## Convection-induced singularity suppression in the Keller-Segel and other non-linear PDEs

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- by Gautam Iyer, Xiaoqian Xu and Andrej Zlatoš PDF
- Trans. Amer. Math. Soc.
**374**(2021), 6039-6058 Request permission

## Abstract:

In this paper we study the effect of the addition of a convective term, and of the resulting increased dissipation rate, on the growth of solutions to a general class of non-linear parabolic PDEs. In particular, we show that blow-up in these models can always be prevented if the added drift has a small enough dissipation time. We also prove a general result relating the dissipation time and the effective diffusivity of stationary cellular flows, which allows us to obtain examples of simple incompressible flows with arbitrarily small dissipation times.

As an application, we show that blow-up in the Keller-Segel model of chemotaxis can always be prevented if the velocity field of the ambient fluid has a sufficiently small dissipation time. We also study reaction-diffusion equations with ignition-type nonlinearities, and show that the reaction can always be quenched by the addition of a convective term with a small enough dissipation time, provided the average initial temperature is initially below the ignition threshold.

## References

- Giovanni Alberti, Gianluca Crippa, and Anna L. Mazzucato,
*Exponential self-similar mixing by incompressible flows*, J. Amer. Math. Soc.**32**(2019), no. 2, 445–490. MR**3904158**, DOI 10.1090/jams/913 - Jacob Bedrossian and Siming He,
*Suppression of blow-up in Patlak-Keller-Segel via shear flows*, SIAM J. Math. Anal.**49**(2017), no. 6, 4722–4766. MR**3730537**, DOI 10.1137/16M1093380 - H. Berestycki.
*The influence of advection on the propagation of fronts in reaction-diffusion equations*, Springer, Netherlands, Dordrecht, 2002, pp. 11–48. - Henri Berestycki, François Hamel, and Nikolai Nadirashvili,
*Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena*, Comm. Math. Phys.**253**(2005), no. 2, 451–480. MR**2140256**, DOI 10.1007/s00220-004-1201-9 - Henri Berestycki, Alexander Kiselev, Alexei Novikov, and Lenya Ryzhik,
*The explosion problem in a flow*, J. Anal. Math.**110**(2010), 31–65. MR**2753290**, DOI 10.1007/s11854-010-0002-7 - S. Childress and A. M. Soward,
*Scalar transport and alpha-effect for a family of cat’s-eye flows*, J. Fluid Mech.**205**(1989), 99–133. MR**1014361**, DOI 10.1017/S0022112089001965 - P. Constantin, A. Kiselev, L. Ryzhik, and A. Zlatoš,
*Diffusion and mixing in fluid flow*, Ann. of Math. (2)**168**(2008), no. 2, 643–674. MR**2434887**, DOI 10.4007/annals.2008.168.643 - Michele Coti Zelati, Matias G. Delgadino, and Tarek M. Elgindi,
*On the relation between enhanced dissipation timescales and mixing rates*, Comm. Pure Appl. Math.**73**(2020), no. 6, 1205–1244. MR**4156602**, DOI 10.1002/cpa.21831 - Tarek M. Elgindi and Andrej Zlatoš,
*Universal mixers in all dimensions*, Adv. Math.**356**(2019), 106807, 33. MR**4008523**, DOI 10.1016/j.aim.2019.106807 - Lawrence C. Evans,
*Partial differential equations*, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR**1625845** - A. Fannjiang, A. Kiselev, and L. Ryzhik,
*Quenching of reaction by cellular flows*, Geom. Funct. Anal.**16**(2006), no. 1, 40–69. MR**2221252**, DOI 10.1007/s00039-006-0554-y - Albert Fannjiang and George Papanicolaou,
*Convection enhanced diffusion for periodic flows*, SIAM J. Appl. Math.**54**(1994), no. 2, 333–408. MR**1265233**, DOI 10.1137/S0036139992236785 - Albert Fannjiang and Lech Wołowski,
*Noise induced dissipation in Lebesgue-measure preserving maps on $d$-dimensional torus*, J. Statist. Phys.**113**(2003), no. 1-2, 335–378. MR**2012983**, DOI 10.1023/A:1025787124437 - Yuanyuan Feng and Gautam Iyer,
*Dissipation enhancement by mixing*, Nonlinearity**32**(2019), no. 5, 1810–1851. MR**3942601**, DOI 10.1088/1361-6544/ab0e56 - Siming He,
*Suppression of blow-up in parabolic-parabolic Patlak-Keller-Segel via strictly monotone shear flows*, Nonlinearity**31**(2018), no. 8, 3651–3688. MR**3826109**, DOI 10.1088/1361-6544/aac1ce - Siming He and Eitan Tadmor,
*Suppressing chemotactic blow-up through a fast splitting scenario on the plane*, Arch. Ration. Mech. Anal.**232**(2019), no. 2, 951–986. MR**3925534**, DOI 10.1007/s00205-018-01336-7 - M. A. Herrero,
*Asymptotic properties of reaction-diffusion systems modeling chemotaxis*, Applied and industrial mathematics, Venice–2, 1998, Kluwer Acad. Publ., Dordrecht, 2000, pp. 89–108. MR**1755323** - M. A. Herrero, E. Medina, and J. J. L. Velázquez,
*Finite-time aggregation into a single point in a reaction-diffusion system*, Nonlinearity**10**(1997), no. 6, 1739–1754. MR**1483563**, DOI 10.1088/0951-7715/10/6/016 - M. A. Herrero, E. Medina, and J. J. L. Velázquez,
*Self-similar blow-up for a reaction-diffusion system*, J. Comput. Appl. Math.**97**(1998), no. 1-2, 99–119. MR**1651769**, DOI 10.1016/S0377-0427(98)00104-6 - Dirk Horstmann,
*From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I*, Jahresber. Deutsch. Math.-Verein.**105**(2003), no. 3, 103–165. MR**2013508** - Dirk Horstmann,
*From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. II*, Jahresber. Deutsch. Math.-Verein.**106**(2004), no. 2, 51–69. MR**2073515** - W. Jäger and S. Luckhaus,
*On explosions of solutions to a system of partial differential equations modelling chemotaxis*, Trans. Amer. Math. Soc.**329**(1992), no. 2, 819–824. MR**1046835**, DOI 10.1090/S0002-9947-1992-1046835-6 - Evelyn F. Keller and Lee A. Segel,
*Initiation of slime mold aggregation viewed as an instability*, J. Theoret. Biol.**26**(1970), no. 3, 399–415. MR**3925816**, DOI 10.1016/0022-5193(70)90092-5 - Evelyn F. Keller and Lee A. Segel.
*Model for chemotaxis*. J. Theor. Biol., 30(2):225 – 234, 1971. - Alexander Kiselev, Roman Shterenberg, and Andrej Zlatoš,
*Relaxation enhancement by time-periodic flows*, Indiana Univ. Math. J.**57**(2008), no. 5, 2137–2152. MR**2463964**, DOI 10.1512/iumj.2008.57.3349 - Alexander Kiselev and Xiaoqian Xu,
*Suppression of chemotactic explosion by mixing*, Arch. Ration. Mech. Anal.**222**(2016), no. 2, 1077–1112. MR**3544323**, DOI 10.1007/s00205-016-1017-8 - Alexander Kiselev and Andrej Zlatoš,
*Quenching of combustion by shear flows*, Duke Math. J.**132**(2006), no. 1, 49–72. MR**2219254**, DOI 10.1215/S0012-7094-06-13212-X - L. Koralov,
*Random perturbations of 2-dimensional Hamiltonian flows*, Probab. Theory Related Fields**129**(2004), no. 1, 37–62. MR**2052862**, DOI 10.1007/s00440-003-0320-0 - Vladimir G. Maz’ja,
*Sobolev spaces*, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. Translated from the Russian by T. O. Shaposhnikova. MR**817985**, DOI 10.1007/978-3-662-09922-3 - Bernt Øksendal,
*Stochastic differential equations*, 6th ed., Universitext, Springer-Verlag, Berlin, 2003. An introduction with applications. MR**2001996**, DOI 10.1007/978-3-642-14394-6 - Clifford S. Patlak,
*Random walk with persistence and external bias*, Bull. Math. Biophys.**15**(1953), 311–338. MR**81586**, DOI 10.1007/bf02476407 - Benoît Perthame,
*Transport equations in biology*, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007. MR**2270822** - Lenya Ryzhik and Andrej Zlatoš,
*KPP pulsating front speed-up by flows*, Commun. Math. Sci.**5**(2007), no. 3, 575–593. MR**2352332** - Jack Xin,
*Front propagation in heterogeneous media*, SIAM Rev.**42**(2000), no. 2, 161–230. MR**1778352**, DOI 10.1137/S0036144599364296 - Jack Xin,
*An introduction to fronts in random media*, Surveys and Tutorials in the Applied Mathematical Sciences, vol. 5, Springer, New York, 2009. MR**2527020**, DOI 10.1007/978-0-387-87683-2 - Yao Yao and Andrej Zlatoš,
*Mixing and un-mixing by incompressible flows*, J. Eur. Math. Soc. (JEMS)**19**(2017), no. 7, 1911–1948. MR**3656475**, DOI 10.4171/JEMS/709 - Andrej Zlatoš,
*Diffusion in fluid flow: dissipation enhancement by flows in 2D*, Comm. Partial Differential Equations**35**(2010), no. 3, 496–534. MR**2748635**, DOI 10.1080/03605300903362546 - Andrej Zlatoš,
*Reaction-diffusion front speed enhancement by flows*, Ann. Inst. H. Poincaré C Anal. Non Linéaire**28**(2011), no. 5, 711–726. MR**2838397**, DOI 10.1016/j.anihpc.2011.05.004

## Additional Information

**Gautam Iyer**- Affiliation: Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
- MR Author ID: 753706
- ORCID: 0000-0001-9638-0455
- Email: gautam@math.cmu.edu
**Xiaoqian Xu**- Affiliation: Zu Chongzhi Center of Mathematical and Computational Science, Duke Kunshan University, People’s Republic of China
- Email: xiaoqian.xu@dukekunshan.edu.cn
**Andrej Zlatoš**- Affiliation: Department of Mathematics, UC San Diego, La Jolla, California 92130
- Email: zlatos@ucsd.edu
- Received by editor(s): August 5, 2019
- Received by editor(s) in revised form: March 27, 2020
- Published electronically: June 7, 2021
- Additional Notes: This work has been partially supported by the National Science Foundation under grants DMS-1652284 and DMS-1900943 to AZ, and DMS-1814147 to GI, as well as by the Center for Nonlinear Analysis.
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**374**(2021), 6039-6058 - MSC (2020): Primary 35B44; Secondary 35B27, 35Q35, 76R05
- DOI: https://doi.org/10.1090/tran/8195
- MathSciNet review: 4302154