Maybe one could try to have a look at the paper Sascha quoted there. I might do it on the weekend.
René Hosfeld, 2020/05/04 14:20, 2020/05/04 14:29
Dear all,
as announced in my mail, I there are some points in habil.pdf which are not totally clear to me. So maybe, you can help me out or we can discuss them.
1) In the proof of Lemma 2.1.2 there is written, that (∂0,ρM(∂0,ρ)+A)∂0,ρ⊆∂0,ρ(∂0,ρM(∂0,ρ)+A) holds. This is not obvious to me.
2) In the proof of Theorem 2.1.3 (almost at the end): What is the point of choosing the component C containing ρ+1 and why does Ω open implies z∈Ω?
3) In the example of the damped wave equation there is C:=ι∗ran(grad0)grad0 and H1=L2(Ω)n. I think it should be H1=ran(grad0) in order to say that C is invertible. In the lectures, this is exercise 11.3.
4) In the example of the dual phase lag heat equation there are some operators commuting and I am wondering whether they are really commuting or whether these are only formal computaions. To be precise, the operators grad and (1+τv∂0,ρ) as well as (1+τq∂0,ρ+1/2τ2q∂20,ρ)−1(1+τv∂0,ρ) and grad or div.
5) In section 2.2. we assume M0,M1 to be bounded. So if we assume the domain of M be some right half-plane with positive distance to 0 and to be contained in the domains of M0,M1 we have M bounded. So, does it make sense to talk about unbounded material laws? Of course it is interessting and it is more general, but is it really needed for our purpose?
Nontheless, I think we should treat some aspects of section 2.1 in our presentation, as the main results from 2.2 are based on them.
Kind regards,
René
Jürgen Voigt, 2020/05/17 22:59
Dear Rene,
I was very busy with something non-mathematical during the last two weeks, but now I will try to answer your comments - one by one at a time.
1) In the proof of Lemma 2.1.2 there is written, that (∂0,ρM(∂0,ρ)+A)∂0,ρ⊆∂0,ρ(∂0,ρM(∂0,ρ)+A) holds. This is not obvious to me.
Maybe you/we should not look at this but rather realise that Lemma 2.1.2 corresponds to Prop. 11.1.2 of the Lecture Notes, and your question corresponds to line 3 of the proof of Prop. 11.1.2. There has been a comment - by myself - on this proof in the forum. The discussion following my post should clear up the matter, I hope.
But let me ask another question, probably stupid: I had commented in the proposal of the project that the operators ∂t,νM(∂t,ν) of the lectures would be of the form ∂0M(∂−10) in [2]. What is the reason that this is not the case in the present instance?
(to be continued soon!)
Best, Jürgen
Jürgen Voigt, 2020/05/18 15:44
So sorry!! I had confused [2] and [3]! It is in [2] that material laws are defined on sets different from the ones used in the Lecture Notes. The difference between the material laws in [3] and the Lecture Notes is that in [3] it is not supposed that M is bounded on a half-plane. I suggest that we could/should stick to the case of material laws treated in the ISem and avoid the case of unbounded material laws.
René Hosfeld, 2020/05/20 11:59
Dear Jürgen,
at first I was also confused by this (for us) unsual definition. But, even it is clear now, we should avoid it. Also, I think we should concentrate on [3]. [2] does not have much more informations for us.
Best wishes,
René
Jürgen Voigt, 2020/05/20 12:31
Dear René,
yes; I agree.
René Hosfeld, 2020/05/22 12:29
Dear Jürgen,
concerning your answer to my first question. I am still puzzled how the fact that the solution operator Sν is of the form S(∂0,ν) for some material law S will help. If I use this, I end up by the following inclusion
S(∂0,ν)∂0,ν⊆∂0,νS(∂0,ν),
which is still not obvious to me. I feel like, this is a stupid question and I am missing an easy argument. Can your clarify what is used here?
Best wishes, René
Jürgen Voigt, 2020/05/24 00:20
Dear René,
it took me some time to understand this in the Lecture Notes, in the proof of Prop. 11.1.2. If S is the material law for Sν, then (abbreviating ∂ν:=∂t,ν)
∂−1νSν=Sν∂−1ν,
because both sides correspond to the material law 1zS(z). This implies that the range of the right hand operator belongs to dom(∂ν), and
Sν=∂νSν∂−1ν.
Composing on the right by ∂ν you get
Sν∂ν=∂νSν∂−1ν∂ν=∂νSν∣dom(∂ν)⊆∂νSν.
Does this answer your question?
Best wishes, Jürgen
René Hosfeld, 2020/05/25 10:47
Thank you Jürgen, this answered my question.
Jürgen Voigt, 2020/05/18 14:55
2) In the proof of Theorem 2.1.3 (almost at the end): What is the point of choosing the component C containing ρ+1 and why does Ω open implies z∈Ω?
Concerning the first question: He wants to show that the equation (2.1) holds everywhere. From (2.1) he knows that ρ+1 belongs to the set. (In principle, he could use any point z with Rez>ρ.) What he then shows is that Ω is open (hence relatively open in D(M)∩CRe>−ν) and relatively(!) closed in D(M)∩CRe>−ν.
Actually, I am a little confused by the statement of Theorem 2.1.3: The requirement that the hypothesis holds for allε>0 implies that one could just omit ε from the hypothesis and suppose that D(M)∩CRe>−ν is connected for all ν<ν0. Probably this is some kind of ``typo'', and what he wants to require that there existsε with the indicated property.
René Hosfeld, 2020/05/21 15:50
Dear Jürgen,
thanks! I missunderstood the closedness argument, but now it is clear to me
Concerning the ϵ argument: I agree that the existence of such an ϵ is enough. This is how I think this works.
At the beginning of the proof where he assumes that the evolutionary problem is exponentially stable, he says:
“We show that s0(M,A)≤−ν for each ν0−ϵ<ν<ν0, which would yield the assertion.”
I think, the argument is, that this implies s0(M,A)≤−ν<−ν0+ϵ for one ϵ>0. Now note, if the condition 'CRe>−ν∩D(M) is connected for all ν0−ϵ<ν<ν0' holds for ϵ, it holds for all 0<˜ϵ<ϵ. Hence, for ϵ→0, we have s0(M,A)≤−ν0.
Do you agree?
Best wishes, René
Jürgen Voigt, 2020/05/22 11:41
Dear René,
I have the impression that you are still making things slightly too complicated. You don't have to vary ε: If s0(M,A)≤−ν for allν∈(ν0−ε,ν0), with someε>0, then s0(M,A)≤−ν0. (Note that you can let ν→ν0.)
Something more general: I see the workshop date coming closer, and we have had (almost) no communication from our friends in Marrakech. I wonder how we can activate them. First thing I will do is writing them a communication, coming weekend.
Best wishes, Jürgen
Jürgen Voigt, 2020/05/18 23:27
Dear Rene,
5) In section 2.2. we assume M0,M1 to be bounded. So if we assume the domain of M be some right half-plane with positive distance to 0 and to be contained in the domains of M0,M1 we have M bounded. So, does it make sense to talk about unbounded material laws? Of course it is interessting and it is more general, but is it really needed for our purpose? Nontheless, I think we should treat some aspects of section 2.1 in our presentation, as the main results from 2.2 are based on them.
I think that, in the context of Sascha's text, it belongs to the definition of a material law M that D(M) contains a right half-plane, and therefore M0 and M1 are supposed to be bounded on a right half-plane, in Section 2.2. This means that in this context there is no need at all to think about unbounded material laws. As I said above, quite generally, we should not consider the case of unbounded material laws: I think that in this respect we can safely stay with the situation that we always had in the ISem.
Best wishes, Jürgen
Jürgen Voigt, 2020/05/19 20:06
Dear Rene,
3) In the example of the damped wave equation there is C:=ι∗ran(grad0)grad0 and H1=L2(Ω)n. I think it should be H1=ran(grad0) in order to say that C is invertible. In the lectures, this is exercise 11.3.
I agree.
Jürgen Voigt, 2020/05/19 20:23, 2020/05/19 20:23
Dear Rene,
4) In the example of the dual phase lag heat equation there are some operators commuting and I am wondering whether they are really commuting or whether these are only formal computaions. To be precise, the operators grad and (1+τv∂0,ρ) as well as (1+τq∂0,ρ+1/2τ2q∂20,ρ)−1(1+τv∂0,ρ) and grad or div.
You are referring to the computation on p. 69, I think. It seems to me that the computations should be interpreted `formally', as no differentiability requirements on θ and q are stated. Nevertheless, one could remark that the differentiations grad etc. and ∂t act on different variables. Incidentally, τq and τθ seem to be just numbers. It would certainly be desirable to provide some explanation concerning the model one is considering here. `dual phase lag' is not of great help for me!
Best wishes, Jürgen
René Hosfeld, 2020/05/21 15:59
Dear Jürgen,
althoug it is clear that the operators act on different variables, it is a good remark.
To be honest, I am not a physicist and I do not understand, how the model can be interpreted. For me it is just a modified version of the heat equation.
Does someone else in our group know more about it?
discussion/project_m.txt · Last modified: 2020/03/07 15:30 by matcs
Discussion on Project M
Maybe one could try to have a look at the paper Sascha quoted there. I might do it on the weekend.
Dear all,
as announced in my mail, I there are some points in habil.pdf which are not totally clear to me. So maybe, you can help me out or we can discuss them.
1) In the proof of Lemma 2.1.2 there is written, that (∂0,ρM(∂0,ρ)+A)∂0,ρ⊆∂0,ρ(∂0,ρM(∂0,ρ)+A) holds. This is not obvious to me.
2) In the proof of Theorem 2.1.3 (almost at the end): What is the point of choosing the component C containing ρ+1 and why does Ω open implies z∈Ω?
3) In the example of the damped wave equation there is C:=ι∗ran(grad0)grad0 and H1=L2(Ω)n. I think it should be H1=ran(grad0) in order to say that C is invertible. In the lectures, this is exercise 11.3.
4) In the example of the dual phase lag heat equation there are some operators commuting and I am wondering whether they are really commuting or whether these are only formal computaions. To be precise, the operators grad and (1+τv∂0,ρ) as well as (1+τq∂0,ρ+1/2τ2q∂20,ρ)−1(1+τv∂0,ρ) and grad or div.
5) In section 2.2. we assume M0,M1 to be bounded. So if we assume the domain of M be some right half-plane with positive distance to 0 and to be contained in the domains of M0,M1 we have M bounded. So, does it make sense to talk about unbounded material laws? Of course it is interessting and it is more general, but is it really needed for our purpose? Nontheless, I think we should treat some aspects of section 2.1 in our presentation, as the main results from 2.2 are based on them.
Kind regards, René
Dear Rene,
I was very busy with something non-mathematical during the last two weeks, but now I will try to answer your comments - one by one at a time.
1) In the proof of Lemma 2.1.2 there is written, that (∂0,ρM(∂0,ρ)+A)∂0,ρ⊆∂0,ρ(∂0,ρM(∂0,ρ)+A) holds. This is not obvious to me.
Maybe you/we should not look at this but rather realise that Lemma 2.1.2 corresponds to Prop. 11.1.2 of the Lecture Notes, and your question corresponds to line 3 of the proof of Prop. 11.1.2. There has been a comment - by myself - on this proof in the forum. The discussion following my post should clear up the matter, I hope.
But let me ask another question, probably stupid: I had commented in the proposal of the project that the operators ∂t,νM(∂t,ν) of the lectures would be of the form ∂0M(∂−10) in [2]. What is the reason that this is not the case in the present instance?
(to be continued soon!)
Best, Jürgen
So sorry!! I had confused [2] and [3]! It is in [2] that material laws are defined on sets different from the ones used in the Lecture Notes. The difference between the material laws in [3] and the Lecture Notes is that in [3] it is not supposed that M is bounded on a half-plane. I suggest that we could/should stick to the case of material laws treated in the ISem and avoid the case of unbounded material laws.
Dear Jürgen,
at first I was also confused by this (for us) unsual definition. But, even it is clear now, we should avoid it. Also, I think we should concentrate on [3]. [2] does not have much more informations for us.
Best wishes, René
Dear René,
yes; I agree.
Dear Jürgen,
concerning your answer to my first question. I am still puzzled how the fact that the solution operator Sν is of the form S(∂0,ν) for some material law S will help. If I use this, I end up by the following inclusion
S(∂0,ν)∂0,ν⊆∂0,νS(∂0,ν),
which is still not obvious to me. I feel like, this is a stupid question and I am missing an easy argument. Can your clarify what is used here?
Best wishes, René
Dear René,
it took me some time to understand this in the Lecture Notes, in the proof of Prop. 11.1.2. If S is the material law for Sν, then (abbreviating ∂ν:=∂t,ν) ∂−1νSν=Sν∂−1ν, because both sides correspond to the material law 1zS(z). This implies that the range of the right hand operator belongs to dom(∂ν), and Sν=∂νSν∂−1ν. Composing on the right by ∂ν you get Sν∂ν=∂νSν∂−1ν∂ν=∂νSν∣dom(∂ν)⊆∂νSν. Does this answer your question?
Best wishes, Jürgen
Thank you Jürgen, this answered my question.
2) In the proof of Theorem 2.1.3 (almost at the end): What is the point of choosing the component C containing ρ+1 and why does Ω open implies z∈Ω?
Concerning the first question: He wants to show that the equation (2.1) holds everywhere. From (2.1) he knows that ρ+1 belongs to the set. (In principle, he could use any point z with Rez>ρ.) What he then shows is that Ω is open (hence relatively open in D(M)∩CRe>−ν) and relatively(!) closed in D(M)∩CRe>−ν.
Actually, I am a little confused by the statement of Theorem 2.1.3: The requirement that the hypothesis holds for all ε>0 implies that one could just omit ε from the hypothesis and suppose that D(M)∩CRe>−ν is connected for all ν<ν0. Probably this is some kind of ``typo'', and what he wants to require that there exists ε with the indicated property.
Dear Jürgen,
thanks! I missunderstood the closedness argument, but now it is clear to me
Concerning the ϵ argument: I agree that the existence of such an ϵ is enough. This is how I think this works. At the beginning of the proof where he assumes that the evolutionary problem is exponentially stable, he says:
“We show that s0(M,A)≤−ν for each ν0−ϵ<ν<ν0, which would yield the assertion.”
I think, the argument is, that this implies s0(M,A)≤−ν<−ν0+ϵ for one ϵ>0. Now note, if the condition 'CRe>−ν∩D(M) is connected for all ν0−ϵ<ν<ν0' holds for ϵ, it holds for all 0<˜ϵ<ϵ. Hence, for ϵ→0, we have s0(M,A)≤−ν0.
Do you agree?
Best wishes, René
Dear René,
I have the impression that you are still making things slightly too complicated. You don't have to vary ε: If s0(M,A)≤−ν for all ν∈(ν0−ε,ν0), with some ε>0, then s0(M,A)≤−ν0. (Note that you can let ν→ν0.)
Something more general: I see the workshop date coming closer, and we have had (almost) no communication from our friends in Marrakech. I wonder how we can activate them. First thing I will do is writing them a communication, coming weekend.
Best wishes, Jürgen
Dear Rene,
5) In section 2.2. we assume M0,M1 to be bounded. So if we assume the domain of M be some right half-plane with positive distance to 0 and to be contained in the domains of M0,M1 we have M bounded. So, does it make sense to talk about unbounded material laws? Of course it is interessting and it is more general, but is it really needed for our purpose? Nontheless, I think we should treat some aspects of section 2.1 in our presentation, as the main results from 2.2 are based on them.
I think that, in the context of Sascha's text, it belongs to the definition of a material law M that D(M) contains a right half-plane, and therefore M0 and M1 are supposed to be bounded on a right half-plane, in Section 2.2. This means that in this context there is no need at all to think about unbounded material laws. As I said above, quite generally, we should not consider the case of unbounded material laws: I think that in this respect we can safely stay with the situation that we always had in the ISem.
Best wishes, Jürgen
Dear Rene,
3) In the example of the damped wave equation there is C:=ι∗ran(grad0)grad0 and H1=L2(Ω)n. I think it should be H1=ran(grad0) in order to say that C is invertible. In the lectures, this is exercise 11.3.
I agree.
Dear Rene,
4) In the example of the dual phase lag heat equation there are some operators commuting and I am wondering whether they are really commuting or whether these are only formal computaions. To be precise, the operators grad and (1+τv∂0,ρ) as well as (1+τq∂0,ρ+1/2τ2q∂20,ρ)−1(1+τv∂0,ρ) and grad or div.
You are referring to the computation on p. 69, I think. It seems to me that the computations should be interpreted `formally', as no differentiability requirements on θ and q are stated. Nevertheless, one could remark that the differentiations grad etc. and ∂t act on different variables. Incidentally, τq and τθ seem to be just numbers. It would certainly be desirable to provide some explanation concerning the model one is considering here. `dual phase lag' is not of great help for me!
Best wishes, Jürgen
Dear Jürgen,
althoug it is clear that the operators act on different variables, it is a good remark.
To be honest, I am not a physicist and I do not understand, how the model can be interpreted. For me it is just a modified version of the heat equation.
Does someone else in our group know more about it?