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Nonlinear aggregation-diffusion equations: radial symmetry and long time asymptotics. (English) Zbl 1427.35136

The authors consider the nonlinear aggregation-diffusion equation \(\partial _{t}\rho =\Delta \rho ^{m}+\nabla \cdot (\rho \nabla (W\ast \rho ))\) posed in \(\mathbb{R}^{d}\times \lbrack 0,\infty )\), which describes the evolution of the mass density of particles or individuals. Here \(m>0\) and the potential \(W\in C^{1}(\mathbb{R}^{d}\setminus \{0\})\) is radially symmetric: \(W(x)=\omega (\left\vert x\right\vert )=\omega (r)\), with \(\omega ^{\prime }(r)>0\) for \(r>0\), \(\omega (1)=0\), \(\omega ^{\prime }(r)\leq C_{w}r^{d-1}\) for \(r\leq 1\), \(\omega ^{\prime }(r)\leq C_{w}\) for \(r>1\) and either \(\omega (r)\) is bounded for \(r\geq 1\) or there exists \(C_{w}>0\) such that for all \( a,b\geq 0\) \(\omega _{+}(a+b)\leq C_{w}(1+\omega (1+a)+\omega (1+b))\). The initial data \(\rho _{0}\in L_{+}^{1}(\mathbb{R}^{d})\cap L^{m}(\mathbb{R} ^{d})\) is imposed. The authors define the notion of stationary state for this problem as a function \(\rho _{s}\in L_{+}^{1}(\mathbb{R}^{d})\cap L^{\infty }(\mathbb{R}^{d})\) such that \(\rho _{s}^{m}\in H_{\mathrm{loc}}^{1}(\mathbb{ R}^{d})\), \(\nabla W\ast \rho _{s}\in L_{\mathrm{loc}}^{1}(\mathbb{R}^{d})\) and \( \nabla \rho _{s}^{m}=-\rho _{s}\nabla (\nabla W\ast \rho _{s})\) in \(\mathbb{R }^{d}\) in the sense of distributions in \(\mathbb{R}^{d}\). The first main result proves that every nonnegative stationary state \(\rho _{s}\) which further satisfies \(\omega (1+\left\vert x\right\vert )\rho _{s}\in L^{1}( \mathbb{R}^{d})\) must be radially decreasing up to a translation, i.e. there exists some \(x_{0}\in \mathbb{R}^{d}\) such that \(\rho _{s}(\cdot -x_{0})\) is radially symmetric and \(\rho _{s}(\left\vert x-x_{0}\right\vert )\) is non-increasing in \(\left\vert x-x_{0}\right\vert \). For the proof, the authors introduce the free energy functional \(\mathcal{E}[\rho ]=\frac{1}{m-1 }\int_{\mathbb{R}^{d}}\rho ^{m}dx+\frac{1}{2}\int_{\mathbb{R}^{d}}\rho (W\ast \rho )dx\) and they proceed by contradiction mainly using the continuous Steiner symmetrization. They then consider the case where the potential \(W\) is such that the above equation has a local-in-time unique gradient flow solution. In a similar way, they prove that if \(\rho _{s}\in L^{\infty }(\mathbb{R}^{d})\cap \mathcal{P}^{2}(\mathbb{R}^{d})\) is a stationary solution of the above equation with \(\mathcal{E}[\rho _{s}]\) finite, then \(\rho _{s}\) must be radially decreasing after a translation. Here \(\mathcal{P}^{2}(\mathbb{R}^{d})\) is the space of nonnegative probability measures with finite second-moment. Considering the equation \( \partial _{t}\rho =\Delta \rho ^{m}+\nabla \cdot (\rho \nabla (W\ast \rho +V))\) posed in \(\mathbb{R}^{d}\times \lbrack 0,\infty \lbrack \) where the potential \(V\in C^{1}(\mathbb{R}^{d})\) is radially symmetric and satisfies \( 0<V^{\prime }(r)\leq C\) for all \(r>0\), or \(V^{\prime }(r)>0\) for all \(r>0\) and \(V^{\prime }(r)\rightarrow \infty \) as \(r\rightarrow \infty \) in the case where \(m=1\), the authors prove that if \(\rho _{s}\in L_{+}^{1}(\mathbb{R }^{d})\cap L^{\infty }(\mathbb{R}^{d})\) is a nonnegative stationary state (in the above sense but with \(\nabla \rho _{s}^{m}=-\rho _{s}\nabla (\nabla W\ast \rho _{s}+V)\)) which satisfies \(\omega (1+\left\vert x\right\vert )\rho _{s}\in L^{1}(\mathbb{R}^{d})\) and \(\rho _{s}V\in L^{1}(\mathbb{R}^{d}) \), then \(\rho _{s}\) is radially decreasing about the origin. The authors then prove the existence of a radially decreasing global minimizer \(\rho \) of the functional \(\mathcal{E}[\rho ]\) over the class of admissible densities \(\mathcal{Y}_{M}=\{\rho \in L_{+}^{1}(\mathbb{R}^{d})\cap L^{m}( \mathbb{R}^{d}):\left\Vert \rho \right\Vert _{L^{1}(\mathbb{R}^{d})}=M\), \( \int_{\mathbb{R}^{d}}x\rho (x)dx=0\), \(\omega (1+\left\vert x\right\vert )\rho (x)\in L^{1}(\mathbb{R}^{d})\}\) assuming further hypotheses on \(W\) and \(\omega \). They prove that all global minimizers are radially symmetric and decreasing. They prove a similar result when a potential \(V\) is added. In the last part of their paper, the authors analyze the long-time behavior of the solutions of the 2D Keller Segel model with nonlinear diffusion written as\(\partial _{t}\rho =\Delta \rho ^{m}-\nabla \cdot (\rho \nabla \mathcal{N }\ast \rho )\) in the case where \(m>1\) and with \(N=-\frac{1}{2\pi }\log \left\vert x\right\vert \). They first prove that for any nonnegative initial data \(\rho _{0}\in L_{\log }^{1}(\mathbb{R}^{2})\cap L^{\infty }( \mathbb{R}^{2})\), there exists a unique global weak solution to this equation which satisfies the energy inequality \(\frac{1}{m-1}\int_{\mathbb{R} ^{2}}\rho ^{m}dx+\frac{1}{4\pi }\int_{\mathbb{R}^{2}}\int_{\mathbb{R} ^{2}}\log \left\vert x-y\right\vert \rho (x)\rho (y)dxdy+\int_{\mathbb{R} ^{2}}\rho \left\vert \nabla h[\rho ]\right\vert ^{2}dx\leq \mathcal{E}[\rho _{0}]\), where \(h[\rho ]=\frac{m}{m-1}\rho ^{m}-\mathcal{N}\ast \rho \). The energy \(\mathcal{E}[\rho ]\) is further bounded from above and below \( \mathcal{E}^{\ast }\leq \mathcal{E}[\rho ](t)\leq \mathcal{E}[\rho _{0}]\) for some (negative) constant \(\mathcal{E}^{\ast }\). The authors prove further properties of \(\rho \) and especially in the case where \(\rho _{0}\in L_{\log }^{1}(\mathbb{R}^{2})\cap L^{\infty }(\mathbb{R}^{2})\) is nonnegative and radially symmetric. They here use a regularizing argument. They then prove the existence of a unique stationary state \(\rho _{M}\) of the Keller Segel model with mass \(M\) and zero center of mass satisfying \( \rho _{M}\in L_{\log }^{1}(\mathbb{R}^{2})\). Moreover, \(\rho _{M}\) is compactly supported, bounded, radially symmetric and non-increasing. This unique stationary state is the unique global minimizer of the free energy functional \(\mathcal{E}[\rho ]\) with mass \(M\). Finally, the authors prove that if \(\rho \) is the weak solution to the Keller Segel model with nonnegative initial data \(\rho _{0}\in L^{1}((1+\left\vert x\right\vert ^{2})dx)\cap L^{\infty }(\mathbb{R}^{2})\), then \(\rho (\cdot ,t)\) converges in \(L^{q}(\mathbb{R}^{2})\) for all \(q\geq 1\) as \(t\rightarrow \infty \) to the unique stationary state with the same mass and center of mass as the initial data, i.e., to \(\rho _{M}^{c}=\rho _{M}(x-x_{c})\) where \(x_{c}=\frac{ 1}{M}\int_{\mathbb{R}^{2}}x\rho _{0}(x)dx\) and \(M=\left\Vert \rho _{0}\right\Vert _{L^{1}(\mathbb{R}^{d})}\).

MSC:

35K59 Quasilinear parabolic equations
35B07 Axially symmetric solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs
49J10 Existence theories for free problems in two or more independent variables
49Q20 Variational problems in a geometric measure-theoretic setting
82C22 Interacting particle systems in time-dependent statistical mechanics
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References:

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