Asymptotics of Chemotaxis Systems with Fractional Dissipation for Small Data in Critical Sobolev Space

Abstract

A chemotaxis system with Newtonian attraction and fractional dissipation of order \(\alpha \in (0,2)\) is considered in \({ \mathbb{R} }^{N}\). For initial data belonging to \(L^{1}\cap H^{4}\) but small in \(L^{\frac{N}{ \alpha }}\), \(N=2,3\), 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.

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Correspondence to Jaewook Ahn.

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JA was supported by NRF, Republic of Korea-2018R1D1A1B07047465. JL was supported by SSTF-BA1701-05 (Samsung Science & Technology Foundation, Republic of Korea).

Appendix: Proof of Lemma 1

Appendix: Proof of Lemma 1

Regarding the proof of Lemma 1 in the two-dimensional case, we refer to Theorem 1.1 and Theorem 1.4–Theorem 1.5 provided by Li–Rodrigo–Zhang [29] (see also [23, Theorem 2.1]). The proofs are based on the a priori estimates and the classical contraction arguments. More precisely, for a given non-negative initial data \(u_{0}\in (L^{1}\cap H^{s})({ \mathbb{R} }^{2})\), \(s>3\), they constructed a unique and non-negative local-in-time solution of (2) in the class \(\mathcal{C}([0,T]; (L^{q} \cap H^{s})( { \mathbb{R} }^{2}))\), \(1< q<2\), satisfying (10)–(11). In our cases, \(u\in \mathcal{C}([0,T]; (L^{q} \cap H^{4})({ \mathbb{R} }^{2}))\), \(1< q<2\) satisfying (2) and (10)–(11) can be constructed as [29] and moreover, it is easy to verify that \(u\in L^{2}([0,T]; H^{4+\frac{ \alpha }{2}}({ \mathbb{R} }^{2}))\) and (9). Indeed, a uniform (in \(n\ge 1\)) bound of \(u_{n}\) in \(L^{2}([0,T]; H^{4+\frac{ \alpha }{2}}({ \mathbb{R} }^{2}))\) was obtained in the approximation scheme [29, (3.30)] using (34), but Li–Rodrigo–Zhang did not state \(u\in L^{2}([0,T]; H^{4+\frac{\alpha }{2}}({ \mathbb{R} }^{2}))\) in their theorem as they also considered (2) with no dissipation term \(\varLambda ^{\alpha }u\) (i.e. inviscid case).

Because the extension of the two-dimensional proofs in [29] to a three-dimensional case is rather straightforward, we shall summarize the main modifications in the following instead of repeating the procedures.

A Priori Estimates, Mass Conservation, Blow-up Criteria

We first note that the a priori estimates on \(\mathcal{C}([0,T]; (L ^{q} \cap H^{4})({ \mathbb{R} }^{3}))\), \(1< q<2\), the blow-up criteria (11), and the total mass conservation property (10) can be obtained as [29] without any serious difficulties. We refer interested readers to [29, Sect. 3.1] for the a priori estimates, to [29, Theorem 1.4] for the blow-up criteria (11), and to [29, Theorem 1.5] for the total mass conservation property (10). Note that the regularity of the solution, (9), can be obtained as in the two-dimensional case above.

Contraction Arguments

We note a simple modification in the contraction arguments because the other parts of the proofs in [29] can be reused in the three-dimensional case. As in [29], we consider a sequence of functions \(\{u_{n}\}_{n=1}^{\infty }\) defined by \(u_{1}(x,t)=u_{0}(x)\),

$$ \left \{ \textstyle\begin{array}{l} \partial _{t}u_{n+1}+(-\Delta )^{\frac{\alpha }{2}}u_{n+1}= -\nabla \cdot (u_{n+1}\nabla (-\Delta )^{-1}u_{n}) \\ u_{n+1}(x,0)=u_{0}(x) \end{array}\displaystyle \right . \quad \mbox{for} \,\,\, n\ge 1. $$
(34)

For a given \(u_{n+1}\in \mathcal{C}([0,T];H^{4}({ \mathbb{R} }^{3}))\), a bound of \(u_{n+1}\) in \(L^{\infty }([0,T];L^{q}({ \mathbb{R} }^{3}))\), \(1< q<2\) can be obtained via

$$\begin{aligned} & \bigl\Vert u_{n+1}(t) \bigr\Vert _{L^{q}} \\ &\quad \lesssim \bigl\Vert G_{\alpha }(t)u_{0} \bigr\Vert _{L^{q}}+ \int _{0}^{t} \bigl\Vert G_{\alpha }(t-\tau )\nabla \cdot \bigl(u_{n+1}\nabla (- \Delta )^{-1}u_{n}\bigr) (\tau ) \bigr\Vert _{L^{q}}d\tau \\ &\quad \lesssim \Vert u_{0} \Vert _{L^{q}}+ \int _{0}^{t} \bigl\Vert \nabla u _{n+1}(\tau ) \bigr\Vert _{L^{3}} \bigl\Vert \nabla (- \Delta )^{-1}u _{n} (\tau ) \bigr\Vert _{L^{\frac{3q}{3-q}}}d\tau \\ & \qquad {}+ \int _{0}^{t} \bigl\Vert u_{n+1}(\tau ) \bigr\Vert _{L^{\infty }} \bigl\Vert u_{n}(\tau ) \bigr\Vert _{L^{q}}d\tau \\ &\quad \lesssim \Vert u_{0} \Vert _{L^{q}}+ \int _{0}^{t} \bigl\Vert u _{n+1}(\tau ) \bigr\Vert _{H^{2}} \bigl\Vert u_{n}(\tau ) \bigr\Vert _{L^{q}} d\tau . \end{aligned} $$

Here, the main modification is the use of the three-dimensional Hardy–Littlewood–Sobolev inequality instead of its two-dimensional version. We refer interested readers to [29, p. 313].

Non-negativity

We provide an elementary proof for \(N=2,3\). For \(u_{-}:=-\min \{u,0\}\), let us note that the weak derivative of \(u_{-}\) is \(-\nabla u\) if \(u<0\) and vanishes otherwise. Taking the \(L^{2}\) scalar product of (2) with \(u_{-}\) yields

$$\frac{1}{2}\frac{d}{dt} \Vert u_{-} \Vert _{L^{2}}^{2}- \int _{{ \mathbb{R} }^{N}}u_{-}\varLambda ^{\alpha }u = \frac{1}{2} \int _{{ \mathbb{R} }^{N}} \nabla (-\Delta )^{-1}u\cdot \nabla (u_{-})^{2}- \int _{{ \mathbb{R} }^{N}}u^{2}u_{-}. $$

Employing a point-wise expression of the fractional Laplacian,

$$\int _{{ \mathbb{R} }^{N}}u_{-}\varLambda ^{\alpha }u =C \iint _{{ \mathbb{R} }^{N}\times { \mathbb{R} }^{N}}\frac{ (u(x)-u(y))(u _{-}(x)-u_{-}(y) ) }{ \vert x-y \vert ^{N+\alpha } }dxdy\le 0 $$

for some positive number \(C\) (see for instance [19, p. 1247]). We also note from the integration by parts that

$$\int _{{ \mathbb{R} }^{N}} \nabla (-\Delta )^{-1}u\cdot \nabla (u_{-})^{2}=- \int _{{ \mathbb{R} }^{N}} u(u_{-})^{2}. $$

Thus, combining above, we have

$$\frac{d}{dt} \Vert u_{-} \Vert _{L^{2}}^{2} \le 3 \Vert u \Vert _{L^{\infty }} \Vert u_{-} \Vert _{L^{2}}^{2}. $$

As \(u\in L^{\infty }([0,T]; L^{\infty }({ \mathbb{R} }^{N}))\) for \(T< T_{\mathrm{max}}\) by (9), the non-negativity of \(u\) can be deduced by the Grönwall lemma and \(u_{0}\ge 0\). This completes the proof.

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Ahn, J., Lee, J. Asymptotics of Chemotaxis Systems with Fractional Dissipation for Small Data in Critical Sobolev Space. Acta Appl Math 169, 199–215 (2020). https://doi.org/10.1007/s10440-019-00296-8

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Keywords

  • Asymptotics
  • Fractional dissipation
  • Kato–Ponce inequality

Mathematics Subject Classification (2010)

  • 35R11
  • 92C17