Fast reaction limit with nonmonotone reaction function
Skrzeczkowski, Jakub
Perthame, Benoît
We analyse fast reaction limit in the reaction-diffusion system \begin{align*} \partial_t u^{\varepsilon} &= \frac{v^{\varepsilon} - F(u^{\varepsilon})}{\varepsilon}, \\ \partial_t v^{\varepsilon} &= \Delta v^{\varepsilon} + \frac{F(u^{\varepsilon}) - v^{\varepsilon}}{\varepsilon}, \end{align*} with nonmonotone reaction function $F$. As speed of reaction tends to infinity, the concentration of non-diffusing component $u^{\varepsilon}$ exhibits fast oscillations. We identify precisely its Young measure which, as a by-product, proves strong convergence of the diffusing component $v^{\varepsilon}$, a result that is not obvious from a priori estimates. Our work is based on analysis of regularization for forward-backward parabolic equations by Plotnikov. We rewrite his ideas in terms of kinetic functions which clarifies the method, brings new insights, relaxes assumptions on model functions and provides a weak formulation for the evolution of the Young measure.
 Okay, so thank you very, very much for introduction. And of course, I would like to thank organizers Agnieszka and Mira for inviting me here. So the title seems to be a little bit biological, but actually it has some connections with conservation laws, with measure valued solutions, with young measures. So it's not entirely biological. And it fits a little bit into the topic of the workshop. So as a warm-up, let's consider system of reaction diffusion equations. So we have two components, u and v. You have diffusion of u and diffusion of v. And you have also this reaction term. And this reaction term is scaled with epsilon. So when epsilon is very small, the reaction is very fast. And the question is, what happens with this system, solutions to this system, when epsilon goes to zero? So let me first consider the classical case that is already done and was done around 20 years ago. So when f is strictly increasing, so f prime is strictly greater than zero, then you can denote by g primitive function of f. Then you multiply the first equation by f of u epsilon, the second by v epsilon. So it's a usual thing in PDEs. And then you obtain two energy equations. So you obtain the change of energy of u and the change of energy of v. Now at this level, this is useless because here you have term one over epsilon, which is in general singular. But what you can do with these two identities, you can sum them up. So this is what I do here. And at this level, these two terms, when they are added together, they result in this term. So this term has this advantage that it is controlled. It's sign is controlled. And this energy equality is now useful for understanding what happens in the limit because you have a lot of a priori estimates. So first, because you have that time derivative of this energy is negative, you have a priori estimates in L infinity. Then if you have a priori estimates in L infinity, then you use the fact that the three terms on the right hand side have negative sign. So f of u epsilon minus v epsilon converges to zero strongly. You also have estimates for gradients of both u and v. So in particular, the sum is bounded. You also have estimates for time derivative because if you sum up these equations, you obtain that the time derivative of the sum is sum of some Laplacian, which is bounded in some negative so-called space by estimation gradients. So you can use Aubame-Lion's lemma to have that the sum converges also strongly. And if you know that the sum converges strongly, you can write the sum of u epsilon and f of u epsilon as combination of two things that you know that converge strongly. And from this, you also have that u epsilon converges strongly, v epsilon converges strongly, and you also have some relation between v and u in the limit. So that's the rather classical things to have done by Betten, Hilhorst in 2003. And now the problem that me and Benoit were working on is the question, what happens when f prime can change sign? So when this reaction function can have some regions when it is decreasing. And now you see the main problem. The main problem is here that all these a priori estimates, they are now useless. They are not, this doesn't work because if f prime changes sign, then you don't have estimates on gradient of u, and you don't have estimates on gradient of v, and you don't have even this strong convergence of f of u epsilon. So now this energy approach doesn't work. So we do not know anything of this. So I will come back to this general problem later on, but then we started to study a simpler problem. So we removed this first diffusion. So we started to consider a system when one component is allowed to diffuse, this is v, and another component cannot diffuse, it's immobile. And this is already difficult problem as you will see in a minute, but it has one advantage that you can still use energy equality to study it, because if d1 is zero, this troublemaker term, it simply vanishes. So now you have, with this, you have estimates on both gradient of v, and you have also the strong convergence of f of u epsilon to v epsilon. So this is a little bit better. And now I will start to, I will now become more precise, and I will first tell you that due to some technicalities that will appear, I consider a special function f. So I assume that it has region where, when the function is increasing, region where it is decreasing, and another region where it is increasing. This is just to focus attention. And now I will show you some numerical simulation because I, before I will state the main result. So numerical simulations are as follows. So the blue line represents the concentration of u, so this is the non-diffusing component, and the red line represents the concentration of the diffusing component. So as you see, this blue line, it looks like an oscillating sequence with a young measure that is some combination of Dirac masses. This is more or less what you would guess if you looked at this picture. So but on the other hand, you see that v behaves rather nicely. And now I will state the result. So again, we have this system. Remember that I removed this one diffusion. And we have a priori estimate from energy equality on gradient of v, and we also have energy estimate for this term. And what we'll prove is that v epsilon converges strongly to v. So strongly, I mean both in time and space. And what is very interesting about this result, what I like a lot is that, you know, the result, this is without any estimates on derivative in time. So you have compactness in space, but you have actually no compactness in time. It's very similar to, for instance, vanishing viscosity in conservation laws. And in fact, if you can multiply this second equation with time derivative of v to realize that this time derivative of v behaves more or less like one over square root of epsilon. So it's singular in general. And some other consequences of this strong convergence of v epsilon of this diffusing component. So we also have that f of u epsilon converges to v, but this is just the consequence of this a priori estimate. Because if f of u epsilon minus v epsilon over square root of epsilon is bounded, then the quantity above converges to zero. This is nothing important now. But another thing that I can have from this strong convergence of diffusing component is young measure of non-diffusing component. So let me recall briefly that what is actually young measure. So if I have a bounded function of u epsilon, then how to pass to the limit under nonlinearity? How to pass to the weak limit under nonlinearity? So you write this g of u epsilon as some artificial integral of function g against this young measure concentrated at u epsilon. And then this measure is bounded in total variation by one. So you can pass to the weak star limit, of course, taking some subsequence. And this measure we call this measure young measure. And now you can use the fact that, OK, the first observation is that since f of u epsilon converges to v epsilon and v epsilon converges strongly to v, then its young measure has to be point musted v. But on the other hand, if you denote by me the young measure of u epsilon, the young measure of f of u epsilon has to be image of the first measure under the map f. So what it follows that me has to be concentrated on the pre-image of v under the map f. And now how this results in these oscillations? Where if you recall the shape of the function f that I assumed, well, it looks like that. It has increasing region. This is this red part. It has decreasing region. This is the blue part. And again, increasing region, this is yellow part. And now if v is somewhere here between f minus and f plus, it hits three points. There are three points such that f of u is equal to v. And this is precisely this point where u is oscillating between. So what follows from strong convergence of the diffusing component is that a young measure of u epsilon is some combination of three Dirac masses, at most three Dirac masses. In numerical simulations, we see two of them. OK, so you see that all these results about this system with one diffusing component and one component which is not diffusing, it all bases about proving the strong convergence of v epsilon without information time derivative of v. That's the main thing. So the way to do that, this is more or less based on the compensated compactness approach. So we'll first obtain some family of energies and then we'll play with weak limits and obtain some identities that will be satisfied for many test functions and then we'll conclude. So how do we go? So recall the system. We have again two equations. One of them is without diffusion. Another is with diffusion. And OK, I fix some smooth function phi and I test it with phi of f of u epsilon. Another equation I test with phi of v epsilon. OK, so the result is as follows. So here nothing changes. I just multiplied. Here I also nothing changes. I just multiplied. When it comes to this term from diffusion, you just play with derivatives to write it like that. And here you have to use primitive functions because to write it again as time derivative of some function. OK, I think it's clear. And again, as with a priori estimates, these terms are still troublemakers because they are singular. But if you sum up these things, you obtain something like that. So we have time derivative. This is the red term. You have Laplacian. It is OK. Now you have these two terms. So as we know from a priori estimate, grad v is bounded. And this is also bounded because if you use some Taylor estimate, you obtain v epsilon minus f of u epsilon squared over epsilon. So this is bounded, say, in L1. OK, so what I get from this is that I have a priori estimates for these energies, namely that time derivative is bounded in some negative sub-left space. I need this h minus 1 because here is this Laplacian. So if I multiply by function h1, I will move one derivative on this test function. And this is exactly why I need h minus 1. And I also know that this is just from a priori estimates because if v epsilon is in h1, then any smooth function of v epsilon is also in h1. OK. So if I know this, I can conclude with the following compensated compactness lemma. So how it goes. I have this function here. So this is precisely a function here. And phi of v epsilon is here. So it's a rather well-known result that if you have one of functions which is compact in space, another one which is compact in time, then the product of weak limits is actually weak limit of the product. And this is more or less like something very similar to difficult lemma. OK. And from this, it's rather easy to obtain some identity on u epsilon. So here I just rewritten what was on the previous slide and how to go now. Well, we have to clean this a little bit. So as you know that v epsilon converges to the same limit as f of u epsilon, you can replace this v epsilon here. Because if test functions are smooth, then they are in particular lip sheets. So I can write it like that. So now I have everywhere u epsilon. And then this level, this is very similar to, for instance, analysis of vanishing viscosity limit for measure solution to conservation law to get that the young measure is Dirac mass. But here is a small difference. And I will point out this difference. So usually you have just some function phi of lambda. And here these functions are composed with f of lambda. So it's not just phi of u epsilon, but for instance, here you have f of u epsilon. And this f is also hidden because c prime is defined as phi of f. So you have also to deal with this f. And here comes another technical assumption on f that we somehow, we couldn't somehow do that without this. So the assumption is quite technical. I will just state it in the simplest possible way. So if you recall the plot of f, it looks like that has one increasing region, one decreasing, and another one, which is increasing. So you can define three differences of f, s1, s2, s3. And now the assumption reads that these three derivatives have to be linearly independent. So it's more or less like f has to be sufficiently curved. In particular, this doesn't hold if f is some piecewise affine functions. Because in this case, all these three inverses are just constants. OK. So I will spend the rest of my time to briefly comment, to briefly give some tribute to the Russian mathematician Plotnikov, because it's more or less his ideas that we applied to this problem. So Plotnikov studied the following equation in 90s. And so he considered forward-backward parabolic equation. So here you have time derivative, here you have Laplacian. And this f has the same monotonicity profile. So two increasing regions and one between them, which is decreasing. So you see that there is this unstable region where the equation is actually backward parabolic. And this makes troubles. And Plotnikov studied the following regularization among many, many, many authors. So he adds this time derivative here with epsilon. So you see that this is the good regularization, because if you move this term on the left-hand side, you will have something like identity minus epsilon Laplace, which is invertible as long as epsilon is greater than zero. And then you can throw it again on the right-hand side, and you will have just simple OD in some Banach spaces. So as long as epsilon is non-zero, you can solve this equation. So it's some kind of regularization. And he also get the same result that the young measure of V epsilon converges. And that the young measure of V epsilon is convex combination of young measures. Precisely each mass is located in one of these regions of f. So this is very similar result. And our problem looks very similar to this Plotnikov regularization, because if you sum up our equation, here we have something like V. OK, here is something different. But I will just tell you that it's not so much different that we could actually apply similar techniques. And OK, this paper is, I didn't find this paper in English. It's in Russian. And quite a big part of this work was to translating his paper from Russian to English. If you are interested, I have English version typed in tech, because it's quite insightful paper. OK, and brief comments on the general problem that we would like to still to understand. So two diffusing components, U and V. OK, so there are some estimates from Japanese mathematical school, which allows us to control, again, this quantity and this quantity. So it's not exactly the same. Now V epsilon, there is no chance for all these convergences that I described to you before, because we have simulations for that. But there is something again similar that if you sum up these equations, what we see in the simulations is that there is a chance for the strong convergence of this quantity. So this is precisely what we get before, because if D1 is 0, this is exactly this quantity. So this, my time. So this is the chance that this will converge strongly. But what you also see in simulations is that both U and V oscillates. So it's not like in this case that at least V epsilon does not oscillate. In this general problem, both of these quantities are oscillating. OK, and with this I would like to thank you. All right.
 Okay, so thank you very, very much for introduction.  And of course, I would like to thank organizers Agnieszka and Mira for inviting me here.  So the title seems to be a little bit biological, but actually it has some connections with  conservation laws, with measure valued solutions, with young measures.  So it's not entirely biological.  And it fits a little bit into the topic of the workshop.  So as a warm-up, let's consider system of reaction diffusion equations.  So we have two components, u and v.  You have diffusion of u and diffusion of v.  And you have also this reaction term.  And this reaction term is scaled with epsilon.  So when epsilon is very small, the reaction is very fast.  And the question is, what happens with this system, solutions to this system, when epsilon  goes to zero?  So let me first consider the classical case that is already done and was done around 20  years ago.  So when f is strictly increasing, so f prime is strictly greater than zero, then you can  denote by g primitive function of f.  Then you multiply the first equation by f of u epsilon, the second by v epsilon.  So it's a usual thing in PDEs.  And then you obtain two energy equations.  So you obtain the change of energy of u and the change of energy of v.  Now at this level, this is useless because here you have term one over epsilon, which  is in general singular.  But what you can do with these two identities, you can sum them up.  So this is what I do here.  And at this level, these two terms, when they are added together, they result in this term.  So this term has this advantage that it is controlled.  It's sign is controlled.  And this energy equality is now useful for understanding what happens in the limit because  you have a lot of a priori estimates.  So first, because you have that time derivative of this energy is negative, you have a priori  estimates in L infinity.  Then if you have a priori estimates in L infinity, then you use the fact that the three terms  on the right hand side have negative sign.  So f of u epsilon minus v epsilon converges to zero strongly.  You also have estimates for gradients of both u and v.  So in particular, the sum is bounded.  You also have estimates for time derivative because if you sum up these equations, you  obtain that the time derivative of the sum is sum of some Laplacian, which is bounded  in some negative so-called space by estimation gradients.  So you can use Aubame-Lion's lemma to have that the sum converges also strongly.  And if you know that the sum converges strongly, you can write the sum of u epsilon and f of  u epsilon as combination of two things that you know that converge strongly.  And from this, you also have that u epsilon converges strongly, v epsilon converges strongly,  and you also have some relation between v and u in the limit.  So that's the rather classical things to have done by Betten, Hilhorst in 2003.  And now the problem that me and Benoit were working on is the question, what happens when  f prime can change sign?  So when this reaction function can have some regions when it is decreasing.  And now you see the main problem.  The main problem is here that all these a priori estimates, they are now useless.  They are not, this doesn't work because if f prime changes sign, then you don't have  estimates on gradient of u, and you don't have estimates on gradient of v, and you don't  have even this strong convergence of f of u epsilon.  So now this energy approach doesn't work.  So we do not know anything of this.  So I will come back to this general problem later on, but then we started to study a simpler  problem.  So we removed this first diffusion.  So we started to consider a system when one component is allowed to diffuse, this is v,  and another component cannot diffuse, it's immobile.  And this is already difficult problem as you will see in a minute, but it has one advantage  that you can still use energy equality to study it, because if d1 is zero, this troublemaker  term, it simply vanishes.  So now you have, with this, you have estimates on both gradient of v, and you have also the  strong convergence of f of u epsilon to v epsilon.  So this is a little bit better.  And now I will start to, I will now become more precise, and I will first tell you that  due to some technicalities that will appear, I consider a special function f.  So I assume that it has region where, when the function is increasing, region where it  is decreasing, and another region where it is increasing.  This is just to focus attention.  And now I will show you some numerical simulation because I, before I will state the main result.  So numerical simulations are as follows.  So the blue line represents the concentration of u, so this is the non-diffusing component,  and the red line represents the concentration of the diffusing component.  So as you see, this blue line, it looks like an oscillating sequence with a young measure  that is some combination of Dirac masses.  This is more or less what you would guess if you looked at this picture.  So but on the other hand, you see that v behaves rather nicely.  And now I will state the result.  So again, we have this system.  Remember that I removed this one diffusion.  And we have a priori estimate from energy equality on gradient of v, and we also have  energy estimate for this term.  And what we'll prove is that v epsilon converges strongly to v.  So strongly, I mean both in time and space.  And what is very interesting about this result, what I like a lot is that, you know, the  result, this is without any estimates on derivative in time.  So you have compactness in space, but you have actually no compactness in time.  It's very similar to, for instance, vanishing viscosity in conservation laws.  And in fact, if you can multiply this second equation with time derivative of v to realize  that this time derivative of v behaves more or less like one over square root of epsilon.  So it's singular in general.  And some other consequences of this strong convergence of v epsilon of this diffusing  component.  So we also have that f of u epsilon converges to v, but this is just the consequence of  this a priori estimate.  Because if f of u epsilon minus v epsilon over square root of epsilon is bounded, then  the quantity above converges to zero.  This is nothing important now.  But another thing that I can have from this strong convergence of diffusing component  is young measure of non-diffusing component.  So let me recall briefly that what is actually young measure.  So if I have a bounded function of u epsilon, then how to pass to the limit under nonlinearity?  How to pass to the weak limit under nonlinearity?  So you write this g of u epsilon as some artificial integral of function g against this young  measure concentrated at u epsilon.  And then this measure is bounded in total variation by one.  So you can pass to the weak star limit, of course, taking some subsequence.  And this measure we call this measure young measure.  And now you can use the fact that, OK, the first observation is that since f of u epsilon  converges to v epsilon and v epsilon converges strongly to v, then its young measure has  to be point musted v.  But on the other hand, if you denote by me the young measure of u epsilon, the young  measure of f of u epsilon has to be image of the first measure under the map f.  So what it follows that me has to be concentrated on the pre-image of v under the map f.  And now how this results in these oscillations?  Where if you recall the shape of the function f that I assumed, well, it looks like that.  It has increasing region.  This is this red part.  It has decreasing region.  This is the blue part.  And again, increasing region, this is yellow part.  And now if v is somewhere here between f minus and f plus, it hits three points.  There are three points such that f of u is equal to v.  And this is precisely this point where u is oscillating between.  So what follows from strong convergence of the diffusing component is that a young measure  of u epsilon is some combination of three Dirac masses, at most three Dirac masses.  In numerical simulations, we see two of them.  OK, so you see that all these results about this system with one diffusing component and  one component which is not diffusing, it all bases about proving the strong convergence  of v epsilon without information time derivative of v.  That's the main thing.  So the way to do that, this is more or less based on the compensated compactness approach.  So we'll first obtain some family of energies and then we'll play with weak limits and obtain  some identities that will be satisfied for many test functions and then we'll conclude.  So how do we go?  So recall the system.  We have again two equations.  One of them is without diffusion.  Another is with diffusion.  And OK, I fix some smooth function phi and I test it with phi of f of u epsilon.  Another equation I test with phi of v epsilon.  OK, so the result is as follows.  So here nothing changes.  I just multiplied.  Here I also nothing changes.  I just multiplied.  When it comes to this term from diffusion, you just play with derivatives to write it  like that.  And here you have to use primitive functions because to write it again as time derivative  of some function.  OK, I think it's clear.  And again, as with a priori estimates, these terms are still troublemakers because they  are singular.  But if you sum up these things, you obtain something like that.  So we have time derivative.  This is the red term.  You have Laplacian.  It is OK.  Now you have these two terms.  So as we know from a priori estimate, grad v is bounded.  And this is also bounded because if you use some Taylor estimate, you obtain v epsilon  minus f of u epsilon squared over epsilon.  So this is bounded, say, in L1.  OK, so what I get from this is that I have a priori estimates for these energies, namely  that time derivative is bounded in some negative sub-left space.  I need this h minus 1 because here is this Laplacian.  So if I multiply by function h1, I will move one derivative on this test function.  And this is exactly why I need h minus 1.  And I also know that this is just from a priori estimates because if v epsilon is in h1,  then any smooth function of v epsilon is also in h1.  OK.  So if I know this, I can conclude with the following compensated compactness lemma.  So how it goes.  I have this function here.  So this is precisely a function here.  And phi of v epsilon is here.  So it's a rather well-known result that if you have one of functions which is compact  in space, another one which is compact in time, then the product of weak limits is actually  weak limit of the product.  And this is more or less like something very similar to difficult lemma.  OK.  And from this, it's rather easy to obtain some identity on u epsilon.  So here I just rewritten what was on the previous slide and how to go now.  Well, we have to clean this a little bit.  So as you know that v epsilon converges to the same limit as f of u epsilon, you can  replace this v epsilon here.  Because if test functions are smooth, then they are in particular lip sheets.  So I can write it like that.  So now I have everywhere u epsilon.  And then this level, this is very similar to, for instance, analysis of vanishing viscosity  limit for measure solution to conservation law to get that the young measure is Dirac  mass.  But here is a small difference.  And I will point out this difference.  So usually you have just some function phi of lambda.  And here these functions are composed with f of lambda.  So it's not just phi of u epsilon, but for instance, here you have f of u epsilon.  And this f is also hidden because c prime is defined as phi of f.  So you have also to deal with this f.  And here comes another technical assumption on f that we somehow, we couldn't somehow  do that without this.  So the assumption is quite technical.  I will just state it in the simplest possible way.  So if you recall the plot of f, it looks like that has one increasing region, one decreasing,  and another one, which is increasing.  So you can define three differences of f, s1, s2, s3.  And now the assumption reads that these three derivatives have to be linearly independent.  So it's more or less like f has to be sufficiently curved.  In particular, this doesn't hold if f is some piecewise affine functions.  Because in this case, all these three inverses are just constants.  OK.  So I will spend the rest of my time to briefly comment, to briefly give some tribute to the  Russian mathematician Plotnikov, because it's more or less his ideas that we applied to  this problem.  So Plotnikov studied the following equation in 90s.  And so he considered forward-backward parabolic equation.  So here you have time derivative, here you have Laplacian.  And this f has the same monotonicity profile.  So two increasing regions and one between them, which is decreasing.  So you see that there is this unstable region where the equation is actually backward parabolic.  And this makes troubles.  And Plotnikov studied the following regularization among many, many, many authors.  So he adds this time derivative here with epsilon.  So you see that this is the good regularization, because if you move this term on the left-hand  side, you will have something like identity minus epsilon Laplace, which is invertible  as long as epsilon is greater than zero.  And then you can throw it again on the right-hand side, and you will have just simple OD in some  Banach spaces.  So as long as epsilon is non-zero, you can solve this equation.  So it's some kind of regularization.  And he also get the same result that the young measure of V epsilon converges.  And that the young measure of V epsilon is convex combination of young measures.  Precisely each mass is located in one of these regions of f.  So this is very similar result.  And our problem looks very similar to this Plotnikov regularization, because if you sum  up our equation, here we have something like V.  OK, here is something different.  But I will just tell you that it's not so much different that we could actually apply  similar techniques.  And OK, this paper is, I didn't find this paper in English.  It's in Russian.  And quite a big part of this work was to translating his paper from Russian to English.  If you are interested, I have English version typed in tech, because it's quite insightful  paper.  OK, and brief comments on the general problem that we would like to still to understand.  So two diffusing components, U and V. OK, so there are some estimates from Japanese  mathematical school, which allows us to control, again, this quantity and this quantity.  So it's not exactly the same.  Now V epsilon, there is no chance for all these convergences that I described to you  before, because we have simulations for that.  But there is something again similar that if you sum up these equations, what we see  in the simulations is that there is a chance for the strong convergence of this quantity.  So this is precisely what we get before, because if D1 is 0, this is exactly this quantity.  So this, my time.  So this is the chance that this will converge strongly.  But what you also see in simulations is that both U and V oscillates.  So it's not like in this case that at least V epsilon does not oscillate.  In this general problem, both of these quantities are oscillating.  OK, and with this I would like to thank you.  All right.
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Okay, so thank you very, very much for introduction. And of course, I would like to thank organizers Agnieszka and Mira for inviting me here. So the title seems to be a little bit biological, but actually it has some connections with conservation laws, with measure valued solutions, with young measures. So it's not entirely biological. And it fits a little bit into the topic of the workshop. So as a warm-up, let's consider system of reaction diffusion equations. So we have two components, u and v. You have diffusion of u and diffusion of v. And you have also this reaction term. And this reaction term is scaled with epsilon. So when epsilon is very small, the reaction is very fast. And the question is, what happens with this system, solutions to this system, when epsilon goes to zero? So let me first consider the classical case that is already done and was done around 20 years ago. So when f is strictly increasing, so f prime is strictly greater than zero, then you can denote by g primitive function of f. Then you multiply the first equation by f of u epsilon, the second by v epsilon. So it's a usual thing in PDEs. And then you obtain two energy equations. So you obtain the change of energy of u and the change of energy of v. Now at this level, this is useless because here you have term one over epsilon, which is in general singular. But what you can do with these two identities, you can sum them up. So this is what I do here. And at this level, these two terms, when they are added together, they result in this term. So this term has this advantage that it is controlled. It's sign is controlled. And this energy equality is now useful for understanding what happens in the limit because you have a lot of a priori estimates. So first, because you have that time derivative of this energy is negative, you have a priori estimates in L infinity. Then if you have a priori estimates in L infinity, then you use the fact that the three terms on the right hand side have negative sign. So f of u epsilon minus v epsilon converges to zero strongly. You also have estimates for gradients of both u and v. So in particular, the sum is bounded. You also have estimates for time derivative because if you sum up these equations, you obtain that the time derivative of the sum is sum of some Laplacian, which is bounded in some negative so-called space by estimation gradients. So you can use Aubame-Lion's lemma to have that the sum converges also strongly. And if you know that the sum converges strongly, you can write the sum of u epsilon and f of u epsilon as combination of two things that you know that converge strongly. And from this, you also have that u epsilon converges strongly, v epsilon converges strongly, and you also have some relation between v and u in the limit. So that's the rather classical things to have done by Betten, Hilhorst in 2003. And now the problem that me and Benoit were working on is the question, what happens when f prime can change sign? So when this reaction function can have some regions when it is decreasing. And now you see the main problem. The main problem is here that all these a priori estimates, they are now useless. They are not, this doesn't work because if f prime changes sign, then you don't have estimates on gradient of u, and you don't have estimates on gradient of v, and you don't have even this strong convergence of f of u epsilon. So now this energy approach doesn't work. So we do not know anything of this. So I will come back to this general problem later on, but then we started to study a simpler problem. So we removed this first diffusion. So we started to consider a system when one component is allowed to diffuse, this is v, and another component cannot diffuse, it's immobile. And this is already difficult problem as you will see in a minute, but it has one advantage that you can still use energy equality to study it, because if d1 is zero, this troublemaker term, it simply vanishes. So now you have, with this, you have estimates on both gradient of v, and you have also the strong convergence of f of u epsilon to v epsilon. So this is a little bit better. And now I will start to, I will now become more precise, and I will first tell you that due to some technicalities that will appear, I consider a special function f. So I assume that it has region where, when the function is increasing, region where it is decreasing, and another region where it is increasing. This is just to focus attention. And now I will show you some numerical simulation because I, before I will state the main result. So numerical simulations are as follows. So the blue line represents the concentration of u, so this is the non-diffusing component, and the red line represents the concentration of the diffusing component. So as you see, this blue line, it looks like an oscillating sequence with a young measure that is some combination of Dirac masses. This is more or less what you would guess if you looked at this picture. So but on the other hand, you see that v behaves rather nicely. And now I will state the result. So again, we have this system. Remember that I removed this one diffusion. And we have a priori estimate from energy equality on gradient of v, and we also have energy estimate for this term. And what we'll prove is that v epsilon converges strongly to v. So strongly, I mean both in time and space. And what is very interesting about this result, what I like a lot is that, you know, the result, this is without any estimates on derivative in time. So you have compactness in space, but you have actually no compactness in time. It's very similar to, for instance, vanishing viscosity in conservation laws. And in fact, if you can multiply this second equation with time derivative of v to realize that this time derivative of v behaves more or less like one over square root of epsilon. So it's singular in general. And some other consequences of this strong convergence of v epsilon of this diffusing component. So we also have that f of u epsilon converges to v, but this is just the consequence of this a priori estimate. Because if f of u epsilon minus v epsilon over square root of epsilon is bounded, then the quantity above converges to zero. This is nothing important now. But another thing that I can have from this strong convergence of diffusing component is young measure of non-diffusing component. So let me recall briefly that what is actually young measure. So if I have a bounded function of u epsilon, then how to pass to the limit under nonlinearity? How to pass to the weak limit under nonlinearity? So you write this g of u epsilon as some artificial integral of function g against this young measure concentrated at u epsilon. And then this measure is bounded in total variation by one. So you can pass to the weak star limit, of course, taking some subsequence. And this measure we call this measure young measure. And now you can use the fact that, OK, the first observation is that since f of u epsilon converges to v epsilon and v epsilon converges strongly to v, then its young measure has to be point musted v. But on the other hand, if you denote by me the young measure of u epsilon, the young measure of f of u epsilon has to be image of the first measure under the map f. So what it follows that me has to be concentrated on the pre-image of v under the map f. And now how this results in these oscillations? Where if you recall the shape of the function f that I assumed, well, it looks like that. It has increasing region. This is this red part. It has decreasing region. This is the blue part. And again, increasing region, this is yellow part. And now if v is somewhere here between f minus and f plus, it hits three points. There are three points such that f of u is equal to v. And this is precisely this point where u is oscillating between. So what follows from strong convergence of the diffusing component is that a young measure of u epsilon is some combination of three Dirac masses, at most three Dirac masses. In numerical simulations, we see two of them. OK, so you see that all these results about this system with one diffusing component and one component which is not diffusing, it all bases about proving the strong convergence of v epsilon without information time derivative of v. That's the main thing. So the way to do that, this is more or less based on the compensated compactness approach. So we'll first obtain some family of energies and then we'll play with weak limits and obtain some identities that will be satisfied for many test functions and then we'll conclude. So how do we go? So recall the system. We have again two equations. One of them is without diffusion. Another is with diffusion. And OK, I fix some smooth function phi and I test it with phi of f of u epsilon. Another equation I test with phi of v epsilon. OK, so the result is as follows. So here nothing changes. I just multiplied. Here I also nothing changes. I just multiplied. When it comes to this term from diffusion, you just play with derivatives to write it like that. And here you have to use primitive functions because to write it again as time derivative of some function. OK, I think it's clear. And again, as with a priori estimates, these terms are still troublemakers because they are singular. But if you sum up these things, you obtain something like that. So we have time derivative. This is the red term. You have Laplacian. It is OK. Now you have these two terms. So as we know from a priori estimate, grad v is bounded. And this is also bounded because if you use some Taylor estimate, you obtain v epsilon minus f of u epsilon squared over epsilon. So this is bounded, say, in L1. OK, so what I get from this is that I have a priori estimates for these energies, namely that time derivative is bounded in some negative sub-left space. I need this h minus 1 because here is this Laplacian. So if I multiply by function h1, I will move one derivative on this test function. And this is exactly why I need h minus 1. And I also know that this is just from a priori estimates because if v epsilon is in h1, then any smooth function of v epsilon is also in h1. OK. So if I know this, I can conclude with the following compensated compactness lemma. So how it goes. I have this function here. So this is precisely a function here. And phi of v epsilon is here. So it's a rather well-known result that if you have one of functions which is compact in space, another one which is compact in time, then the product of weak limits is actually weak limit of the product. And this is more or less like something very similar to difficult lemma. OK. And from this, it's rather easy to obtain some identity on u epsilon. So here I just rewritten what was on the previous slide and how to go now. Well, we have to clean this a little bit. So as you know that v epsilon converges to the same limit as f of u epsilon, you can replace this v epsilon here. Because if test functions are smooth, then they are in particular lip sheets. So I can write it like that. So now I have everywhere u epsilon. And then this level, this is very similar to, for instance, analysis of vanishing viscosity limit for measure solution to conservation law to get that the young measure is Dirac mass. But here is a small difference. And I will point out this difference. So usually you have just some function phi of lambda. And here these functions are composed with f of lambda. So it's not just phi of u epsilon, but for instance, here you have f of u epsilon. And this f is also hidden because c prime is defined as phi of f. So you have also to deal with this f. And here comes another technical assumption on f that we somehow, we couldn't somehow do that without this. So the assumption is quite technical. I will just state it in the simplest possible way. So if you recall the plot of f, it looks like that has one increasing region, one decreasing, and another one, which is increasing. So you can define three differences of f, s1, s2, s3. And now the assumption reads that these three derivatives have to be linearly independent. So it's more or less like f has to be sufficiently curved. In particular, this doesn't hold if f is some piecewise affine functions. Because in this case, all these three inverses are just constants. OK. So I will spend the rest of my time to briefly comment, to briefly give some tribute to the Russian mathematician Plotnikov, because it's more or less his ideas that we applied to this problem. So Plotnikov studied the following equation in 90s. And so he considered forward-backward parabolic equation. So here you have time derivative, here you have Laplacian. And this f has the same monotonicity profile. So two increasing regions and one between them, which is decreasing. So you see that there is this unstable region where the equation is actually backward parabolic. And this makes troubles. And Plotnikov studied the following regularization among many, many, many authors. So he adds this time derivative here with epsilon. So you see that this is the good regularization, because if you move this term on the left-hand side, you will have something like identity minus epsilon Laplace, which is invertible as long as epsilon is greater than zero. And then you can throw it again on the right-hand side, and you will have just simple OD in some Banach spaces. So as long as epsilon is non-zero, you can solve this equation. So it's some kind of regularization. And he also get the same result that the young measure of V epsilon converges. And that the young measure of V epsilon is convex combination of young measures. Precisely each mass is located in one of these regions of f. So this is very similar result. And our problem looks very similar to this Plotnikov regularization, because if you sum up our equation, here we have something like V. OK, here is something different. But I will just tell you that it's not so much different that we could actually apply similar techniques. And OK, this paper is, I didn't find this paper in English. It's in Russian. And quite a big part of this work was to translating his paper from Russian to English. If you are interested, I have English version typed in tech, because it's quite insightful paper. OK, and brief comments on the general problem that we would like to still to understand. So two diffusing components, U and V. OK, so there are some estimates from Japanese mathematical school, which allows us to control, again, this quantity and this quantity. So it's not exactly the same. Now V epsilon, there is no chance for all these convergences that I described to you before, because we have simulations for that. But there is something again similar that if you sum up these equations, what we see in the simulations is that there is a chance for the strong convergence of this quantity. So this is precisely what we get before, because if D1 is 0, this is exactly this quantity. So this, my time. So this is the chance that this will converge strongly. But what you also see in simulations is that both U and V oscillates. So it's not like in this case that at least V epsilon does not oscillate. In this general problem, both of these quantities are oscillating. OK, and with this I would like to thank you. All right.
let's consider system of reaction diffusion if you have a priori estimates in L infinity, sign? So when this reaction function can have line represents the concentration of the young measure of non-diffusing component. So let by one. So you can pass to the weak star limit, of the diffusing component is that a young with one diffusing component and one component which is not diffusing, it all bases about more or less based on the compensated compactness you have these two terms. So as we know from a result that if you have one of functions which is because it's more or less his ideas that we forward-backward parabolic equation. So here you chance for the strong convergence of this