Mixing
Differentiability of solution to ODE
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Consider the problem
$$frac{d X(t,x)}{dt} = f(t, X(t,x))$$
$$X(0,x) = x$$
where $f:[0,T]times mathbb{R}^n to mathbb{R}^n$ and $X:[0,T]times mathbb{R}^n to mathbb{R}^n$. Assume that $f$ is Holder continuous or Lipschitz continuous. How can I prove and where can I find a reference for the fact that $X$ is differentiable and
$$nabla X$$ is the solution to
$$frac{d}{dt} nabla X(t,x) = nabla f(t,X(t,x)) nabla X(t,x)?$$
Do we need to assume that $f$ is differentiable of does the Lipschitz continuity suffice?
Also, does the formula
$$nabla X = e^{int_0^Tnabla f}$$
hold if at least $f in L^1((0,T),W^{1,1}(mathbb{R}^n))$ for example? If not, why?
real-analysis integration differential-equations pde sobolev-spaces
asked Nov 10 at 19:31
Riku
344
This question has an open bounty worth +50
reputation from Riku ending in 6 days.
Looking for an answer drawing from credible and/or official sources.
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show 4 more comments
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Consider the problem
$$frac{d X(t,x)}{dt} = f(t, X(t,x))$$
$$X(0,x) = x$$
where $f:[0,T]times mathbb{R}^n to mathbb{R}^n$ and $X:[0,T]times mathbb{R}^n to mathbb{R}^n$. Assume that $f$ is Holder continuous or Lipschitz continuous. How can I prove and where can I find a reference for the fact that $X$ is differentiable and
$$nabla X$$ is the solution to
$$frac{d}{dt} nabla X(t,x) = nabla f(t,X(t,x)) nabla X(t,x)?$$
Do we need to assume that $f$ is differentiable of does the Lipschitz continuity suffice?
Also, does the formula
$$nabla X = e^{int_0^Tnabla f}$$
hold if at least $f in L^1((0,T),W^{1,1}(mathbb{R}^n))$ for example? If not, why?
real-analysis integration differential-equations pde sobolev-spaces
asked Nov 10 at 19:31
Riku
344
This question has an open bounty worth +50
reputation from Riku ending in 6 days.
Looking for an answer drawing from credible and/or official sources.
Your last statement is in general false, though it is a widely-held misconception, see my answer on MathOverflow.
– user539887
Nov 11 at 13:38
@user539887 Thanks. Then in this case what is the explicit form of the solution to the second ODE in the question? Shouldn't the formula hold if $f in L^1((0,T),W^{1,1}(mathbb{R}^n))$ for example?
– Riku
Nov 11 at 20:50
You need commutativity of all the matrices for the exponential formula. If that is not given, then there is something called a time-ordered exponential. || To write down the derivative of $f$ as you did, you need some kind of differentiability of $f$ in the assumptions.
– LutzL
Nov 11 at 21:37
@LutzL What do you mean by commutativity of the matrices? Which matrices?
– Riku
Nov 12 at 10:14
1
Commutativity is an assumption by itself: $A(t)A(s)=A(s)A(t)$ for all $s,tin(0,T)$. Indeed, it can be relaxed: there is $t_0in(0,T)$ such that $A(s)(intlimits_{t_0}^t A(tau),dtau)=(intlimits_{t_0}^t A(tau),dtau)A(s)$ for all $s,tin(0,T)$. Perhaps you ask what are assumptions on $f$ that guarantee the commutativity of the Jacobian matrices? To such a question I do not know an answer, I am afraid.
– user539887
Nov 12 at 19:36
|
show 4 more comments
up vote
3
down vote
favorite
up vote
3
down vote
favorite
Consider the problem
$$frac{d X(t,x)}{dt} = f(t, X(t,x))$$
$$X(0,x) = x$$
where $f:[0,T]times mathbb{R}^n to mathbb{R}^n$ and $X:[0,T]times mathbb{R}^n to mathbb{R}^n$. Assume that $f$ is Holder continuous or Lipschitz continuous. How can I prove and where can I find a reference for the fact that $X$ is differentiable and
$$nabla X$$ is the solution to
$$frac{d}{dt} nabla X(t,x) = nabla f(t,X(t,x)) nabla X(t,x)?$$
Do we need to assume that $f$ is differentiable of does the Lipschitz continuity suffice?
Also, does the formula
$$nabla X = e^{int_0^Tnabla f}$$
hold if at least $f in L^1((0,T),W^{1,1}(mathbb{R}^n))$ for example? If not, why?
real-analysis integration differential-equations pde sobolev-spaces
asked Nov 10 at 19:31
Riku
344
Consider the problem
$$frac{d X(t,x)}{dt} = f(t, X(t,x))$$
$$X(0,x) = x$$
where $f:[0,T]times mathbb{R}^n to mathbb{R}^n$ and $X:[0,T]times mathbb{R}^n to mathbb{R}^n$. Assume that $f$ is Holder continuous or Lipschitz continuous. How can I prove and where can I find a reference for the fact that $X$ is differentiable and
$$nabla X$$ is the solution to
$$frac{d}{dt} nabla X(t,x) = nabla f(t,X(t,x)) nabla X(t,x)?$$
Do we need to assume that $f$ is differentiable of does the Lipschitz continuity suffice?
Also, does the formula
$$nabla X = e^{int_0^Tnabla f}$$
hold if at least $f in L^1((0,T),W^{1,1}(mathbb{R}^n))$ for example? If not, why?
real-analysis integration differential-equations pde sobolev-spaces
real-analysis integration differential-equations pde sobolev-spaces
asked Nov 10 at 19:31
Riku
344
asked Nov 10 at 19:31
Riku
344
edited Nov 11 at 20:53
asked Nov 10 at 19:31
Riku
344
asked Nov 10 at 19:31
Riku
344
asked Nov 10 at 19:31
Riku
344
344
This question has an open bounty worth +50
reputation from Riku ending in 6 days.
Looking for an answer drawing from credible and/or official sources.
This question has an open bounty worth +50
reputation from Riku ending in 6 days.
Looking for an answer drawing from credible and/or official sources.
Your last statement is in general false, though it is a widely-held misconception, see my answer on MathOverflow.
– user539887
Nov 11 at 13:38
@user539887 Thanks. Then in this case what is the explicit form of the solution to the second ODE in the question? Shouldn't the formula hold if $f in L^1((0,T),W^{1,1}(mathbb{R}^n))$ for example?
– Riku
Nov 11 at 20:50
You need commutativity of all the matrices for the exponential formula. If that is not given, then there is something called a time-ordered exponential. || To write down the derivative of $f$ as you did, you need some kind of differentiability of $f$ in the assumptions.
– LutzL
Nov 11 at 21:37
@LutzL What do you mean by commutativity of the matrices? Which matrices?
– Riku
Nov 12 at 10:14
1
Commutativity is an assumption by itself: $A(t)A(s)=A(s)A(t)$ for all $s,tin(0,T)$. Indeed, it can be relaxed: there is $t_0in(0,T)$ such that $A(s)(intlimits_{t_0}^t A(tau),dtau)=(intlimits_{t_0}^t A(tau),dtau)A(s)$ for all $s,tin(0,T)$. Perhaps you ask what are assumptions on $f$ that guarantee the commutativity of the Jacobian matrices? To such a question I do not know an answer, I am afraid.
– user539887
Nov 12 at 19:36
|
show 4 more comments
Your last statement is in general false, though it is a widely-held misconception, see my answer on MathOverflow.
– user539887
Nov 11 at 13:38
@user539887 Thanks. Then in this case what is the explicit form of the solution to the second ODE in the question? Shouldn't the formula hold if $f in L^1((0,T),W^{1,1}(mathbb{R}^n))$ for example?
– Riku
Nov 11 at 20:50
You need commutativity of all the matrices for the exponential formula. If that is not given, then there is something called a time-ordered exponential. || To write down the derivative of $f$ as you did, you need some kind of differentiability of $f$ in the assumptions.
– LutzL
Nov 11 at 21:37
@LutzL What do you mean by commutativity of the matrices? Which matrices?
– Riku
Nov 12 at 10:14
1
Commutativity is an assumption by itself: $A(t)A(s)=A(s)A(t)$ for all $s,tin(0,T)$. Indeed, it can be relaxed: there is $t_0in(0,T)$ such that $A(s)(intlimits_{t_0}^t A(tau),dtau)=(intlimits_{t_0}^t A(tau),dtau)A(s)$ for all $s,tin(0,T)$. Perhaps you ask what are assumptions on $f$ that guarantee the commutativity of the Jacobian matrices? To such a question I do not know an answer, I am afraid.
– user539887
Nov 12 at 19:36
Your last statement is in general false, though it is a widely-held misconception, see my answer on MathOverflow.
– user539887
Nov 11 at 13:38
Your last statement is in general false, though it is a widely-held misconception, see my answer on MathOverflow.
– user539887
Nov 11 at 13:38
@user539887 Thanks. Then in this case what is the explicit form of the solution to the second ODE in the question? Shouldn't the formula hold if $f in L^1((0,T),W^{1,1}(mathbb{R}^n))$ for example?
– Riku
Nov 11 at 20:50
@user539887 Thanks. Then in this case what is the explicit form of the solution to the second ODE in the question? Shouldn't the formula hold if $f in L^1((0,T),W^{1,1}(mathbb{R}^n))$ for example?
– Riku
Nov 11 at 20:50
You need commutativity of all the matrices for the exponential formula. If that is not given, then there is something called a time-ordered exponential. || To write down the derivative of $f$ as you did, you need some kind of differentiability of $f$ in the assumptions.
– LutzL
Nov 11 at 21:37
You need commutativity of all the matrices for the exponential formula. If that is not given, then there is something called a time-ordered exponential. || To write down the derivative of $f$ as you did, you need some kind of differentiability of $f$ in the assumptions.
– LutzL
Nov 11 at 21:37
@LutzL What do you mean by commutativity of the matrices? Which matrices?
– Riku
Nov 12 at 10:14
@LutzL What do you mean by commutativity of the matrices? Which matrices?
– Riku
Nov 12 at 10:14
1
1
Commutativity is an assumption by itself: $A(t)A(s)=A(s)A(t)$ for all $s,tin(0,T)$. Indeed, it can be relaxed: there is $t_0in(0,T)$ such that $A(s)(intlimits_{t_0}^t A(tau),dtau)=(intlimits_{t_0}^t A(tau),dtau)A(s)$ for all $s,tin(0,T)$. Perhaps you ask what are assumptions on $f$ that guarantee the commutativity of the Jacobian matrices? To such a question I do not know an answer, I am afraid.
– user539887
Nov 12 at 19:36
Commutativity is an assumption by itself: $A(t)A(s)=A(s)A(t)$ for all $s,tin(0,T)$. Indeed, it can be relaxed: there is $t_0in(0,T)$ such that $A(s)(intlimits_{t_0}^t A(tau),dtau)=(intlimits_{t_0}^t A(tau),dtau)A(s)$ for all $s,tin(0,T)$. Perhaps you ask what are assumptions on $f$ that guarantee the commutativity of the Jacobian matrices? To such a question I do not know an answer, I am afraid.
– user539887
Nov 12 at 19:36
|
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Your last statement is in general false, though it is a widely-held misconception, see my answer on MathOverflow.
– user539887
Nov 11 at 13:38
@user539887 Thanks. Then in this case what is the explicit form of the solution to the second ODE in the question? Shouldn't the formula hold if $f in L^1((0,T),W^{1,1}(mathbb{R}^n))$ for example?
– Riku
Nov 11 at 20:50
You need commutativity of all the matrices for the exponential formula. If that is not given, then there is something called a time-ordered exponential. || To write down the derivative of $f$ as you did, you need some kind of differentiability of $f$ in the assumptions.
– LutzL
Nov 11 at 21:37
@LutzL What do you mean by commutativity of the matrices? Which matrices?
– Riku
Nov 12 at 10:14
1
Commutativity is an assumption by itself: $A(t)A(s)=A(s)A(t)$ for all $s,tin(0,T)$. Indeed, it can be relaxed: there is $t_0in(0,T)$ such that $A(s)(intlimits_{t_0}^t A(tau),dtau)=(intlimits_{t_0}^t A(tau),dtau)A(s)$ for all $s,tin(0,T)$. Perhaps you ask what are assumptions on $f$ that guarantee the commutativity of the Jacobian matrices? To such a question I do not know an answer, I am afraid.
– user539887
Nov 12 at 19:36