Difference between revisions of "Flow equation for a differential operator"
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+ | ''This article defines the notion of flow for a scalar differential equation, viz a differential equation with one real variable. For the more general notion, refer [[flow for a vector differential equation]]'' | ||
+ | |||
==Definition== | ==Definition== | ||
− | === | + | Note that a flow equation can be associated to any differential operator on a manifold (and in particular, to one on <math>\R^n</math>). There is a kind of correspondence between the differential operator and the associated flow equation. Thus, any property over flow equations gives a property over differential operators, and vice versa. |
− | Suppose | + | |
+ | ===Definition for Euclidean space=== | ||
+ | |||
+ | Suppose <math>F</math> is a [[differential operator]] on <math>\R^n</math> that is, <math>F</math> takes as input a (sufficiently many times differentiable) function from <math>\R^n</math> to <math>\R</math> and outputs another function from <math>\R^n</math> to <math>\R</math>. | ||
− | + | The flow equation for <math>F</math> is described as follows: | |
<math>\frac{\partial u}{\partial t} = F(u)</math> | <math>\frac{\partial u}{\partial t} = F(u)</math> | ||
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* <math>F(u)</math> on the right side actually means <math>F(y)</math> where <math>y</math> is the map sending <math>x</math> to <math>u(x,t)</math> (that is, <math>y</math> is the map <math>u</math> at a fixed time) | * <math>F(u)</math> on the right side actually means <math>F(y)</math> where <math>y</math> is the map sending <math>x</math> to <math>u(x,t)</math> (that is, <math>y</math> is the map <math>u</math> at a fixed time) | ||
− | + | The solution curves for this (in the space of all possible functions <math>y</math> define the flow of the differential equation) are termed '''trajectories'''. By the property of invariance of the differential equation under time-translation, we conclude that the various trajectories through a point are just time-translaets of each other, so there is a unique trajectory curve through each point. | |
===Definition for a differential manifold=== | ===Definition for a differential manifold=== | ||
− | + | Suppose <math>F</math> is a [[differential operator]] on a [[differential manifold]] <math>M</math> that is, <math>F</math> takes as input a (sufficiently many times differentiable) function from <math>M</math> to <math>\R</math> and outputs another function from <math>M</math> to <math>\R</math>. | |
The flow equation becomes: | The flow equation becomes: | ||
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* <math>u: \R \times M \to \R</math> is the dependent variable, written as <math>u(t,x)</math> | * <math>u: \R \times M \to \R</math> is the dependent variable, written as <math>u(t,x)</math> | ||
* <math>F(u)</math> on the right side actually means <math>F(y)</math> where <math>y</math> is the map sending <math>x</math> to <math>u(t,x)</math> (that is, <math>y</math> is the map <math>u</math> at a fixed time) | * <math>F(u)</math> on the right side actually means <math>F(y)</math> where <math>y</math> is the map sending <math>x</math> to <math>u(t,x)</math> (that is, <math>y</math> is the map <math>u</math> at a fixed time) | ||
+ | |||
+ | The solution curves for this (in the space of all possible functions <math>y</math> define the flow of the differential equation) are termed '''trajectories'''. By the property of invariance of the differential equation under time-translation, we conclude that the various trajectories through a point are just time-translaets of each other, so there is a unique trajectory curve through each point. | ||
+ | |||
+ | ==Properties of the flow equation== | ||
+ | |||
+ | ===Invariance under time-translation=== | ||
+ | |||
+ | The flow equation is autonomous with respect to the time parameter -- the only place where the time parameter occurs is where we differentiate <math>u</math> with respect to time. | ||
+ | |||
+ | ===Dependence on linear change in the differential operator=== | ||
+ | |||
+ | Suppose <math>F</math> is a differential operator, and <math>k \in \R</math>. Then if <math>u</math> is a solution for <math>kF</math>, then the map <math>(t,x) \mapsto u(kt,x)</math> is a solution for <math>F</math>. | ||
==Fixed points and trajectories== | ==Fixed points and trajectories== | ||
+ | |||
+ | ===Notion of trajectory and local solution=== | ||
+ | |||
+ | In the case of the flow equation, we are thinking of the thing flowing as a function <math>y:\R^n \to \R</math>. Thus, a local solution means a specification of <math>y</math> at all points in some neighbourhood. This means a specification of <math>u</math> for ''all'' <math>x \in M</math> and for all <math>t</math> lying inside that time interval. | ||
===Fixed points are solutions of the original equation=== | ===Fixed points are solutions of the original equation=== | ||
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* There exists a limiting point at <math>\infty</math>, which is hence a solution to the original differential equation | * There exists a limiting point at <math>\infty</math>, which is hence a solution to the original differential equation | ||
* {{fillin}} | * {{fillin}} | ||
+ | |||
+ | ==Groups of diffeomorphisms== | ||
+ | ===In case of short-time existence=== | ||
+ | |||
+ | If we have short-time existence at every point, then we get a local one-parameter group of diffeomorphisms acting on the manifold. Namely, we have a local action of <matH>\R</math> on <math>M</math>, where <math>t \in \R</math> takes <math>x \in M</math> to <math>u(t,x)</math> where <math>u</math> is a solution to the flow equation. | ||
+ | |||
+ | The fact that it is a local one-parameter group of diffeomorphisms follows from the fact that the flow equation in invariant under time translation. | ||
+ | |||
+ | ===In case of global existence=== | ||
+ | |||
+ | If we have global existence at every point, then we get a global one-parameter group of diffeomorphisms viz a global action of <math>\R</math> on <math>M</math> via <math>t.x = u(t,x)</math>. This is a group action because of the invariance under time-translation. | ||
+ | |||
+ | ==Supersolutions and subsolutions== | ||
+ | |||
+ | ===Supersolution to the flow equation=== | ||
+ | |||
+ | A function <math>u: \R \times M \to \R</math> is termed a supersolution to the flow equation at a point <math>(t,x)</math> if the following holds at that point: | ||
+ | |||
+ | <math>\frac{\partial u}{\partial t} \ge F(u)</math> | ||
+ | |||
+ | A function is termed a '''supersolution''' if it is a supersolution over the entire time range on which it is defined. | ||
+ | |||
+ | ===Subsolution to the flow equation=== | ||
+ | |||
+ | A function <math>u: \R \times M \to \R</math> is termed a subsolution to the flow equation at a point <math>(t,x)</math> if the following holds at that point: | ||
+ | |||
+ | <math>\frac{\partial u}{\partial t} \le F(u)</math> | ||
+ | |||
+ | A function is termed a '''subsolution''' if it is a subsolution over the entire time range on which it is defined. |
Latest revision as of 19:40, 18 May 2008
This article defines the notion of flow for a scalar differential equation, viz a differential equation with one real variable. For the more general notion, refer flow for a vector differential equation
Contents
Definition
Note that a flow equation can be associated to any differential operator on a manifold (and in particular, to one on ). There is a kind of correspondence between the differential operator and the associated flow equation. Thus, any property over flow equations gives a property over differential operators, and vice versa.
Definition for Euclidean space
Suppose is a differential operator on that is, takes as input a (sufficiently many times differentiable) function from to and outputs another function from to .
The flow equation for is described as follows:
Here:
- , called the time parameter, and , called the space parameter, are the two independent variables
- is the dependent variable, written as
- on the right side actually means where is the map sending to (that is, is the map at a fixed time)
The solution curves for this (in the space of all possible functions define the flow of the differential equation) are termed trajectories. By the property of invariance of the differential equation under time-translation, we conclude that the various trajectories through a point are just time-translaets of each other, so there is a unique trajectory curve through each point.
Definition for a differential manifold
Suppose is a differential operator on a differential manifold that is, takes as input a (sufficiently many times differentiable) function from to and outputs another function from to .
The flow equation becomes:
Here:
- , called the time parameter, and , called the space parameter, are the two independent variables
- is the dependent variable, written as
- on the right side actually means where is the map sending to (that is, is the map at a fixed time)
The solution curves for this (in the space of all possible functions define the flow of the differential equation) are termed trajectories. By the property of invariance of the differential equation under time-translation, we conclude that the various trajectories through a point are just time-translaets of each other, so there is a unique trajectory curve through each point.
Properties of the flow equation
Invariance under time-translation
The flow equation is autonomous with respect to the time parameter -- the only place where the time parameter occurs is where we differentiate with respect to time.
Dependence on linear change in the differential operator
Suppose is a differential operator, and . Then if is a solution for , then the map is a solution for .
Fixed points and trajectories
Notion of trajectory and local solution
In the case of the flow equation, we are thinking of the thing flowing as a function . Thus, a local solution means a specification of at all points in some neighbourhood. This means a specification of for all and for all lying inside that time interval.
Fixed points are solutions of the original equation
A point is a fixed point of the flow if and only if it solves the original equation . This tells us the following thing:
If the limits of the flow are well-defined at or ,, then the limiting points are fixed points of the flow, hence they are solutions for
Trajectories of the flow
To understand the flow, we can look at the trajectory at each point. Note that by the way the flow is defined, the trajectory is invariant under time-translation of the original differential equation.
The trajectories at a point could be of the following types:
- They could be defined over . This phenomenon is called short-time existence of the solution.
- They could be defined over . This phenomenon is called global existence of the solution.
Further, those that satisfy global existence, could satisfy any of the following:
- There exists a limiting point at , which is hence a solution to the original differential equation
- Fill this in later
Groups of diffeomorphisms
In case of short-time existence
If we have short-time existence at every point, then we get a local one-parameter group of diffeomorphisms acting on the manifold. Namely, we have a local action of on , where takes to where is a solution to the flow equation.
The fact that it is a local one-parameter group of diffeomorphisms follows from the fact that the flow equation in invariant under time translation.
In case of global existence
If we have global existence at every point, then we get a global one-parameter group of diffeomorphisms viz a global action of on via . This is a group action because of the invariance under time-translation.
Supersolutions and subsolutions
Supersolution to the flow equation
A function is termed a supersolution to the flow equation at a point if the following holds at that point:
A function is termed a supersolution if it is a supersolution over the entire time range on which it is defined.
Subsolution to the flow equation
A function is termed a subsolution to the flow equation at a point if the following holds at that point:
A function is termed a subsolution if it is a subsolution over the entire time range on which it is defined.