 Yes. Shall I make it full screen? We are good to go. Recording is good. Yeah, I think it's full screen enough. Yeah, you can do that. Yeah, of course. Okay, but first I'll introduce you. So welcome everybody to this ICTP math associate seminar. So we're going to have a talk today and also one week from now, November 19. We'll have the talk by Cecilia Salgado. Thank you very much. Thank you very much. Thank you very much. Thank you for that. The speaker of today is George now from the University of Mumbai. We're very happy to have her speak about her new work on reaction diffusion equations on the evolving domains, regularity and global existence. So far as yours. Thank you very much. I thank that for the invitation and in fact, the entire math group at ICTP presents my work. And I will be speaking on reaction diffusion equations on evolving domain. We'll talk specifically about global existence results, including regularity. I would like to say in beginning that I'm not going to do any or show any technical calculations or computations, but just give an idea or sketch of what goes into the proof and the main result. So maybe it may not, maybe it is shorter talk than what one expects. This is something which I had been reading during last few months and it is a joint work with Dr. Vandana Sharma from IIT Jodhpur, India. It has been very interesting to work with her on this kind of field on reaction diffusion equation. So let me start with a brief introduction of what is known and what has been done. Of course, I'm going to refer to very few of these recent works. And for the detailed references, maybe I can refer to our paper which will be on archive soon. So what is reaction diffusion equation? We cannot. So system of reaction diffusion equations are used to model pattern formation in chemistry and biology. I think it was during who had used these equations to model pattern formations either on shells or the patterns which are formed on animals as they are developed. For example, the stripes on zebra and some other animals. What used to be done initially was like theoretical and numerical studies were done on fixed domain. And now people are kind of moving on to domains which are either growing or shrinking. And when I say domains, then not only not only like domains in the Euclidean space, but also also on general manifolds. So for example, one can consider a sphere or ellipse or one can also think of these kind of equations which are used to model cell growth in mathematical biology. And then you can have different types of surfaces which one can consider. So what is the equation? Here what I am doing is I am directly writing the equation which is which is for the domain which is evolving with time. So instead of what evolving which usually represents something which grows. So here one can also use reforming which means either you grow or shrink in one part or grow in one direction. So one can consider a more general situation. So here this is time derivative and you have this one extra term. So if this middle term doesn't appear and say your C is just a function of Y not depending on T then you would get a reaction diffusion equation on a fixed domain. So if I say T equal to 0 over here then everything becomes on fixed domain. So I will just set up the equations and equation and continue and try to relate with what we are doing soon. So here we look at this system. So though I am writing here only as just think of it as M equal to 1 to begin with. You have D this corresponds to the diffusion term and this is the coupling parameter between the various components of your system. So here one can in general consider C equal to C1 to Cm. This is going to be your diffusion term. So it is for C1 you will have an equation which takes into account C1 to Cm all of them. So here one can consider the way I write it over here. So this is kind of a matrix form. We have C1, C1, C1 and F1 over here where F1 depends on all M and usually one assumes that these coordinate functions are locally Lipschitz. So this is the expression which is considered for which is already existing for evolving domains. So here one needs to know that the term AYT, this term which appears over here it is referred to as the flow velocity and this corresponds to flow velocity and it is the flow which is generated by the evolution of the domain and of course this is the expansion as I said you can just look at M equal to one and you will one can understand that this is the gradient term which comes because of the additional condition that domain omega is deforming and due to one usually assumes I think that is called as Reynolds transport theorem that flow velocity is same as the domain velocity. So Y is going to be a function of rho T times X. So basically one can think of the fact that the flow velocity is separable in T variable and the space variable. That kind of simplifies later computations also as we will see. So let me just speak about some of the works as I said these I am mentioning very few of them. One of the earlier works was in 1999 by Cramping et al. and they considered uniform and isotropic domain growth in one dimension reaction diffusion system and here as I have written over here like they allow like sometimes the time to grow fast or time to be slow like they control the velocity of growth and try to see how the patterns change. So many of these existing works they are just mostly focusing on numerics and then trying to observe how the patterns emerge after so they start with an initial pattern they allow they apply the differential equation they allow the program to run for certain period of time and then they try to see how the patterns have developed at certain later time capital T that is what is usually done using the numerics. Then also one of the important papers was in 2004 by Plaza Padilla and there were a couple of more collaborators where they developed a general formulation of reaction diffusion theory on isotropically evolving one and two dimensional manifolds. So they derived the equation for general manifolds using the Laplacian and with the first motivation from biological settings they had a problem where they were considering how the curvature of the surface where the evolution is taking place or where the pattern is forming affects the resulting pattern after the growth. So for example like you may have a small shell and there is an initial pattern and over a period of time the patterns develop and then they see what kind of patterns are formed in the shell after it has completed it's both period that's what they are interested in. So the latest result which I found and in fact this is kind of quite interesting and these are results by Cross and his collaborators where they generalized the modeling of reaction diffusion systems on doing manifolds which was developed by in the previous work of Plaza to allow for dilational and isotropic growth. So one thing which we observe is that if you saw we had the assumption that the flow velocity is proportional to is of the form roti times X so it is like dilational in the space variable right you have roti and your X is your usual coordinates. So here you can allow an isotropic growth so in different directions one can allow different parameters to be depending on the T variable. So at this point I would like to show the figure from this particular paper so I will escape from the yes yes so this is the figure which I have taken from their paper and as you can see they have studied so here they are studying the reaction on so this is the initial pattern which they have and then this is the pattern which they have on an ellipse this is a two dimensional figure or manifold one can say this is a two dimensional manifold and then you allow it to grow so as we see that here after certain period of time so actually this is more visible in this second figure where the initial pattern is on the sphere then they allow it to grow in this axial direction and then they have a resulting pattern here here in this figure C they have allowed it to grow in X direction so one can easily see that you are having an isotropic because you are going more in the X direction and less in the Y direction or Z direction the three dimensional plot which they have got and then this is the final means you do this kind of growth actually or say in X direction, Y direction and then again you come back to the sphere and these are the patterns which appear over here so most of the existing works have been about the stability of the stability results and usually apply to do numerics and study how the patterns grow on different manifolds so here ellipse and here on the sphere so so that is in fact that paper contains other numerics also and there are quite a few nice pictures which one can see also as one can see that this is the most recent work and yeah so they studied how rate of deformation type of growth or extremity of growth affect the patterns formed on the domain in case there is if the pattern changes at all then they studied what are the final results after applying the numerics okay so as I mentioned that many existing results focus on sorry yeah focus on growth and curvature focus on effects of growth and curvature of the domain on the resulting patterns and their stability but there are very few works where people have studied the question of existence and regularity of solutions of systems of reaction diffusion equation on domains which are evolving with time now one of the results which we have come across is by Chandrasekhar and his collaborators Chandrasekhar and Akita and maybe one more there is one more person where they had included in their work of course they again focus on stability and you know the effect of curvature and growth of the domain and how the patterns change but here so but they had also studied the global existence of a solution of this reaction diffusion equation on going domains and their proof was based on the work of Morgan Jeff Morgan on stationary domain and what I would like to remark is that they did this work which of course one can see if one looks at any of the previous papers also one sees that the usual method is that you take your problem on the stationary domain and you reduce it to or transform it to a fixed domain where you now look for a solution which is also depending on time and there are certain extra terms which also involve the fact that your domain is changing with time now I will show you in the next few slides what I mean and I write down the equation so this is one of the work which we found and in our work so yes this is my collaborator so she has been working on reaction diffusion system of reaction diffusion equations on stable domain or stationary domain so what we will do here is that the results that she has developed in couple of papers recent papers we are going to adapt them to the domain which is deforming with time so yes so let me go back to the notations my here is the domain which is changing with time and it is with boundaries C2 boundary and I am going to consider omega t in Rn and if we denote C as the chemical concentration in the domain omega t then this is the equation which drives the diffusion process so here as you see means it really doesn't matter whether I write C as one function or as a vector but for simplicity I am just going to write it in this simple form if one wants to write think of it as a vector one can write it as a vector also it's just one equation for to avoid the technical complications so here so this is d of omega t so here this is the measure on omega which is of course also changing with time so this is this derivative is outside this integral and this is the normal derivative on the boundary again which is changing with time so this is the governing equation for the diffusion process of chemical concentration on a domain which is changing with time so here we have the initial domain we fix or we take initial domain omega 0 is equal to omega and we assume as we usually do in geometry that your omega t is the white of omega so basically we want to say that omega t is generated by flow of omega under given diffeomorphism so then one can see that your your measure is just going to be the determinant of this particular flow which I have defined here times the usual Euclidean measure on omega so if one solves the above relation then one gets this particular equation so here this is what we get as flow velocity the equation that A, X, T whatever term which I had got before so this is your flow velocitiveness and now since we have simply written omega t as y t of omega one can see the justification that why we get this term over here and then the equation is one can work out and equate the terms on the left hand side and right hand side it is nice because we are getting the linear term but it comes with negative sign and it does not cause too much problem I think usually that is what I would say so now this is where I transform I am going to say that my original equation which was represented on omega t if I define my u x t as my previous equation becomes now equation on a stationary domain so if I solve for u in this particular equation then I have solved for c in omega t so are there any questions are there any questions so I think there was a question in the chat but I minimized this so Boris is saying I would like to know if there is someone studying the effects of time delay in the reaction diffusion system time delay I do not know actually I do not know because as I said this is for something which I am learning so sorry I am not sure okay because it becomes visible here so I do not know how to take care of it so yes so this is so now what we have done we have done is reduce the problem to stationary domain but we see that the fact that we have this flow term now it is going to be a different Laplace operator actually it is the pullback of the metric of whatever we have on omega t to omega we have this additional linear term which comes from the because of the flow term over there so here yes so what we do as I said is I take we take a simpler situation where we just look at the 80 times x one can probably introduce a complicated matrix and go through the entire work but for sake of simplicity we are just trying to work with a simple case where the matrix A is diagonal and as you can see we do allow for an isotropic flow so you don't need to have 1 equal to lambda 2 equal to lambda 3 one can have different flow, different speeds in different directions and it is quite simple we start with the initial one identity and these are diffeomorphisms so these are non-singular matrices and I have written it down for 3 by 3 but of course same thing can be generalized I think there is a type over here because so here this is determinant of n product divided by n product which should be n product here because this equation which I have written I have written for n dimension so yes your delta t once you do the transformation you get it because you have a simple representation here you have it to be 1 upon lambda i t square d square by dx i square so if this is identity you are going to get your usual laplacian so when t equal to 0 you are back to usual laplacian yes so yes this is what we finally come to so what we are doing is studying this system of equation on the stationary domain as I said that if one does not have growth then of course this operator I have sorry I think there is another mistake I am so bad this is a type over here it should be l which is given by di delta t minus at where at is this term the term which we had got in the previous one here so I apologize for this type over here d t u i equal to yeah so l times u i I think I should write it in vector form this is the Neumann boundary condition here eta now is the unit normal vector, outward vector on the fixed domain omega so the usual normal derivative condition on omega t translates to di gradient t sub t u i so gradient sub t u i is nothing but 1 by lambda i times del by del u i so yes yeah so this is my new gradient vector so it is going to be 80 inverse of the nabla which I have and it is of this particular form and for say for calculations that we require to do I think this is a natural condition that we assume that these coefficients which appear over here are bounded so since here in this paper we are only interested in terms and equality so we are assuming everything is kind of fine so because in practice of course one can expect that with time maybe the domain breaks up or the boundary no longer remains smooth so all these kind of possibilities one can expect but for sake of our analysis we are assuming that everything remains fine over the entire period of time so the domain omega t evolves nicely in fact what we have assumed is that as t goes to infinity my omega t remains in neighborhood of a fixed domain omega infinity this is just a notation and that also ensures that say if gamma t is the boundary of my domain then gamma t also remains in neighborhood of one fixed boundary so it doesn't maybe it oscillates but you do not have the smoothness doesn't break or the domain doesn't split there is no crossing over no singularity is developing so we are just trying to work in kind of ideal situation so that's one remark which I wanted to make so coming back here this is the equation sorry for this error here and these are the assumptions so here we assume that the initial function on the stationary domain at the time t equal to zero is in c2 omega bar and it satisfies the compatibility condition on the boundary this is the non-homogeneous boundary condition here we also assume that the non-linear functions on the right hand side so these remember the FIs are on the domain and GIs are on the boundary so these are posi positive so what it means is that if I take Fi and if I take ui equal to zero then sorry on all other conditions the condition is non-negative and this condition ensures that if we start with initial non-negative data then the solution of the system continues to remain non-negative and this is a condition which is what we refer to as VL1 so here one sees that you have linear combinations of Fi and Gi so again boundary and domain and there exist a fixed constant so this is strictly greater than zero otherwise it's redone and it's missing so sorry so you should have L1 over here times this particular sum L1 over here times this particular sum so this is condition VL1 and this is the condition which Fi and Gi are polynomially bounded so there are there exist M and national number L such that these functions are less than this summation of the coordinates power L so here these are usually this condition is used for for deriving high regularity of the solutions so these are the common conditions which I have put over here let me say over here that my collaborators paper it has got recently accepted in communications and applicable analysis I can see pure and applied analysis I believe and our work over here relies on the work in that particular paper so many of the assumptions which I have taken in fact all the assumptions are compatible with the assumptions which she has assumed on the stationary domain so the difference is that if you have gradient to there then we have to kind of generalize it to Nabla T over here where you have a slightly modified gradient term and also the operator is different it is delta T in our case so yes so I will state the first result which is talks about local existence so here if we have the functions f i n g i are locally live sheets and satisfies v n that is the Neumann boundary condition and this is posi positive then one can find T max strictly greater than zero such that the system this system this system has this system has unique maximal component wise non-negative solution so here you can see that I have written it for me to what it is for m it is also true and if T max is finite this limit is pos to infinity so here as T approaches T max from the left it is pos to infinity so this is the first result which shows the existence of non-negative local existence and for the proof let me mention the main results so here the next we have also proved the global existence result and here what I have done is instead of writing it as two theorems I have combined them what we have shown is that in the case so this is again Neumann boundary condition non-negative and polynomial polynomial bounded I should say and for m equal to 2 so this is if I have two components u1 and u2 or which I refer to as uv then if we assume vl1 and suppose that there exists a non-decreasing function h such that one of the component functions is bounded and so remember vl1 gave a fixed constant l1 whereas here we have one more condition on the boundary on the gis that whenever there exists k greater than zero such that wherever a is bigger than equal to k you can find la such that this estimate holds so you have one over here and this is similar to l1 but as I said l1 condition say gives us fixed constant whereas here you have one can one can one has a choice to make so either you assume one component is bounded by a suitable function non-decreasing function together with this condition or for any m if this condition what is called as vl holes there exists k such that for this particular vector with am equal to 1 and all these ai's strictly greater than equal to zero there exists la greater than zero so this is a linear combination of f i as well as linear combination of g i is less than this particular sum then the equation the system has a unique component wise non-negative global solution so for m equal to 2 if we assume that one of the component is bounded we are able to arrive at global existence but in general for any m greater than equal to 2 if we assume this we are still able to arrive at global existence one thing which I would like to mention is that for m equal to 2 the vl1 condition is of course contained in this one so let me say few words about the proof I just wanted to mention so about local existence the proof of local existence relies on holder estimates for the linearized problem and so in this particular case since our operator has changed we have to look at the linearized problem corresponding to this particular operator over here here I have written delta t is equal to a i t d square by d xi square in our particular case a i t is 1 upon lambda i t whole square so but one can one can see that the proof works in general for these kind of operators it's suitable condition of boundedness that's what I would like to mention so in fact for the so this is the result of holder estimate which I require for local existence so again this is related to the linearized operator so we have this equation with the initial condition initial normal initial Neumann condition the function phi naught satisfying this then we have the following estimate where the constant is independent of theta phi 1 and phi naught so this is the Neumann condition function which appears there this is the boundary condition the initial function is equal to 0 and this is your function which appears on the right hand side of here phi naught phi 1 and theta over here so the function is independent of this and you get holder estimates for a solution of the linearized problem so what I would like to mention here the proof of the linearized problem uses extensions of the estimates of fapes and river paper to the operator delta t together with the first alignment boundary the nabla t which we have and in fact this was done because we have used the techniques and results of Sharma and Sharma Morgan this paper has appeared already in SIAM and they required similar result for stationary domain so we extend those results here to our operator and we get nice estimates and get the holder estimates that we require over here so this is one of the classic parameters of fapes river and what we do is for the proof we first show that if fngr then there exists a unique solution using the solution of the linearized problem and if they are just locally then we use cutoff function and arrive at the conclusion by taking limit so here the condition that fngr quasi positive ensure as I said initially that the solutions remain non-negative so next what we also obtain is L1 estimates again I prefer to show it separately because these hold for both the results for m equal to 2 or m greater than equal to 2 this particular step is kind of common so here again as you see I have written only for two components so you let uv be the unique maximal non-negative solutions to 5 and suppose that tmax is finite so this is Neumann condition this is Lipschitz and Vl1 Vl1 holds then there exists this constant which depends on the diffusion coefficients such that for each fixed t in the interval we obtain this particular bound we see that the L1 norms are all bounded so one sees that this constant is going to depend on t so the proof of L1 estimates is like again I wouldn't say it is difficult but of course one needs to add the solutions and then one uses the given data particularly this L1 condition and uses ground wall inequality to arrive at the required application of ground wall inequalities to arrive at the required result so here I have written for for 2 functions remember for m equal to 2 I am assuming one of them is bounded so assuming one of the say so here I am already so it already means that V is in L1 so here I am saying that U is going to be in L1 on boundary also there is L1 bound for V also the L1 bound holds also let me mention that this one really doesn't require one of the functions to be bounded over here I have not used that property over here so this L1 estimate is a common feature for either m equal to 2 or m greater than equal to 2 and it does not use one of the functions or any component function to be bounded so is there any question because I am not seeing anyone is there any question I don't know because I don't think that of yours yeah so there was a question in the chat again so now I wasn't sure if Claudio also had one but the question again from Boli is in the chat he is saying that I am actually trying to improve reaction diffusion in epidemic systems are there are biological implications of an isotropy in the epidemic model I don't know if this is maybe I would I mean often I cannot say anything unless I see the equation okay and then we have another question so Claudio would like to see again equation 5 yeah you please stop me I am not looking at this I cannot see chat or anything here so you can stop me and ask yes now this question just came in this from Claudio so yeah yeah here yes there is an error there there is an error here please it is this I should have written here so I think you can go on yeah so I was talking about the proof yes so for m equal to 2 okay so why did we do m equal to 2 separate and m greater than equal to separate so again as I said since our work is based on my collaborators work recent work and I think it is quite interesting because there is a paper by Martin Hollis where they are dealing with two components and where they assume one of them is bounded and they prove global existence again only for stationary case they prove for stationary case and they use different techniques I think it is semi-group they use techniques of semi-group if I am correct to prove the result so in her recent work my collaborator proved this result that is when m is equal to 2 one of them is bounded she used duality and lay up on our function type of functional to prove this so without using the semi-group theory methods yeah so it was natural that we extend the similar ideas to the evolving or deforming domain case so that is what I have written here that we prove is combination of duality argument so the duality argument gives you LP bounds for the component functions and then if you want to extend so what one usually does is that you assume T max is finite if you want to show T max is infinity so you assume by contradiction T max is finite and then the duality argument gives you LP bounds and what one does is one uses lay up on our type of functional which is defined by this particular polynomial so as you can see I have written it only for two components case so one uses this kind of functional and derives you differentiate it with respect to T you derive one more type of inequality using this functional and then you try to not try then you show that the solution is bounded up to the boundary T equal to T max okay so then then that would be a contradiction to the local existence where we have already shown that the component solution go to infinity goes to infinity as T approaches T max so that is how one arrives at a contradiction to the fact that T max is finite so the novelty in her work was this lay up on our functional which is nice it is much simpler than a similar functional which was used for general M components by in work of Koachi and his collaborator where they had slightly more complicated polynomial it is still a polynomial equation but yeah so thetas here are constants so here there is just one theta so this is a constant there is a choice of theta involved over here where the condition which are given on G that is used so that is the proof and of course what I would like to say is that for M in general in general if we don't assume that any of the component is bounded when you don't make any assumption on any component then one can still use this this is something which we have just recently done one can extend this polynomial to general M so here instead of just uv you can write it down for u1, u2, um and again use Lyapunov type of arguments like a differentiated and then try to estimate get LP bounds and then you use trace theorems embeddings and you will get contradiction with the assumption that Tmax is finite so this is something that we have been working on I think last couple of months and yeah I think I have not really shown you any proof but this is the thing which I would like to end with over here so let me show these are the references this is not comprehensive so this is the paper of Fitt's review and these are some of the references this is paper of cross as you see that it's very recent they are quite beautiful pictures over here and yeah this is Plaza's paper if someone needs reference for Sandra Shekhar's paper maybe I can give so this is the recent work which has been already accepted of my collaborator on which our results are based I would like to make a remark here that as I mentioned Sandra Shekhar had also Sandra Shekhar and his collaborators had also proved global existence for evolving domains but what they had done is that in their analysis they had changed, used a transformation to change your instead of Delta T you work with Delta T as we have tried to do in our books that's what that's one I think one major difference which we have also I think that is all that I would like to say there are a lot of things which we are planning to work on because we plan to develop this for interactions surface and bulk surface interaction of reaction diffusion equations and that is in progress so the paper will be on archive maybe in couple of days so yeah I think I would like to end over here if someone wants to see more details I can share with them and I think I would like to end here okay so thank you very much for a very interesting talk so do we have any questions if so you may feel free to unmute yourself and ask for you can also type in the chat and I will relay the question because I cannot see the yes so we'll see so perhaps while people are thinking if you have a question or not I can ask so from the point of view of a completely non-expert of course in this I was just trying to understand a bit better your assumptions in the global existence PRM so yeah I don't understand them very well so maybe if you could explain a bit more see how this for example this assumption here yeah some should be well what does it mean sort of more at least speaking let's say not just technically yeah I think this is relating all the non-linear functions which appears on the right hand side so it probably gives you a combined condition that you cannot so this is like a diagonal matrix acting on the vector F1 F2 Fm okay you have a diagonal matrix A1 A2 Am which acts on F1 F2 Fm and in fact I should mention that this condition I think it has its origin in math biology or the problems which they consider some of these conditions appear naturally okay since you asked in fact one of the original conditions with which my collaborator has worked with is what is called as intermediate some condition where instead of a diagonal matrix you have a lower triangular matrix and that condition was formulated by Morgan so if one sees their papers one has references to that condition which is slightly more so I should say these conditions naturally occur in the real life applications it's possible to explain why or how they occur naturally or something just very basic way for example I don't know if my I am too new to the subject to comment on these but the examples which are there the functions on the right hand side whatever nonlinearities they put they do satisfy such conditions as I said this is slightly stronger the intermediate some condition is slightly more common it covers more general general varied kind of applications so I am sorry I am also but at least that is what I have seen that in the examples which are cited or where the works are there these conditions are satisfied yes okay so any other questions okay we cannot hear you well or not at all actually at least I cannot so okay so glad you will type I think and in the meantime I have a question in the chat from Johanna who is asking are there uniform in time L1 estimates no I don't think so so we have the answer yes from I think is this your collaborator or mention in the talk yeah good okay okay they said this is work in progress but the way I have mentioned here the estimate depends on time the way I have mentioned in this result here the way we have it so also if you want uniform estimates I think we require one has to be here it is time that's what yeah so that's my collaborator so thank you one then okay and then we have another question I did not understand how the domain evolves in equation 5 so we go back to this evolves in the so actually actually what I have done here is we have this is my original equation this is my original equation where so what I have done is that I have I have written so now if I do uxt equals to we have this uxt is once I substitute this my x depends x is in domain omega so if I want to go to the c I will have to use this relation to go back to c the usual reaction diffusion equation does not involve this term it does not involve delta t have I answered this question I am not sure I cannot see the chat okay yes we have no answer yet from Claudia but they think he's typing something okay well in the meantime we have people saying thank you for the web conference so for the talk so you're welcome for listening to me but yeah here I in this equation it is the evolution is visible in the operator and this linear term okay so we have a follow up question here so from Claudia saying but do you prescribe the evolution of omega t prior to finding the solution of u or not yes yes so it is given by 80x the flow is given by this omega t is 80 times omega so it is prescribed the flow is prescribed okay so any more questions if not I think we can thank the speaker again for this very interesting talk and I will hope to see you all next week on November 19 for the next speaker and here is also