## Osaka Journal of Mathematics

### Ill-posedness issue for the drift diffusion system in the homogeneous Besov spaces

#### Abstract

We consider the ill-posedness issue for the drift-diffusion system of bipolar type by showing that the continuous dependence on initial data does not hold generally in the scaling invariant Besov spaces. The scaling invariant Besov spaces are $\dot{B}_{p, \sigma}^{-2+ n/p} (\mathbb{R}^{n})$ with $1 \leq p, \sigma \leq \infty$ and we show the optimality of the case $p = 2n$ to obtain the well-posedness and the ill-posedness for the drift-diffusion system of bipolar type.

#### Article information

Source
Osaka J. Math. Volume 53, Number 4 (2016), 919-939.

Dates
First available in Project Euclid: 4 October 2016

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1475601824

Mathematical Reviews number (MathSciNet)
MR3554849

Zentralblatt MATH identifier
06654656

Subjects
Primary: 35K55: Nonlinear parabolic equations
Secondary: 35K08: Heat kernel

#### Citation

Iwabuchi, Tsukasa; Ogawa, Takayoshi. Ill-posedness issue for the drift diffusion system in the homogeneous Besov spaces. Osaka J. Math. 53 (2016), no. 4, 919--939. https://projecteuclid.org/euclid.ojm/1475601824.

#### References

• P. Biler and J. Dolbeault: Long time behavior of solutions of Nernst–Planck and Debye–Hückel drift-diffusion systems, Ann. Henri Poincaré 1 (2000), 461–472.
• J. Bourgain and N. Pavlović: Ill-posedness of the Navier–Stokes equations in a critical space in 3D, J. Funct. Anal. 255 (2008), 2233–2247.
• P. Biler, M. Cannone, I.A. Guerra and G. Karch: Global regular and singular solutions for a model of gravitating particles, Math. Ann. 330 (2004), 693–708.
• M. Cannone and F. Planchon: Self-similar solutions for Navier–Stokes equations in $\mathbf{R}^{3}$, Comm. Partial Differential Equations 21 (1996), 179–193.
• L. Corrias, B. Perthame and H. Zaag: Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math. 72 (2004), 1–28.
• W. Fang and K. Ito: Global solutions of the time-dependent drift-diffusion semiconductor equations, J. Differential Equations 123 (1995), 523–566.
• H.G. Feichtinger: Modulation spaces on locally compact Abelian groups, Technical Report, University of Vienna, 1983, in; Proc. Internat. Conf. on Wavelets and Applications, New Delhi Allied Publishers, 1–56, 2003.
• H. Gajewski and K. Gröger: On the basic equations for carrier transport in semiconductors, J. Math. Anal. Appl. 113 (1986), 12–35.
• T. Iwabuchi: Global well-posedness for Keller–Segel system in Besov type spaces, J. Math. Anal. Appl. 379 (2011), 930–948.
• T. Iwabuchi and T. Ogawa: Ill-posedness for the nonlinear Schrödinger equation with quadratic non-linearity in low dimensions, Trans. Amer. Math. Soc. 367 (2015), 2613–2630.
• A. Jüngel: Qualitative behavior of solutions of a degenerate nonlinear drift-diffusion model for semiconductors, Math. Models Methods Appl. Sci. 5 (1995), 497–518.
• E.F. Keller and L.A. Segel: Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399–415.
• H. Kozono, T. Ogawa and Y. Taniuchi: Navier–Stokes equations in the Besov space near $L^{\infty}$ and BMO, Kyushu J. Math. 57 (2003), 303–324.
• H. Kozono and Y. Sugiyama: Local existence and finite time blow-up of solutions in the 2-D Keller–Segel system, J. Evol. Equ. 8 (2008), 353–378.
• H. Kozono and Y. Sugiyama: Strong solutions to the Keller–Segel system with the weak $L^{n/2}$ initial data and its application to the blow-up rate, Math. Nachr. 283 (2010), 732–751.
• H. Kozono and M. Yamazaki: Semilinear heat equations and the Navier–Stokes equation with distributions in new function spaces as initial data, Comm. Partial Differential Equations 19 (1994), 959–1014.
• M. Kurokiba and T. Ogawa: Finite time blow-up of the solution for a nonlinear parabolic equation of drift-diffusion type, Differential Integral Equations 16 (2003), 427–452.
• M. Kurokiba and T. Ogawa: Well-posedness for the drift-diffusion system in $L^{p}$ arising from the semiconductor device simulation, J. Math. Anal. Appl. 342 (2008), 1052–1067.
• M.S. Mock: An initial value problem from semiconductor device theory, SIAM J. Math. Anal. 5 (1974), 597–612.
• T. Nagai: Behavior of solutions to a parabolic-elliptic system modelling chemotaxis, J. Korean Math. Soc. 37 (2000), 721–733.
• T. Nagai, T. Senba and K. Yoshida: Application of the Trudinger–Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac. 40 (1997), 411–433.
• T. Ogawa and S. Shimizu: The drift-diffusion system in two-dimensional critical Hardy space, J. Funct. Anal. 255 (2008), 1107–1138.
• T. Ogawa and S. Shimizu: End-point maximal regularity and its application to two-dimensional Keller–Segel system, Math. Z. 264 (2010), 601–628.
• T.I. Seidman and G.M. Troianiello: Time-dependent solutions of a nonlinear system arising in semiconductor theory, Nonlinear Anal. 9 (1985), 1137–1157.
• J. Toft: Continuity properties for modulation spaces, with applications to pseudo-differential calculus, I, J. Funct. Anal. 207 (2004), 399–429.
• H. Triebel: Theory of Function Spaces, Birkhäuser, Basel, 1983.
• W. Baoxiang, Z. Lifeng and G. Boling: Isometric decomposition operators, function spaces $E^{\lambda}_{p,q}$ and applications to nonlinear evolution equations, J. Funct. Anal. 233 (2006), 1–39.
• A. Yagi: Norm behavior of solutions to a parabolic system of chemotaxis, Math. Japon. 45 (1997), 241–265.
• T. Yoneda: Ill-posedness of the 3D-Navier–Stokes equations in a generalized Besov space near $BMO^{-1}$, J. Funct. Anal. 258 (2010), 3376–3387.
• J. Zhao, Q. Liu and S. Cui: Existence of solutions for the Debye–Hückel system with low regularity initial data, Acta Appl. Math. 125 (2013), 1–10.