Browse > Article
http://dx.doi.org/10.4134/BKMS.b191054

GEVREY REGULARITY AND TIME DECAY OF THE FRACTIONAL DEBYE-HÜCKEL SYSTEM IN FOURIER-BESOV SPACES  

Cui, Yiwen (School of Applied Mathematics Nanjing University of Finance and Economics)
Xiao, Weiliang (School of Applied Mathematics Nanjing University of Finance and Economics)
Publication Information
Bulletin of the Korean Mathematical Society / v.57, no.6, 2020 , pp. 1393-1408 More about this Journal
Abstract
In this paper we mainly study existence and regularity of mild solutions to the parabolic-elliptic system of drift-diffusion type with small initial data in Fourier-Besov spaces. To be more detailed, we will explain that global-in-time mild solutions are well-posed and Gevrey regular by means of multilinear singular integrals and Fourier localization argument. Furthermore, we can get time decay rate estimate of mild solutions in Fourier-Besov spaces.
Keywords
Debye-$H{\ddot{u}}ckel$ system; Gevrey regularity; time decay; Fourier-Besov spaces;
Citations & Related Records
연도 인용수 순위
  • Reference
1 A. B. Ferrari and E. S. Titi, Gevrey regularity for nonlinear analytic parabolic equations, Comm. Partial Differential Equations 23 (1998), no. 1-2, 1-16. https://doi.org/10.1080/03605309808821336   DOI
2 C. Foia, s and R. Temam, Some analytic and geometric properties of the solutions of the evolution Navier-Stokes equations, J. Math. Pures Appl. (9) 58 (1979), no. 3, 339-368.
3 C. Foias, Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal. 87 (1989), no. 2, 359-369. https://doi.org/10.1016/0022-1236(89)90015-3   DOI
4 H. Gajewski, On existence, uniqueness and asymptotic behavior of solutions of the basic equations for carrier transport in semiconductors, Z. Angew. Math. Mech. 65 (1985), no. 2, 101-108. https://doi.org/10.1002/zamm.19850650210   DOI
5 H. Gajewski and K. Groger, On the basic equations for carrier transport in semiconductors, J. Math. Anal. Appl. 113 (1986), no. 1, 12-35. https://doi.org/10.1016/0022-247X(86)90330-6   DOI
6 G. Karch, Scaling in nonlinear parabolic equations, J. Math. Anal. Appl. 234 (1999), no. 2, 534-558. https://doi.org/10.1006/jmaa.1999.6370   DOI
7 M. Kurokiba and T. Ogawa, Well-posedness for the drift-diffusion system in Lp arising from the semiconductor device simulation, J. Math. Anal. Appl. 342 (2008), no. 2, 1052-1067. https://doi.org/10.1016/j.jmaa.2007.11.017   DOI
8 P. G. Lemarie-Rieusset, Recent developments in the Navier-Stokes problem, Chapman & Hall/CRC Research Notes in Mathematics, 431, Chapman & Hall/CRC, Boca Raton, FL, 2002. https://doi.org/10.1201/9781420035674   DOI
9 W. Liu, One-dimensional steady-state Poisson-Nernst-Planck systems for ion channels with multiple ion species, J. Differential Equations 246 (2009), no. 1, 428-451. https://doi.org/10.1016/j.jde.2008.09.010   DOI
10 W. Liu and B. Wang, Poisson-Nernst-Planck systems for narrow tubular-like membrane channels, J. Dynam. Differential Equations 22 (2010), no. 3, 413-437. https://doi.org/10.1007/s10884-010-9186-x   DOI
11 Y. Luo, Well-posedness of a Cauchy problem involving nonlinear fractal dissipative equations, Appl. Math. E-Notes 10 (2010), 112-118.
12 S. Selberherr, Analysis and Simulation of Semiconductor Devices, Springer Science & Business Media, 2012.
13 C. Miao, B. Yuan, and B. Zhang, Well-posedness of the Cauchy problem for the fractional power dissipative equations, Nonlinear Anal. 68 (2008), no. 3, 461-484. https://doi.org/10.1016/j.na.2006.11.011   DOI
14 M. S. Mock, An initial value problem from semiconductor device theory, SIAM J. Math. Anal. 5 (1974), 597-612. https://doi.org/10.1137/0505061   DOI
15 T. Ogawa and M. Yamamoto, Asymptotic behavior of solutions to drift-diffusion system with generalized dissipation, Math. Models Methods Appl. Sci. 19 (2009), no. 6, 939-967. https://doi.org/10.1142/S021820250900367X   DOI
16 N. Ben Abdallah, F. Mehats, and N. Vauchelet, A note on the long time behavior for the drift-diffusion-Poisson system, C. R. Math. Acad. Sci. Paris 339 (2004), no. 10, 683-688. https://doi.org/10.1016/j.crma.2004.09.025   DOI
17 P. Biler and J. Dolbeault, Long time behavior of solutions of Nernst-Planck and Debye-Huckel drift-diffusion systems, Ann. Henri Poincare 1 (2000), no. 3, 461-472. https://doi.org/10.1007/s000230050003   DOI
18 P. Biler, W. Hebisch, and T. Nadzieja, The Debye system: existence and large time behavior of solutions, Nonlinear Anal. 23 (1994), no. 9, 1189-1209. https://doi.org/10.1016/0362-546X(94)90101-5   DOI
19 I. Chueshov, M. Polat, and S. Siegmund, Gevrey regularity of global attractor for generalized Benjamin-Bona-Mahony equation, Mat. Fiz. Anal. Geom. 11 (2004), no. 2, 226-242.
20 G. Wu and J. Yuan, Well-posedness of the Cauchy problem for the fractional power dissipative equation in critical Besov spaces, J. Math. Anal. Appl. 340 (2008), no. 2, 1326-1335. https://doi.org/10.1016/j.jmaa.2007.09.060   DOI
21 J. Zhao, Gevrey regularity of mild solutions to the parabolic-elliptic system of driftdiffusion type in critical Besov spaces, J. Math. Anal. Appl. 448 (2017), no. 2, 1265-1280. https://doi.org/10.1016/j.jmaa.2016.11.050   DOI
22 J. Zhao, Q. Liu, and S. Cui, Existence of solutions for the Debye-Huckel system with low regularity initial data, Acta Appl. Math. 125 (2013), 1-10. https://doi.org/10.1007/s10440-012-9777-0   DOI