American Institute of Mathematical Sciences

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

On a class of diffusion-aggregation equations

Yuming Paul Zhang ,

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 $.

Keywords: Aggregation equation, degenerate diffusion, a priori estimates, uniform boundedness, singular integral.
Mathematics Subject Classification: Primary: 35K55, 45G05; Secondary: 35K65, 35B65.
Citation: Yuming Paul Zhang. On a class of diffusion-aggregation equations. Discrete & Continuous Dynamical Systems - A, 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.  Google Scholar

[2]

J. Bedrossian, N. 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.  Google Scholar

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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.  Google Scholar

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A. Blanchet, V. 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.  Google Scholar

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A. Blanchet, J. 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.  Google Scholar

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S. Boi, V. 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.  Google Scholar

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L. Caffarelli, F. 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.  Google Scholar

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V. Calvez, J. 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.  Google Scholar

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J. A. Carrillo, K. 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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L. Chayes, I. 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.  Google Scholar

[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.  Google Scholar

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

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H. Hajaiej, L. Molinet, T. 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.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

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

[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.  Google Scholar

[29]

C. M. Topaz, A. 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.  Google Scholar

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

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.  Google Scholar

[2]

J. Bedrossian, N. 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.  Google Scholar

[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.  Google Scholar

[4]

A. Blanchet, V. 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.  Google Scholar

[5]

A. Blanchet, J. 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.  Google Scholar

[6]

S. Boi, V. 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.  Google Scholar

[7]

L. Caffarelli, F. 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.  Google Scholar

[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.  Google Scholar

[9]

V. Calvez, J. 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.  Google Scholar

[10]

J. A. Carrillo, K. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

L. Chayes, I. 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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

H. Hajaiej, L. Molinet, T. 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

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

[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.  Google Scholar

[29]

C. M. Topaz, A. 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.  Google Scholar

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