory, we prove that global-in-time mild solutions are Gevrey regular. As a consequence of analyticity, we get time decay of mild solutions in Fourier-BesovMorrey spaces. Finally, we show a blow-up criterion of the local-in-time mild solutions of the Debye-Huckel system.
Referencias
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9. Benameur, J.: Long time decay to the Lei-Lin solution of 3D Navier-Stokes equations. Journal of Mathematical Analysis and Applications. 422 , 424-434 (2015). https://doi.org/10.1016/j.jmaa.2014.08.039
10. Benameur, J., Benhamed, M.: Global existence of the two-dimensional QGE with sub-critical dissipation. Journal of Mathematical Analysis and Applications. 423, 1330-1347 (2015). https://doi.org/10.1016/j.jmaa.2014.10.066
11. Benhamed, M., Abusalim, S. M.: Long Time Behavior of the Solution of the Two-Dimensional Dissipative QGE in Lei-Lin Spaces. International Journal of Mathematics and Mathematical Sciences. 2020, 1-6 (2020). https://doi.org/10.1155/2020/6409609
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13. Biswas, A.: Gevrey regularity for a class of dissipative equations with applications to decay. Journal of Differential Equations. 253, 2739-2764 (2012). https://doi.org/10.1016/j.jde.2012.08.003
14. Cannone, M., Wu, G.: Global well-posedness for Navier-Stokes equations in critical Fourier-Herz spaces. Nonlinear Anal, 75 (2012). https://doi.org/10.1016/j.na.2012.01.029
15. Chae, D.: On the well-posedness of the Euler equations in the Triebel-Lizorkin spaces. Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences. 55, 654-678 (2002). https://doi.org/10.1002/cpa.10029
16. Chen, X.: Well-Posedness of the Keller-Segel System in Fourier-Besov-Morrey Spaces. Zeitschrift f¨ur Analysis und ihre Anwendungen. 37, 417-434 (2018). https://doi.org/10.4171/ZAA/1621
17. Cui, Y., Xiao, W.: Gevrey regularity and time decay of the fractional Debey-Huckel system in Fourrier-Besov spaces. Bulletin of the Korean Mathematical Society. 57, 1393-1408 (2020).
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20. Ferreira, L. C. F., Lidiane S. M. L: Self-similar solutions for active scalar equations in Fourier-Besov-Morrey spaces. Monatshefte fur Mathematik, 175, 491-509 (2014). https://doi.org/10.1007/s00605-014-0659-6
21. Ferreira, L. C.: On the uniqueness of mild solutions for the parabolic-elliptic Keller-Segel system in the critical Lp-space. Mathematics in Engineering. 4, 1-14 (2022). https://doi.org/10.3934/mine.2022048
22. Ferreira, L. C., Precioso, J. C.: Existence and asymptotic behaviour for the parabolic-parabolic Keller-Segel system with singular data. Nonlinearity. 24, 1433 (2011). https://doi.org/10.1088/0951-7715/24/5/003
23. Foias, C., Temam, R.: Gevrey class regularity for the solutions of the Navier-Stokes equations. Journal of Functional Analysis. 87, 359-369 (1989). https://doi.org/10.1016/0022-1236(89)90015-3
24. Gruji'c, Z., Kukavica, I.: Space analyticity for the Navier-Stokes and related equations with initial data inLp. journal of functional analysis. 152, 447-466 (1998). https://doi.org/10.1006/jfan.1997.3167
25. Iwabuchi, T., Takada, R.: Global well-posedness and ill-posedness for the Navier-Stokes equations with the Coriolis force in function spaces of Besov type. J. Funct. Anal. 267, 1321-1337 (2014). https://doi.org/10.1016/j.jfa.2014.05.022
26. Iwabuchi, T.: Global well-posedness for Keller-Segel system in Besov type spaces. Journal of Mathematical Analysis and Applications. 379, 930-948 (2011). https://doi.org/10.1016/j.jmaa.2011.02.010
27. Iwabuchi, T., and Makoto N.: Small solutions for nonlinear heat equations, the Navier-Stokes equation and the KellerSegel system in Besov and Triebel-Lizorkin spaces. Advances in Differential Equations. 18, 687-736 (2013). https://doi.org/10.57262/ade/1369057711
28. Karch, G.: Scaling in nonlinear parabolic equations. Journal of mathematical analysis and applications. 234, 534-558 (1999). https://doi.org/10.1006/jmaa.1999.6370
29. Kato, T.: Strong Lp-solutions of the Navier-Stokes equation in Rm, with applications to weak solutions. Math. Z. 187, 471-480 (1984). https://doi.org/10.1007/BF01174182
30. Kato, T.: Strong solutions of the Navier-Stokes equation in Morrey spaces. Boletim da Sociedade Brasileira de Matem'etica-Bulletin/Brazilian Mathematical Society. 22, 127-155 (1992). https://doi.org/10.1007/BF01232939
31. Koch, H., Tataru, D.: Well-posedness for the Navier-Stokes equations, Adv. Math. 157, 22-35 (2001). https://doi.org/10.1006/aima.2000.1937
32. Kozono, H., Yamazaki, M.: Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data. Comm. Partial Differential Equations. 19, 959-1014 (1994). https://doi.org/10.1080/03605309408821042
33. Kurokiba, M., Ogawa, T.: Well-posedness for the drift-diffusion system in Lp arising from the semiconductor device simulation. Journal of Mathematical Analysis and Applications. 342, 1052-1067 (2008). https://doi.org/10.1016/j.jmaa.2007.11.017
34. Lemari'e-Rieusset, P. G.: Recent developments in the Navier-Stokes problem. CRC Press, (2002). https://doi.org/10.1201/9780367801656
35. Liu, Q., Zhao, J., Cui, S.: Existence and regularizing rate estimates of solutions to a generalized magneto-hydrodynamic system in pseudomeasure spaces. Annali di Matematica Pura ed Applicata. 191, 293-309 (2012). https://doi.org/10.1007/s10231-010-0184-8
36. Luo, Y.: Well-Posedness Of A Cauchy Problem Involving Nonlinear Fractal Dissipative Equations. Applied Mathematics E-Notes. 10, 112-118 (2010).
37. Miao, C., Yuan, B., Zhang, B.: Well-posedness of the Cauchy problem for the fractional power dissipative equations. Nonlinear Analysis: Theory, Methods and Applications. 68 (2008), 461-484. https://doi.org/10.1016/j.na.2006.11.011
38. Wang, W., Wu, G.: Global mild solution of the generalized Navier-Stokes equations with the Coriolis force. Applied Mathematics Letters. 76, 181-186 (2018) https://doi.org/10.1016/j.aml.2017.09.001
39. Yamamoto, M.: Spatial analyticity of solutions to the drift-diffusion equation with generalized dissipation. Arch. Math. 97, 261-270 (2011). https://doi.org/10.1007/s00013-011-0302-x
40. Yamazaki, M.: The Navier-Stokes equations in the weak-Ln space with time-dependent external force. Math. Ann. 317, 635-675 (2000). https://doi.org/10.1007/PL00004418
41. Zhao, J.: Well-posedness and Gevrey analyticity of the generalized Keller-Segel system in critical Besov spaces. Annali di Matematica. 197, 521-548 (2018). https://doi.org/10.1007/s10231-017-0691-y
42. Zhao, J.: Gevrey regularity of mild solutions to the parabolic-elliptic system of drift-diffusion type in critical Besov spaces. Journal of Mathematical Analysis and Applications, 448, 1265-1280 (2017). https://doi.org/10.1016/j.jmaa.2016.11.050
43. Zhao, J., Liu, Q., Cui, S.: Existence of solutions for the Debye-H¨uckel system with low regularity initial data. Acta applicandae mathematicae. 125, 1-10 (2013). https://doi.org/10.1007/s10440-012-9777-0
44. Zhou, X., Xiao, W.: Algebra Properties in Fourier-Besov Spaces and Their Applications. Journal of Function Spaces, 2018, (2018). https://doi.org/10.1155/2018/3629179
2. Azanzal, A., Allalou, C., A., Abbassi,: Well-posedness and analyticity for generalized Navier-Stokes equations in critical Fourier-Besov-Morrey spaces. J. Nonlinear Funct. Anal. 2021 (2021), Article ID 24.
3. Azanzal, A., Abbassi, A., Allalou, C., Existence of Solutions for the Debye-H¨uckel System with Low Regularity Initial Data in Critical Fourier-Besov-Morrey Spaces. Nonlinear Dynamics and Systems Theory, 21, 367-380 (2021).
4. Azanzal, A., Abbassi, A., Allalou, C.: On the Cauchy problem for the fractional drift-diffusion system in critical Fourier-Besov-Morrey spaces. International Journal On Optimization and Applications, 1, pp.28 (2021).
5. Azanzal, A., Allalou, C., Melliani, S.: Well-posedness and blow-up of solutions for the 2D dissipative quasi-geostrophic equation in critical Fourier-Besov-Morrey spaces. J Elliptic Parabol Equ (2021). https://doi.org/10.1007/s41808-021-00140-x
6. Bae, H.: Existence and analyticity of Lei-Lin solution to the Navier-Stokes equations. Proceedings of the American Mathematical Society, 2887-2892 (2015). https://doi.org/10.1090/S0002-9939-2015-12266-6
7. Bahouri, H., Chemin, J. Y., Danchin, R.: Fourier analysis and nonlinear partial differential equations. Springer Science and Business Media. 343 (2011). https://doi.org/10.1007/978-3-642-16830-7
8. Bahouri, H.: The Littlewood-Paley theory: a common thread of many words in nonlinear analysis. European Mathematical Society Newsletter (2019). https://doi.org/10.4171/NEWS/112/4
9. Benameur, J.: Long time decay to the Lei-Lin solution of 3D Navier-Stokes equations. Journal of Mathematical Analysis and Applications. 422 , 424-434 (2015). https://doi.org/10.1016/j.jmaa.2014.08.039
10. Benameur, J., Benhamed, M.: Global existence of the two-dimensional QGE with sub-critical dissipation. Journal of Mathematical Analysis and Applications. 423, 1330-1347 (2015). https://doi.org/10.1016/j.jmaa.2014.10.066
11. Benhamed, M., Abusalim, S. M.: Long Time Behavior of the Solution of the Two-Dimensional Dissipative QGE in Lei-Lin Spaces. International Journal of Mathematics and Mathematical Sciences. 2020, 1-6 (2020). https://doi.org/10.1155/2020/6409609
12. Biler, Piotr.: Existence and asymptotics of solutions for a parabolic-elliptic system with nonlinear no-flux boundary conditions. Nonlinear Analysis: Theory, Methods and Applications. 19, 1121-1136 (1992). https://doi.org/10.1016/0362-546X(92)90186-I
13. Biswas, A.: Gevrey regularity for a class of dissipative equations with applications to decay. Journal of Differential Equations. 253, 2739-2764 (2012). https://doi.org/10.1016/j.jde.2012.08.003
14. Cannone, M., Wu, G.: Global well-posedness for Navier-Stokes equations in critical Fourier-Herz spaces. Nonlinear Anal, 75 (2012). https://doi.org/10.1016/j.na.2012.01.029
15. Chae, D.: On the well-posedness of the Euler equations in the Triebel-Lizorkin spaces. Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences. 55, 654-678 (2002). https://doi.org/10.1002/cpa.10029
16. Chen, X.: Well-Posedness of the Keller-Segel System in Fourier-Besov-Morrey Spaces. Zeitschrift f¨ur Analysis und ihre Anwendungen. 37, 417-434 (2018). https://doi.org/10.4171/ZAA/1621
17. Cui, Y., Xiao, W.: Gevrey regularity and time decay of the fractional Debey-Huckel system in Fourrier-Besov spaces. Bulletin of the Korean Mathematical Society. 57, 1393-1408 (2020).
18. de Almeida, M. F., Ferreira, L. C. F., Lima, L. S. M.: Uniform global well-posedness of the Navier-Stokes-Coriolis system in a new critical space. Mathematische Zeitschrift. 287, 735-750 (2017). https://doi.org/10.1007/s00209-017-1843-x
19. Duvaut, G., Lions, J. L.: In'equations en thermo'elasticite et magnetohydrodynamique. Archive for Rational Mechanics and Analysis. 46, 241-279 (1972). https://doi.org/10.1007/BF00250512
20. Ferreira, L. C. F., Lidiane S. M. L: Self-similar solutions for active scalar equations in Fourier-Besov-Morrey spaces. Monatshefte fur Mathematik, 175, 491-509 (2014). https://doi.org/10.1007/s00605-014-0659-6
21. Ferreira, L. C.: On the uniqueness of mild solutions for the parabolic-elliptic Keller-Segel system in the critical Lp-space. Mathematics in Engineering. 4, 1-14 (2022). https://doi.org/10.3934/mine.2022048
22. Ferreira, L. C., Precioso, J. C.: Existence and asymptotic behaviour for the parabolic-parabolic Keller-Segel system with singular data. Nonlinearity. 24, 1433 (2011). https://doi.org/10.1088/0951-7715/24/5/003
23. Foias, C., Temam, R.: Gevrey class regularity for the solutions of the Navier-Stokes equations. Journal of Functional Analysis. 87, 359-369 (1989). https://doi.org/10.1016/0022-1236(89)90015-3
24. Gruji'c, Z., Kukavica, I.: Space analyticity for the Navier-Stokes and related equations with initial data inLp. journal of functional analysis. 152, 447-466 (1998). https://doi.org/10.1006/jfan.1997.3167
25. Iwabuchi, T., Takada, R.: Global well-posedness and ill-posedness for the Navier-Stokes equations with the Coriolis force in function spaces of Besov type. J. Funct. Anal. 267, 1321-1337 (2014). https://doi.org/10.1016/j.jfa.2014.05.022
26. Iwabuchi, T.: Global well-posedness for Keller-Segel system in Besov type spaces. Journal of Mathematical Analysis and Applications. 379, 930-948 (2011). https://doi.org/10.1016/j.jmaa.2011.02.010
27. Iwabuchi, T., and Makoto N.: Small solutions for nonlinear heat equations, the Navier-Stokes equation and the KellerSegel system in Besov and Triebel-Lizorkin spaces. Advances in Differential Equations. 18, 687-736 (2013). https://doi.org/10.57262/ade/1369057711
28. Karch, G.: Scaling in nonlinear parabolic equations. Journal of mathematical analysis and applications. 234, 534-558 (1999). https://doi.org/10.1006/jmaa.1999.6370
29. Kato, T.: Strong Lp-solutions of the Navier-Stokes equation in Rm, with applications to weak solutions. Math. Z. 187, 471-480 (1984). https://doi.org/10.1007/BF01174182
30. Kato, T.: Strong solutions of the Navier-Stokes equation in Morrey spaces. Boletim da Sociedade Brasileira de Matem'etica-Bulletin/Brazilian Mathematical Society. 22, 127-155 (1992). https://doi.org/10.1007/BF01232939
31. Koch, H., Tataru, D.: Well-posedness for the Navier-Stokes equations, Adv. Math. 157, 22-35 (2001). https://doi.org/10.1006/aima.2000.1937
32. Kozono, H., Yamazaki, M.: Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data. Comm. Partial Differential Equations. 19, 959-1014 (1994). https://doi.org/10.1080/03605309408821042
33. Kurokiba, M., Ogawa, T.: Well-posedness for the drift-diffusion system in Lp arising from the semiconductor device simulation. Journal of Mathematical Analysis and Applications. 342, 1052-1067 (2008). https://doi.org/10.1016/j.jmaa.2007.11.017
34. Lemari'e-Rieusset, P. G.: Recent developments in the Navier-Stokes problem. CRC Press, (2002). https://doi.org/10.1201/9780367801656
35. Liu, Q., Zhao, J., Cui, S.: Existence and regularizing rate estimates of solutions to a generalized magneto-hydrodynamic system in pseudomeasure spaces. Annali di Matematica Pura ed Applicata. 191, 293-309 (2012). https://doi.org/10.1007/s10231-010-0184-8
36. Luo, Y.: Well-Posedness Of A Cauchy Problem Involving Nonlinear Fractal Dissipative Equations. Applied Mathematics E-Notes. 10, 112-118 (2010).
37. Miao, C., Yuan, B., Zhang, B.: Well-posedness of the Cauchy problem for the fractional power dissipative equations. Nonlinear Analysis: Theory, Methods and Applications. 68 (2008), 461-484. https://doi.org/10.1016/j.na.2006.11.011
38. Wang, W., Wu, G.: Global mild solution of the generalized Navier-Stokes equations with the Coriolis force. Applied Mathematics Letters. 76, 181-186 (2018) https://doi.org/10.1016/j.aml.2017.09.001
39. Yamamoto, M.: Spatial analyticity of solutions to the drift-diffusion equation with generalized dissipation. Arch. Math. 97, 261-270 (2011). https://doi.org/10.1007/s00013-011-0302-x
40. Yamazaki, M.: The Navier-Stokes equations in the weak-Ln space with time-dependent external force. Math. Ann. 317, 635-675 (2000). https://doi.org/10.1007/PL00004418
41. Zhao, J.: Well-posedness and Gevrey analyticity of the generalized Keller-Segel system in critical Besov spaces. Annali di Matematica. 197, 521-548 (2018). https://doi.org/10.1007/s10231-017-0691-y
42. Zhao, J.: Gevrey regularity of mild solutions to the parabolic-elliptic system of drift-diffusion type in critical Besov spaces. Journal of Mathematical Analysis and Applications, 448, 1265-1280 (2017). https://doi.org/10.1016/j.jmaa.2016.11.050
43. Zhao, J., Liu, Q., Cui, S.: Existence of solutions for the Debye-H¨uckel system with low regularity initial data. Acta applicandae mathematicae. 125, 1-10 (2013). https://doi.org/10.1007/s10440-012-9777-0
44. Zhou, X., Xiao, W.: Algebra Properties in Fourier-Besov Spaces and Their Applications. Journal of Function Spaces, 2018, (2018). https://doi.org/10.1155/2018/3629179
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