Global well-posedness for a generalized Keller-Segel system with degenerate dissipation and mixing

We study the mixing effect for a generalized Keller-Segel system with degenerate dissipation and advection by a weakly mixing. Here the attractive operator has weak singularity, namely, the negative derivative appears in the nonlinear term by singular integral. Without advection, the solution of equation blows up in finite time. We show that the global well-posedness of solution with large advection. Since dissipation term degenerate into the damping, the enhanced dissipation effect of mixing no longer occurs, we prove that the mixing effect can weak the influence of nonlinear term. In this case, the mixing effect is similar with inviscid damping of shear flow. Combining to the mixing effect and damping effect of degenerate dissipation, the global L∞ estimate of solution is established.

[1]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[2]  Zhifei Zhang,et al.  Enhanced dissipation for the Kolmogorov flow via the hypocoercivity method , 2019, Science China Mathematics.

[3]  C. Zillinger.  Linear Inviscid Damping for Monotone Shear Flows in a Finite Periodic Channel, Boundary Effects, Blow-up and Critical Sobolev Regularity , 2015, 1506.04010.

[4]  Chongchun Zeng,et al.  Inviscid Dynamical Structures Near Couette Flow , 2010, 1004.5149.

[5]  P. Constantin,et al.  Diffusion and mixing in fluid flow , 2005 .

[6]  Dongyi Wei,et al.  Linear Inviscid Damping for a Class of Monotone Shear Flow in Sobolev Spaces , 2015, 1509.08228.

[7]  Dongyi Wei.  Diffusion and mixing in fluid flow via the resolvent estimate , 2018, Science China Mathematics.

[8]  B. Fayad.  Weak mixing for reparameterized linear flows on the torus , 2002, Ergodic Theory and Dynamical Systems.

[9]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[10]  Takashi Suzuki,et al.  Weak Solutions to a Parabolic-Elliptic System of Chemotaxis , 2002 .

[11]  Zhiwu Lin,et al.  Metastability of Kolmogorov Flows and Inviscid Damping of Shear Flows , 2017, Archive for Rational Mechanics and Analysis.

[12]  Hans L. Cycon,et al.  Schrodinger Operators: With Application to Quantum Mechanics and Global Geometry , 1987 .

[13]  H. Jia.  Linear Inviscid Damping in Gevrey Spaces , 2019, Archive for Rational Mechanics and Analysis.

[14]  Weike Wang,et al.  Global classical solutions to the Cauchy problem of conservation laws with degenerate diffusion , 2016 .

[15]  Zhifei Zhang,et al.  Linear Inviscid Damping and Vorticity Depletion for Shear Flows , 2017, Annals of PDE.

[16]  小澤 徹,et al.  Nonlinear dispersive equations , 2006 .

[17]  Dong Li,et al.  Exploding solutions for a nonlocal quadratic evolution problem , 2010 .

[18]  Weike Wang,et al.  Suppression of blow up by mixing in generalized Keller–Segel system with fractional dissipation , 2019, 2103.04484.

[19]  A. Ionescu,et al.  Inviscid Damping Near the Couette Flow in a Channel , 2018, Communications in Mathematical Physics.

[20]  Vlad Vicol,et al.  Enhanced Dissipation and Inviscid Damping in the Inviscid Limit of the Navier–Stokes Equations Near the Two Dimensional Couette Flow , 2014, 1408.4754.

[21]  Y. Ascasibar,et al.  An approximate treatment of gravitational collapse , 2012, 1211.5392.

[22]  Christian Zillinger,et al.  Linear Inviscid Damping for Monotone Shear Flows , 2014, 1410.7341.

[23]  B. Fayad.  SMOOTH MIXING FLOWS WITH PURELY SINGULAR SPECTRA , 2006 .

[24]  Benoît Perthame,et al.  Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions , 2006 .

[25]  Michele Coti Zelati,et al.  Enhanced Dissipation in the Navier–Stokes Equations Near the Poiseuille Flow , 2019, 1901.01571.

[26]  Ruizhao Zi,et al.  Suppression of blow-up in Patlak-Keller-Segel-Navier-Stokes system via the Couette flow , 2021, Journal of Functional Analysis.

[27]  Weike Wang,et al.  The Pointwise Estimates of Solutions for Euler Equations with Damping in Multi-Dimensions , 2001 .

[28]  A. Kiselev,et al.  Suppression of Chemotactic Explosion by Mixing , 2015, 1508.05333.

[29]  Zhifei Zhang,et al.  Linear inviscid damping and enhanced dissipation for the Kolmogorov flow , 2017, Advances in Mathematics.

[30]  Benoît Perthame,et al.  Global Solutions of Some Chemotaxis and Angiogenesis Systems in High Space Dimensions , 2004 .

[31]  P. Biler,et al.  Blowup of solutions to generalized Keller–Segel model , 2008, 0812.4982.

[32]  N. Masmoudi,et al.  Enhanced dissipation for the 2D couette flow in critical space , 2019, Communications in Partial Differential Equations.

[33]  J. Bedrossian,et al.  Enhanced Dissipation, Hypoellipticity, and Anomalous Small Noise Inviscid Limits in Shear Flows , 2015, 1510.08098.

[34]  Samir Salem,et al.  Fractional Keller–Segel equation: Global well-posedness and finite time blow-up , 2018, Communications in Mathematical Sciences.

[35]  Siming He,et al.  Suppression of Blow-Up in Patlak-Keller-Segel Via Shear Flows , 2016, SIAM J. Math. Anal..

[36]  N. Masmoudi,et al.  Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations , 2013, 1306.5028.