Abstract
In this paper we study existence of solutions for the Cauchy problem of the Debye-Hückel system with low regularity initial data. By using the Chemin-Lerner time-space estimate for the heat equation, we prove that there exists a unique local solution if the initial data belongs to the Besov space \(\dot{B}^{s}_{p,q}(\mathbb{R}^{n})\) for \(-\frac{3}{2}<s\leq-2+\frac{n}{2}\), \(p=\frac{n}{s+2}\) and 1≤q≤∞, and furthermore, if the initial data is sufficiently small then the solution is global. This result improves the regularity index of the initial data space in previous results on this system. The blow-up criterion of solutions is also established.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ben Abdallah, N., Méhats, F., Vauchelet, N.: A note on the long time behavior for the drift-diffusion-Poisson system. C. R. Math. Acad. Sci. Paris 339(10), 683–688 (2004)
Biler, P., Dolbeault, J.: Long time behavior of solutions to Nernst-Planck and Debye-Hückel drift-diffusion systems. Ann. Henri Poincaré 1, 461–472 (2000)
Biler, P., Hebisch, W., Nadzieja, T.: The Debye system: existence and large time behavior of solutions. Nonlinear Anal. 23, 1189–1209 (1994)
Chemin, J.-Y.: Perfect Incompressible Fluids. Oxford Lecture Series in Mathematics and Its Applications, vol. 14. The Clarendon Press/Oxford University Press, New York (1998)
Dachin, R.: Fourier analysis methods for PDE’s (2005). http://perso-math.univ-mlv.fr/users/danchin.raphael/courschine.pdf
Debye, P., Hückel, E.: Zur Theorie der Elektrolyte. II. Das Grenzgesetz für die elektrische Leitfähigkeit. Phys. Z. 24, 305–325 (1923)
Gajewski, H.: On existence, uniqueness and asymptotic behavior of solutions of the basic equations for carrier transport in semiconductors. Z. Angew. Math. Mech. 65, 101–108 (1985)
Gajewski, H., Gröger, K.: On the basic equations for carrier transport in semiconductors. J. Math. Anal. Appl. 113, 12–35 (1986)
Iwabuchi, T.: Global wellposedness for Keller-Segel system in Besov type spaces. J. Math. Anal. Appl. 379, 930–948 (2011)
Karch, G.: Scaling in nonlinear parabolic equations. J. Math. Anal. Appl. 234, 534–558 (1999)
Kurokiba, M., Ogawa, T.: Well-posedness for the drift-diffusion system in L p arising from the semiconductor device simulation. J. Math. Anal. Appl. 342, 1052–1067 (2008)
Lemarié-Rieusset, P.-G.: Recent Developments in the Navier-Stokes Problem. Research Notes in Mathematics. Chapman & Hall/CRC, London (2002)
Mock, M.S.: An initial value problem from semiconductor device theory. SIAM J. Math. Anal. 5, 597–612 (1974)
Ogawa, T., Shimizu, S.: The drift-diffusion system in two-dimensional critical hardy space. J. Funct. Anal. 255, 1107–1138 (2008)
Ogawa, T., Shimizu, S.: End-point maximal regularity and wellposedness of the two dimensional Keller-Segel system in a critical Besov space. Math. Z. 264, 601–628 (2010)
Runst, T., Sickel, W.: Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations. de Gruyter Series in Nonlinear Analysis and Applications, vol. 3. de Gruyter, Berlin (1996)
Selberherr, S.: Analysis and Simulation of Semiconductor Devices. Springer, Berlin (1983)
Wu, G., Yuan, J.: Well-posedness of the cauchy problem for the fractional power dissipative equation in critical Besov spaces. J. Math. Anal. Appl. 340, 1326–1335 (2008)
Zhao, J., Liu, Q., Cui, S.: Regularizing and decay rate estimates for solutions to the cauchy problem of the Debye-Hückel system. Nonlinear Differ. Equ. Appl. 19, 1–18 (2012)
Acknowledgements
The authors wish to acknowledge their sincere thanks to the anonymous referee for informing us some important references and valuable suggestions on improving the writing of this paper. This work is supported by the NSF of China (11171357) and the Doctor Fund of Northwest A&F University (Z109021118).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhao, J., Liu, Q. & Cui, S. Existence of Solutions for the Debye-Hückel System with Low Regularity Initial Data. Acta Appl Math 125, 1–10 (2013). https://doi.org/10.1007/s10440-012-9777-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10440-012-9777-0