Asymptotics of Chemotaxis Systems with Fractional Dissipation for Small Data in Critical Sobolev Space
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Asymptotics of Chemotaxis Systems with Fractional Dissipation for Small Data in Critical Sobolev Space Jaewook Ahn1
· Jihoon Lee1
Received: 20 June 2019 / Accepted: 24 October 2019 © Springer Nature B.V. 2019
Abstract A chemotaxis system with Newtonian attraction and fractional dissipation of orN der α ∈ (0, 2) is considered in RN . For initial data belonging to L1 ∩ H 4 but small in L α , N = 2, 3, the temporal decay and the asymptotic behavior of a global classical solution are established. In particular, we derive a precise decay estimate for higher Sobolev norms. Keywords Asymptotics · Fractional dissipation · Kato–Ponce inequality Mathematics Subject Classification (2010) 35R11 · 92C17
1 Introduction Recent experiments [2, 27, 32] suggested that certain chemotactic cells follow a Lévy jump process. This process is a random process with the property that whose marginal distribution is α-stable, i.e. with heavy tail. Therefore, Lévy jump process exhibits a non-negligible probability of “long positional jumps” in contrast with a standard Brownian process [34]. According to [27], the bacterium E. coli can sometimes perform “long jumps” when it is exposed to an increasing chemoattractant gradient in a certain environment. For another interpretation on the origin of “long jumps”, see [38]. Note that these “long jumps” in a biological context correspond to maintaining a single direction of motion for a longer interval of time than normal random walks [18]. Motivated by aforementioned and related experimental works, many theoretical efforts have been made to derive and analyze a macroscopic chemotaxis model in the form α ∂t u + (−) 2 u = −∇ · uB(u) . (1) JA was supported by NRF, Republic of Korea-2018R1D1A1B07047465. JL was supported by SSTF-BA1701-05 (Samsung Science & Technology Foundation, Republic of Korea).
B J. Ahn
[email protected] J. Lee [email protected]
1
Department of Mathematics, Chung-Ang University, Seoul 06974, Republic of Korea
J. Ahn, J. Lee α
Here, the anomalous diffusion operator (−) 2 (which corresponds to Lévy jump process) is defined by the Fourier transform
α
F (−) 2 f = |ξ |α F [f ],
0 < α < 2.
Under assumptions relevant for the run and tumble motion of E. coli, Estrada-Rodriguez et al. [18] derived a fractional model of type (1) from a microscopic model. Perthame– Sun–Tang [34] also derived a fractional chemotaxis model from a kinetic equation under proper scaling and conditions on the tumbling frequency and the form of noise. For more mathematical derivation of (1) from a microscopic particle system, we refer to [1, 3, 18, 34, 35] and references therein. We also refer the reader to [4, 5, 9, 24, 28, 30] for the results regarding solution existence, uniqueness and asymptotics related to (1). In this study, we investigate the asymptotics of the chemotaxis model of type (1). More precisely, generalizing the classical Patlak–Keller–Segel system with the Newtonian attraction, we consider
α
∂t u + (−) 2 u = −∇ · (u∇(−)−1 u) u(x, 0) = u0 (x),
x ∈ RN , t > 0, x ∈ RN
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