Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces
Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces
复制标题
Wasserstein 空间中梯度流的变系数非线性扩散方程
DOI：
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
S. Lisini
中科院分区：
文献类型：
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作者：
S. Lisini
We study existence and approximation of non-negative solutions of partial differential equations of the type $$\partial_t u - \div (A(\nabla (f(u))+u\nabla V )) = 0 \qquad \mbox{in } (0,+\infty )\times \mathbb{R}^n,\qquad\qquad (0.1)$$
where A is a symmetric matrix-valued function of the spatial variable satisfying a uniform ellipticity condition,
$f:[0,+\infty) \rightarrow[0,+\infty)$
is a suitable non decreasing function, $V:\mathbb{R}^n \rightarrow\mathbb{R}$
is a convex function.
Introducing the energy functional $\phi(u)=\int_{\mathbb{R}^n} F(u(x))\,{\rm d}x+\int_{\mathbb{R}^n}V(x)u(x)\,{\rm d}x$
,
where F is a convex function linked to f by $f(u) = uF'(u)-F(u)$
,
we show that u is the “gradient flow” of ϕ with respect to the
2-Wasserstein distance between probability measures on
the space $\mathbb{R}^n$
, endowed with the Riemannian distance induced by $A^{-1}.$
In the case of uniform convexity of V , long time asymptotic behaviour and decay rate to the stationary state
for solutions of equation (0.1) are studied.
A contraction property in Wasserstein distance for solutions of equation (0.1)
is also studied in a particular case.
We study existence and approximation of non-negative solutions of partial differential equations of the type $$\partial_t u - \div (A(\nabla (f(u))+u\nabla V )) = 0 \qquad \mbox{in } (0,+\infty )\times \mathbb{R}^n,\qquad\qquad (0.1)$$
where A is a symmetric matrix-valued function of the spatial variable satisfying a uniform ellipticity condition,
$f:[0,+\infty) \rightarrow[0,+\infty)$
is a suitable non decreasing function, $V:\mathbb{R}^n \rightarrow\mathbb{R}$
is a convex function.
Introducing the energy functional $\phi(u)=\int_{\mathbb{R}^n} F(u(x))\,{\rm d}x+\int_{\mathbb{R}^n}V(x)u(x)\,{\rm d}x$
,
where F is a convex function linked to f by $f(u) = uF'(u)-F(u)$
,
we show that u is the “gradient flow” of ϕ with respect to the
2-Wasserstein distance between probability measures on
the space $\mathbb{R}^n$
, endowed with the Riemannian distance induced by $A^{-1}.$
In the case of uniform convexity of V , long time asymptotic behaviour and decay rate to the stationary state
for solutions of equation (0.1) are studied.
A contraction property in Wasserstein distance for solutions of equation (0.1)
is also studied in a particular case.