Aggregation equation - Revision history

imported>Russell at 13:57, 14 March 2012

2012-03-14T13:57:56Z

← Older revision		Revision as of 08:57, 14 March 2012
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	always positive, respectively), or whether the interaction is isotropic, or whether it enjoys a homogeneous scaling (when say $K$ is homogeneous). Accordingly, different assumptions on kernel lead to finite time blow up or extinction. In the biological literature a common kernel is $K(x) = \|x\|$.		always positive, respectively), or whether the interaction is isotropic, or whether it enjoys a homogeneous scaling (when say $K$ is homogeneous). Accordingly, different assumptions on kernel lead to finite time blow up or extinction. In the biological literature a common kernel is $K(x) = \|x\|$.

	* Note that these equations can be seen as a Wasserstein Gradient Flow		* Note that these equations can be seen as a Wasserstein Gradient Flow (only possibly for some choices of $K$)

	* Note the similarities and differences when the energy generating these dynamics is written in a [[Nonlocal Phase Field]] model		* Note the similarities and differences when the energy generating these dynamics is written in a [[Nonlocal Phase Field]] model

	Note the close connection with the [[nonlocal porous medium equation \| nonlocal porous media equation]], the key difference being that $\Delta K$ is not locally integrable in that case.		Note the close connection with the [[nonlocal porous medium equation \| nonlocal porous media equation]], the key difference being that $\Delta K$ is not locally integrable in that case.

imported>Russell at 03:16, 14 March 2012

2012-03-14T03:16:39Z

← Older revision		Revision as of 22:16, 13 March 2012
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	where the kernel $K$ satisfies several properties but always being such that $\Delta K \in L^1_{\text{loc}}(\mathbb{R}^d)$. This equation arises in many models in biology, where $u_0$ represents the density of some population that is self-interacting through the vector field $v$, in this context, $K$ determines many properties of the interaction between different "agents" within the population. For instance the sign of $K$ determines whether there is a tendency to aggregate or segregate (depending on whether $-x \cdot \nabla K(x)$ is always negative or		where the kernel $K$ satisfies several properties but always being such that $\Delta K \in L^1_{\text{loc}}(\mathbb{R}^d)$. This equation arises in many models in biology, where $u_0$ represents the density of some population that is self-interacting through the vector field $v$, in this context, $K$ determines many properties of the interaction between different "agents" within the population. For instance the sign of $K$ determines whether there is a tendency to aggregate or segregate (depending on whether $-x \cdot \nabla K(x)$ is always negative or
	always positive, respectively), or whether the interaction is isotropic, or whether it enjoys a homogeneous scaling (when say $K$ is homogeneous). Accordingly, different assumptions on kernel lead to finite time blow up or extinction. In the biological literature a common kernel is $K(x) = \|x\|$.		always positive, respectively), or whether the interaction is isotropic, or whether it enjoys a homogeneous scaling (when say $K$ is homogeneous). Accordingly, different assumptions on kernel lead to finite time blow up or extinction. In the biological literature a common kernel is $K(x) = \|x\|$.

			* Note that these equations can be seen as a Wasserstein Gradient Flow

			* Note the similarities and differences when the energy generating these dynamics is written in a [[Nonlocal Phase Field]] model

	Note the close connection with the [[nonlocal porous medium equation \| nonlocal porous media equation]], the key difference being that $\Delta K$ is not locally integrable in that case.		Note the close connection with the [[nonlocal porous medium equation \| nonlocal porous media equation]], the key difference being that $\Delta K$ is not locally integrable in that case.

imported>Nestor at 01:16, 13 March 2012

2012-03-13T01:16:16Z

← Older revision		Revision as of 20:16, 12 March 2012
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	\end{array} \]		\end{array} \]

	where the kernel $K$ satisfies several properties but always being such that $\Delta K \in L^1_{\text{loc}}(\mathbb{R}^d)$. This equation arises in many models in biology, where $u_0$ represents the density of some population that is self-interacting through the vector field $v$, in this context, $K$ determines many properties of the interaction between different "agents" within the population. For instance the sign of $K$ determines whether there is a tendency to aggregate ~~($K\geq 0$ above)~~ or segregate ($K~~\leq 0~~$) ~~and~~ whether the interaction is isotropic. Accordingly, different assumptions on ~~$K$~~ lead to finite time blow up (~~aggregation~~) ~~as well as to finite time extinction~~.		where the kernel $K$ satisfies several properties but always being such that $\Delta K \in L^1_{\text{loc}}(\mathbb{R}^d)$. This equation arises in many models in biology, where $u_0$ represents the density of some population that is self-interacting through the vector field $v$, in this context, $K$ determines many properties of the interaction between different "agents" within the population. For instance the sign of $K$ determines whether there is a tendency to aggregate or segregate (depending on whether $-x \cdot \nabla K(x)$ is always negative or
			always positive, respectively), or whether the interaction is isotropic, or whether it enjoys a homogeneous scaling (when say $K$ is homogeneous). Accordingly, different assumptions on kernel lead to finite time blow up or extinction. In the biological literature a common kernel is $K(x) = \|x\|$.

	Note the close connection with the [[nonlocal porous medium equation \| nonlocal porous media equation]], the key difference being that $\Delta K$ is not locally integrable in that case.		Note the close connection with the [[nonlocal porous medium equation \| nonlocal porous media equation]], the key difference being that $\Delta K$ is not locally integrable in that case.

imported>Nestor: Created page with "The aggregation equation consists in the scalar equation \[\begin{array}{rll} u_t+\text{div}(uv) & = 0 & \text{ in } \mathbb{R}^d \times \mathbb{R}_+\\ u(x,0) & = u_0\\ v (x,t..."

2012-03-13T00:41:48Z

Created page with "The aggregation equation consists in the scalar equation \[\begin{array}{rll} u_t+\text{div}(uv) & = 0 & \text{ in } \mathbb{R}^d \times \mathbb{R}_+\\ u(x,0) & = u_0\\ v (x,t..."

New page

The aggregation equation consists in the scalar equation

\[\begin{array}{rll}
u_t+\text{div}(uv) & = 0 & \text{ in } \mathbb{R}^d \times \mathbb{R}_+\\
u(x,0) & = u_0\\
v (x,t) & = -\nabla (K*u(.,t))(x) &
\end{array} \]

where the kernel $K$ satisfies several properties but always being such that $\Delta K \in L^1_{\text{loc}}(\mathbb{R}^d)$. This equation arises in many models in biology, where $u_0$ represents the density of some population that is self-interacting through the vector field $v$, in this context, $K$ determines many properties of the interaction between different "agents" within the population. For instance the sign of $K$ determines whether there is a tendency to aggregate ($K\geq 0$ above) or segregate ($K\leq 0$) and whether the interaction is isotropic. Accordingly, different assumptions on $K$ lead to finite time blow up (aggregation) as well as to finite time extinction.

Note the close connection with the [[nonlocal porous medium equation | nonlocal porous media equation]], the key difference being that $\Delta K$ is not locally integrable in that case.