On Euler's equation and `EPDiff'
On Euler's equation and `EPDiff'
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关于欧拉方程和“EPDiff”
DOI：
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
P. Michor
中科院分区：
文献类型：
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作者：
D. Mumford;P. Michor
我们研究了一个家庭的近似欧拉方程取决于两个参数$\varepsilon，\eta \ge 0$。当$\vareps =\eta=0$时，我们有欧拉方程，当两者都是正值时，我们有一类在成像科学中称为EPDiff的积分微分方程的实例。这些方程都是在全同态群$\operatorname{Diff}_{H^\infty}（\mathbb R^n）$上的测地线方程，或者如果$\vareps = 0$，它的保体积子群上的测地线方程。它们是由向量场上的范数所诱导的右不变度量定义的：$$v\|_{\varepsilon，\eta} = \int_{\mathbb R^n} dx $$其中$L_{\varepsilon，\eta} =（I-\tfrac{\eta^2}{p} \triangle）^p \circ（I-\tfrac {\varepsilon^2} \nabla \circ \div）$。所有的测地线方程都是局部适定的，并且当$\eta>0$和$p\ge（n+3）/2$时，$L_{\vareps，\eta}$-方程有解.我们绑在一起的所有这些方程的解决方案的估计，然而，只有当地的时间。这种方法导致了一个新的概念，动量是由流动和作为一个概括的涡。我们还讨论了三角洲分布动量如何导致“涡孤子”，也称为“地标”在成像科学，和新的数值近似流体。
We study a family of approximations to Euler's equation depending on two parameters $\varepsilon,\eta \ge 0$. When $\varepsilon=\eta=0$ we have Euler's equation and when both are positive we have instances of the class of integro-differential equations called EPDiff in imaging science. These are all geodesic equations on either the full diffeomorphism group $\operatorname{Diff}_{H^\infty}(\mathbb R^n)$ or, if $\varepsilon = 0$, its volume preserving subgroup. They are defined by the right invariant metric induced by the norm on vector fields given by $$ \|v\|_{\varepsilon,\eta} = \int_{\mathbb R^n} dx $$ where $L_{\varepsilon,\eta} = (I-\tfrac{\eta^2}{p} \triangle)^p \circ (I-\tfrac1{\varepsilon^2} \nabla \circ \div)$. All geodesic equations are locally well-posed, and the $L_{\varepsilon,\eta}$-equation admits solutions for all time if $\eta>0$ and $p\ge (n+3)/2$. We tie together solutions of all these equations by estimates which, however, are only local in time. This approach leads to a new notion of momentum which is transported by the flow and serves as a generalization of vorticity. We also discuss how delta distribution momenta lead to "vortex-solitons", also called "landmarks" in imaging science, and to new numeric approximations to fluids.
DOI：
10.1073/pnas.90.24.11944
发表时间：
1993-12-15
影响因子：
11.1
作者：
MILLER, MI;CHRISTENSEN, GE;GRENANDER, U
通讯作者：
GRENANDER, U