February  2020, 40(2): 907-932. doi: 10.3934/dcds.2020066

On a class of diffusion-aggregation equations

Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90025, USA

* Corresponding author: Yuming Paul Zhang

Received  March 2019 Revised  August 2019 Published  November 2019

We investigate the diffusion-aggregation equations with degenerate diffusion $ \Delta u^m $ and singular interaction kernel $ \mathcal{K}_s = (-\Delta)^{-s} $ with $ s\in(0,\frac{d}{2}) $. The equation is related to biological aggregation models. We analyze the regime where the diffusive force is stronger than the aggregation effect. In such regime, we show the existence and uniform boundedness of solutions in the case either $ s>\frac{1}{2} $ or $ m<2 $. Hölder regularity is obtained when $ d\geq3, s>1/2 $ and uniqueness is proved when $ s\geq 1 $.

Citation: Yuming Paul Zhang. On a class of diffusion-aggregation equations. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 907-932. doi: 10.3934/dcds.2020066
References:
[1]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, volume 343, Springer Science & Business Media, 2011. doi: 10.1007/978-3-642-16830-7.

[2]

J. BedrossianN. Rodríguez and A. L Bertozzi, Local and global well-posedness for aggregation equations and patlak–keller–segel models with degenerate diffusion, Nonlinearity, 24 (2011), 1683-1714.  doi: 10.1088/0951-7715/24/6/001.

[3]

A. L Bertozzi and D. Slepcev, Existence and uniqueness of solutions to an aggregation equation with degenerate diffusion, Communications on Pure and Applied Analysis, 9 (2010), 1617-1637.  doi: 10.3934/cpaa.2010.9.1617.

[4]

A. BlanchetV. Calvez and J. A Carrillo, Convergence of the mass-transport steepest descent scheme for the subcritical patlak–keller–segel model, SIAM Journal on Numerical Analysis, 46 (2008), 691-721.  doi: 10.1137/070683337.

[5]

A. BlanchetJ. A Carrillo and P. Laurençot, Critical mass for a patlak–keller–segel model with degenerate diffusion in higher dimensions, Calculus of Variations and Partial Differential Equations, 35 (2009), 133-168.  doi: 10.1007/s00526-008-0200-7.

[6]

S. BoiV. Capasso and D. Morale, Modeling the aggregative behavior of ants of the species polyergus rufescens, Nonlinear Analysis: Real World Applications, 1 (2000), 163-176.  doi: 10.1016/S0362-546X(99)00399-5.

[7]

L. CaffarelliF. Soria and J. L. Vázquez, Regularity of solutions of the fractional porous medium flow, Journal of the European Mathematical Society, 15 (2013), 1701-1746.  doi: 10.4171/JEMS/401.

[8]

L. Caffarelli and J. L. Vazquez, Nonlinear porous medium flow with fractional potential pressure, Archive for Rational Mechanics and Analysis, 202 (2011), 537-565.  doi: 10.1007/s00205-011-0420-4.

[9]

V. CalvezJ. A. Carrillo and F. Hoffmann, Equilibria of homogeneous functionals in the fair-competition regime, Nonlinear Analysis, 159 (2017), 85-128.  doi: 10.1016/j.na.2017.03.008.

[10]

J. A. CarrilloK. Craig and Y. Yao, Aggregation-diffusion equations: Dynamics, asymptotics, and singular limits, Active Particles, 2 (2019), 65-108.  doi: 10.1007/978-3-030-20297-2_3.

[11]

J. A. Carrillo, F. Hoffmann, E. Mainini and B. Volzone, Ground states in the diffusion-dominated regime, Calculus of Variations and Partial Differential Equations, 57 (2018), Art. 127, 28 pp. doi: 10.1007/s00526-018-1402-2.

[12]

J. A. Carrillo and G. Toscani, Asymptotic l 1-decay of solutions of the porous medium equation to self-similarity, Indiana University Mathematics Journal, 49 (2000), 113-142.  doi: 10.1512/iumj.2000.49.1756.

[13]

J. A. Carrillo and J. Wang, Uniform in time $ l^{\infty}$-estimates for nonlinear aggregation-diffusion equations, Acta Applicandae Mathematicae, (2018), 1–19. doi: 10.1007/s10440-018-0221-y.

[14]

L. ChayesI. Kim and Y. Yao, An aggregation equation with degenerate diffusion: Qualitative property of solutions, SIAM Journal on Mathematical Analysis, 45 (2013), 2995-3018.  doi: 10.1137/120874965.

[15]

J. Dolbeault and B. Perthame, Optimal critical mass in the two dimensional keller–segel model in $\mathbb{R}^2$, Comptes Rendus Mathematique, 339 (2014), 611-616.  doi: 10.1016/j.crma.2004.08.011.

[16]

L. Grafakos, Classical Fourier Analysis, volume 2, Springer, 2008.

[17]

H. HajaiejL. MolinetT. Ozawa and B. Wang, Necessary and sufficient conditions for the fractional gagliardo-nirenberg inequalities and applications to navier-stokes and generalized boson equations (harmonic analysis and nonlinear partial differential equations), RIMS Kokyuroku Bessatsu, 26 (2011), 159-175. 

[18]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, Journal of Differential Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.

[19]

H. E. Huppert and A. W. Woods, Gravity-driven flows in porous layers, Journal of Fluid Mechanics, 292 (1995), 55-69.  doi: 10.1017/S0022112095001431.

[20]

I. Kim and Y. P. Zhang, Regularity properties of degenerate diffusion equations with drifts, SIAM Journal on Mathematical Analysis, 50 (2018), 4371-4406.  doi: 10.1137/17M1159749.

[21]

R. Kowalczyk, Preventing blow-up in a chemotaxis model, Journal of Mathematical Analysis and Applications, 305 (2005), 566-588.  doi: 10.1016/j.jmaa.2004.12.009.

[22]

M. Kwaśnicki, Ten equivalent definitions of the fractional laplace operator, Fractional Calculus and Applied Analysis, 20 (2017), 7-51.  doi: 10.1515/fca-2017-0002.

[23]

Q.-H. Nguyen and J. L. Vázquez, Porous medium equation with nonlocal pressure in a bounded domain, Communications in Partial Differential Equations, 43 (2018), 1502-1539.  doi: 10.1080/03605302.2018.1475492.

[24]

L. Nirenberg, On elliptic partial differential equations, In Il Principio di Minimo e sue Applicazioni Alle Equazioni Funzionali, Springer, 2011, 1–48. doi: 10.1007/978-3-642-10926-3_1.

[25]

M. Riesz, L'intégrale de riemann-liouville et le problème de cauchy, Acta mathematica, 81 (1949), 1-222.  doi: 10.1007/BF02395016.

[26] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, N.J., 1970. 
[27]

Y. Sugiyama et al, Time global existence and asymptotic behavior of solutions to degenerate quasi-linear parabolic systems of chemotaxis, Differential and Integral Equations, 20 (2007), 133-180. 

[28]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic–parabolic keller–segel system with subcritical sensitivity, Journal of Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.

[29]

C. M. TopazA. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bulletin of Mathematical Biology, 68 (2006), 1601-1623.  doi: 10.1007/s11538-006-9088-6.

[30] J. L. Vázquez, The Porous Medium Equation: Mathematical Theory, Oxford University Press, 2007. 

show all references

References:
[1]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, volume 343, Springer Science & Business Media, 2011. doi: 10.1007/978-3-642-16830-7.

[2]

J. BedrossianN. Rodríguez and A. L Bertozzi, Local and global well-posedness for aggregation equations and patlak–keller–segel models with degenerate diffusion, Nonlinearity, 24 (2011), 1683-1714.  doi: 10.1088/0951-7715/24/6/001.

[3]

A. L Bertozzi and D. Slepcev, Existence and uniqueness of solutions to an aggregation equation with degenerate diffusion, Communications on Pure and Applied Analysis, 9 (2010), 1617-1637.  doi: 10.3934/cpaa.2010.9.1617.

[4]

A. BlanchetV. Calvez and J. A Carrillo, Convergence of the mass-transport steepest descent scheme for the subcritical patlak–keller–segel model, SIAM Journal on Numerical Analysis, 46 (2008), 691-721.  doi: 10.1137/070683337.

[5]

A. BlanchetJ. A Carrillo and P. Laurençot, Critical mass for a patlak–keller–segel model with degenerate diffusion in higher dimensions, Calculus of Variations and Partial Differential Equations, 35 (2009), 133-168.  doi: 10.1007/s00526-008-0200-7.

[6]

S. BoiV. Capasso and D. Morale, Modeling the aggregative behavior of ants of the species polyergus rufescens, Nonlinear Analysis: Real World Applications, 1 (2000), 163-176.  doi: 10.1016/S0362-546X(99)00399-5.

[7]

L. CaffarelliF. Soria and J. L. Vázquez, Regularity of solutions of the fractional porous medium flow, Journal of the European Mathematical Society, 15 (2013), 1701-1746.  doi: 10.4171/JEMS/401.

[8]

L. Caffarelli and J. L. Vazquez, Nonlinear porous medium flow with fractional potential pressure, Archive for Rational Mechanics and Analysis, 202 (2011), 537-565.  doi: 10.1007/s00205-011-0420-4.

[9]

V. CalvezJ. A. Carrillo and F. Hoffmann, Equilibria of homogeneous functionals in the fair-competition regime, Nonlinear Analysis, 159 (2017), 85-128.  doi: 10.1016/j.na.2017.03.008.

[10]

J. A. CarrilloK. Craig and Y. Yao, Aggregation-diffusion equations: Dynamics, asymptotics, and singular limits, Active Particles, 2 (2019), 65-108.  doi: 10.1007/978-3-030-20297-2_3.

[11]

J. A. Carrillo, F. Hoffmann, E. Mainini and B. Volzone, Ground states in the diffusion-dominated regime, Calculus of Variations and Partial Differential Equations, 57 (2018), Art. 127, 28 pp. doi: 10.1007/s00526-018-1402-2.

[12]

J. A. Carrillo and G. Toscani, Asymptotic l 1-decay of solutions of the porous medium equation to self-similarity, Indiana University Mathematics Journal, 49 (2000), 113-142.  doi: 10.1512/iumj.2000.49.1756.

[13]

J. A. Carrillo and J. Wang, Uniform in time $ l^{\infty}$-estimates for nonlinear aggregation-diffusion equations, Acta Applicandae Mathematicae, (2018), 1–19. doi: 10.1007/s10440-018-0221-y.

[14]

L. ChayesI. Kim and Y. Yao, An aggregation equation with degenerate diffusion: Qualitative property of solutions, SIAM Journal on Mathematical Analysis, 45 (2013), 2995-3018.  doi: 10.1137/120874965.

[15]

J. Dolbeault and B. Perthame, Optimal critical mass in the two dimensional keller–segel model in $\mathbb{R}^2$, Comptes Rendus Mathematique, 339 (2014), 611-616.  doi: 10.1016/j.crma.2004.08.011.

[16]

L. Grafakos, Classical Fourier Analysis, volume 2, Springer, 2008.

[17]

H. HajaiejL. MolinetT. Ozawa and B. Wang, Necessary and sufficient conditions for the fractional gagliardo-nirenberg inequalities and applications to navier-stokes and generalized boson equations (harmonic analysis and nonlinear partial differential equations), RIMS Kokyuroku Bessatsu, 26 (2011), 159-175. 

[18]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, Journal of Differential Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.

[19]

H. E. Huppert and A. W. Woods, Gravity-driven flows in porous layers, Journal of Fluid Mechanics, 292 (1995), 55-69.  doi: 10.1017/S0022112095001431.

[20]

I. Kim and Y. P. Zhang, Regularity properties of degenerate diffusion equations with drifts, SIAM Journal on Mathematical Analysis, 50 (2018), 4371-4406.  doi: 10.1137/17M1159749.

[21]

R. Kowalczyk, Preventing blow-up in a chemotaxis model, Journal of Mathematical Analysis and Applications, 305 (2005), 566-588.  doi: 10.1016/j.jmaa.2004.12.009.

[22]

M. Kwaśnicki, Ten equivalent definitions of the fractional laplace operator, Fractional Calculus and Applied Analysis, 20 (2017), 7-51.  doi: 10.1515/fca-2017-0002.

[23]

Q.-H. Nguyen and J. L. Vázquez, Porous medium equation with nonlocal pressure in a bounded domain, Communications in Partial Differential Equations, 43 (2018), 1502-1539.  doi: 10.1080/03605302.2018.1475492.

[24]

L. Nirenberg, On elliptic partial differential equations, In Il Principio di Minimo e sue Applicazioni Alle Equazioni Funzionali, Springer, 2011, 1–48. doi: 10.1007/978-3-642-10926-3_1.

[25]

M. Riesz, L'intégrale de riemann-liouville et le problème de cauchy, Acta mathematica, 81 (1949), 1-222.  doi: 10.1007/BF02395016.

[26] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, N.J., 1970. 
[27]

Y. Sugiyama et al, Time global existence and asymptotic behavior of solutions to degenerate quasi-linear parabolic systems of chemotaxis, Differential and Integral Equations, 20 (2007), 133-180. 

[28]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic–parabolic keller–segel system with subcritical sensitivity, Journal of Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.

[29]

C. M. TopazA. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bulletin of Mathematical Biology, 68 (2006), 1601-1623.  doi: 10.1007/s11538-006-9088-6.

[30] J. L. Vázquez, The Porous Medium Equation: Mathematical Theory, Oxford University Press, 2007. 
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