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| The aggregation equation consists in the scalar equation | | The Monge-Ampere equation refers to |
| | \[ \det D^2 u = f(x). \] |
| | The right hand side $f$ is always nonnegative. The unknown function $u$ should be convex. |
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| \[\begin{array}{rll}
| | The equation is uniformly elliptic when restricted to a subset of functions which are uniformly $C^{1,1}$ and strictly convex. Moreover, the operator |
| u_t+\text{div}(uv) & = 0 & \text{ in } \mathbb{R}^d \times \mathbb{R}_+\\
| | \[ MA(D^2 u) := \left( \det (D^2 u) \right)^{1/n} = \inf \{ a_{ij} \partial_{ij} u : \det \{a_{ij}\} = 1 \},\] |
| u(x,0) & = u_0\\
| | is concave. |
| v (x,t) & = -\nabla (K*u(.,t))(x) &
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| \end{array} \]
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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
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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 $K(x) = |x|$.
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| * Note that these equations can be seen as a Wasserstein Gradient Flow (only possibly for some choices of $K$)
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| * Note the similarities and differences when the energy generating these dynamics is written in a [[Nonlocal Phase Field]] model
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| 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.
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Latest revision as of 16:47, 8 April 2015
The Monge-Ampere equation refers to
\[ \det D^2 u = f(x). \]
The right hand side $f$ is always nonnegative. The unknown function $u$ should be convex.
The equation is uniformly elliptic when restricted to a subset of functions which are uniformly $C^{1,1}$ and strictly convex. Moreover, the operator
\[ MA(D^2 u) := \left( \det (D^2 u) \right)^{1/n} = \inf \{ a_{ij} \partial_{ij} u : \det \{a_{ij}\} = 1 \},\]
is concave.
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