Non-uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial data
Kreml, Ondřej
Chiodaroli, Elisabetta
We consider the isentropic Euler equations of gas dynamics in the whole two-dimensional space and we prove the existence of a $C^\infty$ initial datum which admits infinitely many bounded admissible weak solutions. Taking advantage of the relation between smooth solutions to the Euler system and to the Burgers equation we construct a smooth compression wave which collapses into a perturbed Riemann state at some time instant $T > 0$. In order to continue the solution after the formation of the discontinuity, we adjust and apply the theory developed by De Lellis and SzÃ
 So non uniqueness of admissible weak solutions to the compressible oil air equations with smooth initial data. Yeah, so thanks, Eduard, and thanks to the organizers for inviting me to this workshop. It's a pity that we cannot meet in Banff, but okay, I hope that we meet someday there again. So yeah, so this workshop should be about complex fluids. So I apologize that my talk is not really about complex fluids. But at least I'm glad that Alexis spoke ahead of me and also not really on the topic, so I don't feel that bad. Yeah, so my system of equations is also a very simple one. It's the compressible isentropic oil air system in the 2D space. So here the unknowns are the density and the velocity of the fluid. And the pressure which which appears here is a given function of rho and throughout my talk, the pressure will be actually rho squared. But it doesn't mean that the results I will be speaking about do not hold for for other pressures. And they hold for other power low pressures as well. So this is just a choice for simplicity. And yeah, so this is hyperbolic system of conservation laws. And as you may know, the existence theory for hyperbolic systems of conservation laws is still quite open and there are some satisfactory results only in case of 1D or in case of systems in 1D or in a single equation in several space dimensions. However, as you see, I will be talking about the system in multiple space dimensions. And there the existence theory has some gaps still. But in particular, already for a simple equation as is the Burgers equation, it is well known that the hyperbolic equations do not need to have unique solutions and some additional conditions have to be imposed in order to select a unique solution. So in our case as well, we will talk about admissible solutions. So these will be solutions that satisfy this sort of energy inequality, which in the terminology of hyperbolic system of conservation laws is actually the entropy inequality because this quantity here, which is the total energy of the system, plays the role of the mathematical entropy. So here this epsilon of rho is related to the pressure through this formula. Okay, so the whole content of my talk stems from the results of Camillo de Lellis and Laszlo Sekalihidi. So following their groundbreaking works in 2009 and 2010, there appeared a whole tree of various results related to their results. And I had the privilege to be involved in a couple of branches of this tree. And one of the branches of this tree is actually the following. So already in their second paper on the topic in 2010, they actually showed that there exists initial data for the compressible or large system, rho zero and v zero, which are just bounded such that there are infinitely many bounded admissible solutions. So this in particular, suggesting that the admissibility condition, the energy inequality or entropy inequality, if you wish, does not play the same role as, for example, in one dimension, and it's not enough to use this to select a unique solution. So this was for initial data in L infinity. And then concerning the questions about how regular the initial data can be in such a way that this pill poseness appear. The next step was due to Elizabeth Accio Daroli and also Edward Fieryzel, who showed in 2014 that you can have actually a regular density rho zero in C1 and find an irregular initial velocity v zero, which is still just L infinity in such a way that you get essentially the same result, which is ill poseness of bounded admissible solutions. And then together with Elizabeth Accio Daroli and Camilo Del Alice, we have shown that actually you can have ellipsoid initial data, rho zero and v zero, such that you have still the same result, that you have infinitely many bounded admissible solutions. So at this point, you may say, okay, well, this is weird because we have weak strong uniqueness which holds here. And you will be right. The point is that you have the weak strong uniqueness which holds only on the finite time interval as long as the strong solution exists. And once the strong solution fails to exist and some discontinuity appears, then of course you don't have a weak strong uniqueness anymore. And actually you can show that there may exist infinitely many weak solutions, starting from still the same initial data, which is exactly what is written here in our theorem from 2015. So we have shown that there exists ellipsoid initial data such that there are infinitely many bounded admissible weak solutions. However, these solutions all are locally ellipsoid on some finite interval of time and they all coincide with the unique classical solution here. So our proof was based on the analysis of the Riemann problem and the suitable application of the theory of Camillo and Laszlo, which was developed originally for the incompressible Euler equations. So what is the Riemann problem? I just recall here that the Riemann problem is a special type of initial value problem where the initial values are actually piecewise constant. So for negative x2s we have a pair of constants rho minus and v minus and for positive x2s we have another pair of constants rho plus and v plus and we call this the Riemann initial data and as you can see these initial data are one dimensional, they depend only on x2 and not on x1. And you can use the 1D theory which is a well-established theory to show that there exists self-similar solutions to this Riemann problem, which are all of them of course one dimensional and actually they depend only on the single variable x2 over t. But of course you may ask whether you have also other solutions which are not one dimensional and the answer is for certain values of these constants that indeed you have other solutions as well. So here is our main theorem which is due in our joint paper with Elisabetta Ciodaroli, Valslav Macha and Sebastian Schwarzacher, which says the following that we can actually have as was hinted in the title of the talk the initial data which are the infinity such that there exist infinitely many bounded admissible weak solutions to the Euler system. There is a slight catch here because our solutions we cannot prove or at least we didn't succeed in proving that you have these solutions global in time, so our solutions are defined on time interval between 0 and some capital T plus delta 0, where this capital T is the time of existence of the unique smooth solutions which starts from this initial data. So essentially what happens is that at time capital T the smooth solution which is unique develops discontinuity and you can continue this solution in a highly non-unique way but only at least we are able to prove it only for a short period of time of the length delta 0. So what are the ideas of our proof? The idea is the following. So we have the Lipschitz theorem, we have the Riemann data there and the idea there was to take Riemann data which were generated by a compression wave. So the compression wave is Lipschitz and therefore of course that the Riemann worked for Lipschitz initial data and we know that for this we have infinitely many weak solutions. So the idea is somehow natural, we smooth out the compression wave and see what happens and indeed this yields a sort of generalized Riemann problem where you no longer have piecewise constant initial data at the time of the discontinuity and therefore you have to generalize the proof, introduce some generalized definitions of substitution and so on and calculate a lot but in the end you can use the key convex integration lemma from Delalis and Czecklihidi to show that to prove the existence of infinitely many admissible weak solutions it is enough to find a single admissible found subsolution where of course the subsolution is an object which is suitably defined. Okay so of course I don't have time to go through the whole proof here so I will just speak about what was maybe for me at least the more fun part of the proof which was actually the first part how to construct the compression wave which is C infinity instead of Lipschitz in order to make the proof work because then again the convex integration part is somehow technical and not really that interesting. So what is a smooth compression wave? So here we use the well-known relation between the Euler equation and the Burgers equation namely the fact that if we take one function here called w1 which I think is called Riemann invariant which if we take this to be constant then another quantity which is called here lambda1 which is the characteristic speed for the Euler equation then this quantity solves the Burgers equation and the point is that if you have smooth solutions to Burgers and you define this w1 to be constant then of course you produce smooth solutions to the Euler equation and in particular since these are smooth objects they satisfy the energy inequality with the equality sign so there is no problem there. So this is what we call the compression wave I mean the simple picture you have two states lambda minus which is bigger than lambda plus and they are connected with a straight line so you can see that this guy is Lipschitz and it's not smoother due to these points here and here and if you let evolve this through the Burgers equation you know that at a certain time, finite time which is given by the slope of this line you develop discontinuity, discontinuity between lambda minus and lambda plus. So what we need to do is to smooth this out so we introduce the initial data which will be in the end the initial data for our Euler equations as is written here so maybe instead of looking for a long time on this formula I can show you the picture what happens here and what happens is you keep the shape but instead of these corners which were not smooth you introduce some smooth functions which I call here f0 minus and f0 plus which of course are such that what you create here is the infinity so that's the idea how to proceed and then the question of course is what happens if you let this evolve through the Burgers equation again and the point is what happens is that at time t equals capital T again a discontinuity appears and the solution at the time capital T looks like this where again I can simply show you the picture how it looks like so you have a jump here between these two values and there are neighborhoods of this point on the left and on the right where you have some functions which okay I call here inaccurately f0 but f0 will be a little bit different thing next. So essentially the question now was how this functions here in the neighborhood of this continuity look like and the answer is that one type of a function which you can get there is this one so if you take the function here which is written here 1 minus 2 over pi of arc of tangents of log x then you can use this function to construct backwards somehow see infinity initial data so the lemma is here and again it's maybe a little bit technical but what you can see is that you use this function f0 which is this 1 minus arc of tangents of log x and say that if you have on the neighborhood of the discontinuity these functions then there exists initial data which are c infinity which produce solutions to the burgers equation which are c infinity up to the time t and at time capital T the solution to burgers has the following has the form which was which was written before so this was a good function to to look for and this function has all the good properties we need and this may be a maybe a well-known well-known property somewhere but we were looking for this in the literature and we were not able to find actually examples of such functions and we had to sort of figure out ourselves how to how to do it and as I said this was the more fun part of the proof to actually actually find this because what we needed later in the proof was indeed to know exactly how the function behaves around this discontinuity and so now we know now we know that using the burgers equation we can use our our nice function which is written here and then okay let's go back to the to the idea sorry let's go back to the to the ideas of the proof and okay so we have now our smooth compression wave and we end up at the time t equals capital T with the sort of generalized Riemann problem so this is the problem where you have this discontinuity already and you ask yourself how to proceed further with solutions to the Euler equations and here comes the part where you where you use this convex integration stuff so just briefly because I will run out of time in like three minutes I will show you so first of all we instead of velocity we use we use momentum here but this is not really a big big issue because all the way through through the through the paper we always assume that the density is bounded from below by some constant so this is not really an issue and we introduce first of all something which is called a generalized fun partition of our space time so the generalized fun partition consists of three sets so essentially what we do is we split our space time starting from the time where we have the discontinuity into three into three regions so one of the region is P minus and the other one is P plus which are here and they are separated by curves which okay which are functions of time here so this new minus tilde and new plus tilde they define the generalized fun partition and then using this fun partition we introduce something which is called generalized fun subsolution and I will not go through the details here but it is a piecewise continuous object or piecewise differentiable object which has jumps across these curves so actually what happens is that the fun the fun subsolution actually coincides with the solution to the Euler equation in this P minus region and in this P plus region and it has a jump discontinuity across these curves and here in this region P1 it is some object which which we called a strict subsolution which satisfies a relaxed version of the Euler equations which is written here together with some point wise inequality between these functions Mu and rho and C which are all parts of the generalized fun subsolution and then the rest of the story is is simple yeah and moreover of course you need to you need to define the subsolution to be admissible if you want to if you want to end up with admissible solutions to Euler and then the rest of the story is simply to find the admissible fun subsolution because the work of Delalis and Sekalihidi tells you that if you find just one subsolution then you have infinitely many solutions and yeah after a lot of technical stuff you can indeed do that and find the admissible fun subsolution with the initial data which I which I told you before which are generated by the smooth compression way and you end up with the result and that is actually the end of my talk so thank you for your attention. Thanks a lot Andrei.
 So non uniqueness of admissible weak solutions to the compressible oil air equations with  smooth initial data.  Yeah, so thanks, Eduard, and thanks to the organizers for inviting me to this workshop.  It's a pity that we cannot meet in Banff, but okay, I hope that we meet someday there  again.  So yeah, so this workshop should be about complex fluids.  So I apologize that my talk is not really about complex fluids.  But at least I'm glad that Alexis spoke ahead of me and also not really on the topic, so  I don't feel that bad.  Yeah, so my system of equations is also a very simple one.  It's the compressible isentropic oil air system in the 2D space.  So here the unknowns are the density and the velocity of the fluid.  And the pressure which which appears here is a given function of rho and throughout  my talk, the pressure will be actually rho squared.  But it doesn't mean that the results I will be speaking about do not hold for for other  pressures.  And they hold for other power low pressures as well.  So this is just a choice for simplicity.  And yeah, so this is hyperbolic system of conservation laws.  And as you may know, the existence theory for hyperbolic systems of conservation laws  is still quite open and there are some satisfactory results only in case of 1D or in case of systems  in 1D or in a single equation in several space dimensions.  However, as you see, I will be talking about the system in multiple space dimensions.  And there the existence theory has some gaps still.  But in particular, already for a simple equation as is the Burgers equation, it is well known  that the hyperbolic equations do not need to have unique solutions and some additional  conditions have to be imposed in order to select a unique solution.  So in our case as well, we will talk about admissible solutions.  So these will be solutions that satisfy this sort of energy inequality, which in the terminology  of hyperbolic system of conservation laws is actually the entropy inequality because  this quantity here, which is the total energy of the system, plays the role of the mathematical  entropy.  So here this epsilon of rho is related to the pressure through this formula.  Okay, so the whole content of my talk stems from the results of Camillo de Lellis and  Laszlo Sekalihidi.  So following their groundbreaking works in 2009 and 2010, there appeared a whole tree  of various results related to their results.  And I had the privilege to be involved in a couple of branches of this tree.  And one of the branches of this tree is actually the following.  So already in their second paper on the topic in 2010, they actually showed that there exists  initial data for the compressible or large system, rho zero and v zero, which are just  bounded such that there are infinitely many bounded admissible solutions.  So this in particular, suggesting that the admissibility condition, the energy inequality  or entropy inequality, if you wish, does not play the same role as, for example, in one  dimension, and it's not enough to use this to select a unique solution.  So this was for initial data in L infinity.  And then concerning the questions about how regular the initial data can be in such a  way that this pill poseness appear.  The next step was due to Elizabeth Accio Daroli and also Edward Fieryzel, who showed in 2014  that you can have actually a regular density rho zero in C1 and find an irregular initial  velocity v zero, which is still just L infinity in such a way that you get essentially the  same result, which is ill poseness of bounded admissible solutions.  And then together with Elizabeth Accio Daroli and Camilo Del Alice, we have shown that actually  you can have ellipsoid initial data, rho zero and v zero, such that you have still the same  result, that you have infinitely many bounded admissible solutions.  So at this point, you may say, okay, well, this is weird because we have weak strong  uniqueness which holds here.  And you will be right.  The point is that you have the weak strong uniqueness which holds only on the finite  time interval as long as the strong solution exists.  And once the strong solution fails to exist and some discontinuity appears, then of course  you don't have a weak strong uniqueness anymore.  And actually you can show that there may exist infinitely many weak solutions, starting from  still the same initial data, which is exactly what is written here in our theorem from 2015.  So we have shown that there exists ellipsoid initial data such that there are infinitely  many bounded admissible weak solutions.  However, these solutions all are locally ellipsoid on some finite interval of time and they all  coincide with the unique classical solution here.  So our proof was based on the analysis of the Riemann problem and the suitable application  of the theory of Camillo and Laszlo, which was developed originally for the incompressible  Euler equations.  So what is the Riemann problem?  I just recall here that the Riemann problem is a special type of initial value problem  where the initial values are actually piecewise constant.  So for negative x2s we have a pair of constants rho minus and v minus and for positive x2s  we have another pair of constants rho plus and v plus and we call this the Riemann initial  data and as you can see these initial data are one dimensional, they depend only on x2  and not on x1.  And you can use the 1D theory which is a well-established theory to show that there exists self-similar  solutions to this Riemann problem, which are all of them of course one dimensional and  actually they depend only on the single variable x2 over t.  But of course you may ask whether you have also other solutions which are not one dimensional  and the answer is for certain values of these constants that indeed you have other solutions  as well.  So here is our main theorem which is due in our joint paper with Elisabetta Ciodaroli,  Valslav Macha and Sebastian Schwarzacher, which says the following that we can actually  have as was hinted in the title of the talk the initial data which are the infinity such  that there exist infinitely many bounded admissible weak solutions to the Euler system.  There is a slight catch here because our solutions we cannot prove or at least we didn't succeed  in proving that you have these solutions global in time, so our solutions are defined on time  interval between 0 and some capital T plus delta 0, where this capital T is the time of  existence of the unique smooth solutions which starts from this initial data.  So essentially what happens is that at time capital T the smooth solution which is unique  develops discontinuity and you can continue this solution in a highly non-unique way but  only at least we are able to prove it only for a short period of time of the length delta  0.  So what are the ideas of our proof?  The idea is the following.  So we have the Lipschitz theorem, we have the Riemann data there and the idea there  was to take Riemann data which were generated by a compression wave.  So the compression wave is Lipschitz and therefore of course that the Riemann worked for Lipschitz  initial data and we know that for this we have infinitely many weak solutions.  So the idea is somehow natural, we smooth out the compression wave and see what happens  and indeed this yields a sort of generalized Riemann problem where you no longer have piecewise  constant initial data at the time of the discontinuity and therefore you have to generalize the proof,  introduce some generalized definitions of substitution and so on and calculate a lot  but in the end you can use the key convex integration lemma from Delalis and Czecklihidi  to show that to prove the existence of infinitely many admissible weak solutions it is enough  to find a single admissible found subsolution where of course the subsolution is an object  which is suitably defined.  Okay so of course I don't have time to go through the whole proof here so I will just  speak about what was maybe for me at least the more fun part of the proof which was actually  the first part how to construct the compression wave which is C infinity instead of Lipschitz  in order to make the proof work because then again the convex integration part is somehow  technical and not really that interesting.  So what is a smooth compression wave?  So here we use the well-known relation between the Euler equation and the Burgers equation  namely the fact that if we take one function here called w1 which I think is called Riemann  invariant which if we take this to be constant then another quantity which is called here  lambda1 which is the characteristic speed for the Euler equation then this quantity  solves the Burgers equation and the point is that if you have smooth solutions to Burgers  and you define this w1 to be constant then of course you produce smooth solutions to  the Euler equation and in particular since these are smooth objects they satisfy the  energy inequality with the equality sign so there is no problem there.  So this is what we call the compression wave I mean the simple picture you have two states  lambda minus which is bigger than lambda plus and they are connected with a straight line  so you can see that this guy is Lipschitz and it's not smoother due to these points  here and here and if you let evolve this through the Burgers equation you know that  at a certain time, finite time which is given by the slope of this line you develop discontinuity,  discontinuity between lambda minus and lambda plus. So what we need to do is to smooth this  out so we introduce the initial data which will be in the end the initial data for our  Euler equations as is written here so maybe instead of looking for a long time on this  formula I can show you the picture what happens here and what happens is you keep the shape  but instead of these corners which were not smooth you introduce some smooth functions  which I call here f0 minus and f0 plus which of course are such that what you create here  is the infinity so that's the idea how to proceed and then the question of course is  what happens if you let this evolve through the Burgers equation again and the point is  what happens is that at time t equals capital T again a discontinuity appears and the solution  at the time capital T looks like this where again I can simply show you the picture how  it looks like so you have a jump here between these two values and there are neighborhoods  of this point on the left and on the right where you have some functions which okay I  call here inaccurately f0 but f0 will be a little bit different thing next. So essentially  the question now was how this functions here in the neighborhood of this continuity look  like and the answer is that one type of a function which you can get there is this one  so if you take the function here which is written here 1 minus 2 over pi of arc of  tangents of log x then you can use this function to construct backwards somehow see infinity  initial data so the lemma is here and again it's maybe a little bit technical but what  you can see is that you use this function f0 which is this 1 minus arc of tangents of  log x and say that if you have on the neighborhood of the discontinuity these functions then there  exists initial data which are c infinity which produce solutions to the burgers equation  which are c infinity up to the time t and at time capital T the solution to burgers  has the following has the form which was which was written before so this was a good function  to to look for and this function has all the good properties we need and this may be a  maybe a well-known well-known property somewhere but we were looking for this in the literature  and we were not able to find actually examples of such functions and we had to sort of figure  out ourselves how to how to do it and as I said this was the more fun part of the proof  to actually actually find this because what we needed later in the proof was indeed to  know exactly how the function behaves around this discontinuity and so now we know now we  know that using the burgers equation we can use our our nice function which is written  here and then okay let's go back to the to the idea sorry let's go back to the to the  ideas of the proof and okay so we have now our smooth compression wave and we end up  at the time t equals capital T with the sort of generalized Riemann problem so this is  the problem where you have this discontinuity already and you ask yourself how to proceed  further with solutions to the Euler equations and here comes the part where you where you  use this convex integration stuff so just briefly because I will run out of time in  like three minutes I will show you so first of all we instead of velocity we use we use  momentum here but this is not really a big big issue because all the way through through  the through the paper we always assume that the density is bounded from below by some  constant so this is not really an issue and we introduce first of all something which  is called a generalized fun partition of our space time so the generalized fun partition  consists of three sets so essentially what we do is we split our space time starting  from the time where we have the discontinuity into three into three regions so one of the  region is P minus and the other one is P plus which are here and they are separated by curves  which okay which are functions of time here so this new minus tilde and new plus tilde  they define the generalized fun partition and then using this fun partition we introduce  something which is called generalized fun subsolution and I will not go through the details here  but it is a piecewise continuous object or piecewise differentiable object which has jumps  across these curves so actually what happens is that the fun the fun subsolution actually  coincides with the solution to the Euler equation in this P minus region and in this  P plus region and it has a jump discontinuity across these curves and here in this region  P1 it is some object which which we called a strict subsolution which satisfies a relaxed  version of the Euler equations which is written here together with some point wise inequality  between these functions Mu and rho and C which are all parts of the generalized fun subsolution  and then the rest of the story is is simple yeah and moreover of course you need to you  need to define the subsolution to be admissible if you want to if you want to end up with  admissible solutions to Euler and then the rest of the story is simply to find the admissible  fun subsolution because the work of Delalis and Sekalihidi tells you that if you find  just one subsolution then you have infinitely many solutions and yeah after a lot of technical  stuff you can indeed do that and find the admissible fun subsolution with the initial  data which I which I told you before which are generated by the smooth compression way  and you end up with the result and that is actually the end of my talk so thank you for  your attention.  Thanks a lot Andrei.
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So non uniqueness of admissible weak solutions to the compressible oil air equations with smooth initial data. Yeah, so thanks, Eduard, and thanks to the organizers for inviting me to this workshop. It's a pity that we cannot meet in Banff, but okay, I hope that we meet someday there again. So yeah, so this workshop should be about complex fluids. So I apologize that my talk is not really about complex fluids. But at least I'm glad that Alexis spoke ahead of me and also not really on the topic, so I don't feel that bad. Yeah, so my system of equations is also a very simple one. It's the compressible isentropic oil air system in the 2D space. So here the unknowns are the density and the velocity of the fluid. And the pressure which which appears here is a given function of rho and throughout my talk, the pressure will be actually rho squared. But it doesn't mean that the results I will be speaking about do not hold for for other pressures. And they hold for other power low pressures as well. So this is just a choice for simplicity. And yeah, so this is hyperbolic system of conservation laws. And as you may know, the existence theory for hyperbolic systems of conservation laws is still quite open and there are some satisfactory results only in case of 1D or in case of systems in 1D or in a single equation in several space dimensions. However, as you see, I will be talking about the system in multiple space dimensions. And there the existence theory has some gaps still. But in particular, already for a simple equation as is the Burgers equation, it is well known that the hyperbolic equations do not need to have unique solutions and some additional conditions have to be imposed in order to select a unique solution. So in our case as well, we will talk about admissible solutions. So these will be solutions that satisfy this sort of energy inequality, which in the terminology of hyperbolic system of conservation laws is actually the entropy inequality because this quantity here, which is the total energy of the system, plays the role of the mathematical entropy. So here this epsilon of rho is related to the pressure through this formula. Okay, so the whole content of my talk stems from the results of Camillo de Lellis and Laszlo Sekalihidi. So following their groundbreaking works in 2009 and 2010, there appeared a whole tree of various results related to their results. And I had the privilege to be involved in a couple of branches of this tree. And one of the branches of this tree is actually the following. So already in their second paper on the topic in 2010, they actually showed that there exists initial data for the compressible or large system, rho zero and v zero, which are just bounded such that there are infinitely many bounded admissible solutions. So this in particular, suggesting that the admissibility condition, the energy inequality or entropy inequality, if you wish, does not play the same role as, for example, in one dimension, and it's not enough to use this to select a unique solution. So this was for initial data in L infinity. And then concerning the questions about how regular the initial data can be in such a way that this pill poseness appear. The next step was due to Elizabeth Accio Daroli and also Edward Fieryzel, who showed in 2014 that you can have actually a regular density rho zero in C1 and find an irregular initial velocity v zero, which is still just L infinity in such a way that you get essentially the same result, which is ill poseness of bounded admissible solutions. And then together with Elizabeth Accio Daroli and Camilo Del Alice, we have shown that actually you can have ellipsoid initial data, rho zero and v zero, such that you have still the same result, that you have infinitely many bounded admissible solutions. So at this point, you may say, okay, well, this is weird because we have weak strong uniqueness which holds here. And you will be right. The point is that you have the weak strong uniqueness which holds only on the finite time interval as long as the strong solution exists. And once the strong solution fails to exist and some discontinuity appears, then of course you don't have a weak strong uniqueness anymore. And actually you can show that there may exist infinitely many weak solutions, starting from still the same initial data, which is exactly what is written here in our theorem from 2015. So we have shown that there exists ellipsoid initial data such that there are infinitely many bounded admissible weak solutions. However, these solutions all are locally ellipsoid on some finite interval of time and they all coincide with the unique classical solution here. So our proof was based on the analysis of the Riemann problem and the suitable application of the theory of Camillo and Laszlo, which was developed originally for the incompressible Euler equations. So what is the Riemann problem? I just recall here that the Riemann problem is a special type of initial value problem where the initial values are actually piecewise constant. So for negative x2s we have a pair of constants rho minus and v minus and for positive x2s we have another pair of constants rho plus and v plus and we call this the Riemann initial data and as you can see these initial data are one dimensional, they depend only on x2 and not on x1. And you can use the 1D theory which is a well-established theory to show that there exists self-similar solutions to this Riemann problem, which are all of them of course one dimensional and actually they depend only on the single variable x2 over t. But of course you may ask whether you have also other solutions which are not one dimensional and the answer is for certain values of these constants that indeed you have other solutions as well. So here is our main theorem which is due in our joint paper with Elisabetta Ciodaroli, Valslav Macha and Sebastian Schwarzacher, which says the following that we can actually have as was hinted in the title of the talk the initial data which are the infinity such that there exist infinitely many bounded admissible weak solutions to the Euler system. There is a slight catch here because our solutions we cannot prove or at least we didn't succeed in proving that you have these solutions global in time, so our solutions are defined on time interval between 0 and some capital T plus delta 0, where this capital T is the time of existence of the unique smooth solutions which starts from this initial data. So essentially what happens is that at time capital T the smooth solution which is unique develops discontinuity and you can continue this solution in a highly non-unique way but only at least we are able to prove it only for a short period of time of the length delta 0. So what are the ideas of our proof? The idea is the following. So we have the Lipschitz theorem, we have the Riemann data there and the idea there was to take Riemann data which were generated by a compression wave. So the compression wave is Lipschitz and therefore of course that the Riemann worked for Lipschitz initial data and we know that for this we have infinitely many weak solutions. So the idea is somehow natural, we smooth out the compression wave and see what happens and indeed this yields a sort of generalized Riemann problem where you no longer have piecewise constant initial data at the time of the discontinuity and therefore you have to generalize the proof, introduce some generalized definitions of substitution and so on and calculate a lot but in the end you can use the key convex integration lemma from Delalis and Czecklihidi to show that to prove the existence of infinitely many admissible weak solutions it is enough to find a single admissible found subsolution where of course the subsolution is an object which is suitably defined. Okay so of course I don't have time to go through the whole proof here so I will just speak about what was maybe for me at least the more fun part of the proof which was actually the first part how to construct the compression wave which is C infinity instead of Lipschitz in order to make the proof work because then again the convex integration part is somehow technical and not really that interesting. So what is a smooth compression wave? So here we use the well-known relation between the Euler equation and the Burgers equation namely the fact that if we take one function here called w1 which I think is called Riemann invariant which if we take this to be constant then another quantity which is called here lambda1 which is the characteristic speed for the Euler equation then this quantity solves the Burgers equation and the point is that if you have smooth solutions to Burgers and you define this w1 to be constant then of course you produce smooth solutions to the Euler equation and in particular since these are smooth objects they satisfy the energy inequality with the equality sign so there is no problem there. So this is what we call the compression wave I mean the simple picture you have two states lambda minus which is bigger than lambda plus and they are connected with a straight line so you can see that this guy is Lipschitz and it's not smoother due to these points here and here and if you let evolve this through the Burgers equation you know that at a certain time, finite time which is given by the slope of this line you develop discontinuity, discontinuity between lambda minus and lambda plus. So what we need to do is to smooth this out so we introduce the initial data which will be in the end the initial data for our Euler equations as is written here so maybe instead of looking for a long time on this formula I can show you the picture what happens here and what happens is you keep the shape but instead of these corners which were not smooth you introduce some smooth functions which I call here f0 minus and f0 plus which of course are such that what you create here is the infinity so that's the idea how to proceed and then the question of course is what happens if you let this evolve through the Burgers equation again and the point is what happens is that at time t equals capital T again a discontinuity appears and the solution at the time capital T looks like this where again I can simply show you the picture how it looks like so you have a jump here between these two values and there are neighborhoods of this point on the left and on the right where you have some functions which okay I call here inaccurately f0 but f0 will be a little bit different thing next. So essentially the question now was how this functions here in the neighborhood of this continuity look like and the answer is that one type of a function which you can get there is this one so if you take the function here which is written here 1 minus 2 over pi of arc of tangents of log x then you can use this function to construct backwards somehow see infinity initial data so the lemma is here and again it's maybe a little bit technical but what you can see is that you use this function f0 which is this 1 minus arc of tangents of log x and say that if you have on the neighborhood of the discontinuity these functions then there exists initial data which are c infinity which produce solutions to the burgers equation which are c infinity up to the time t and at time capital T the solution to burgers has the following has the form which was which was written before so this was a good function to to look for and this function has all the good properties we need and this may be a maybe a well-known well-known property somewhere but we were looking for this in the literature and we were not able to find actually examples of such functions and we had to sort of figure out ourselves how to how to do it and as I said this was the more fun part of the proof to actually actually find this because what we needed later in the proof was indeed to know exactly how the function behaves around this discontinuity and so now we know now we know that using the burgers equation we can use our our nice function which is written here and then okay let's go back to the to the idea sorry let's go back to the to the ideas of the proof and okay so we have now our smooth compression wave and we end up at the time t equals capital T with the sort of generalized Riemann problem so this is the problem where you have this discontinuity already and you ask yourself how to proceed further with solutions to the Euler equations and here comes the part where you where you use this convex integration stuff so just briefly because I will run out of time in like three minutes I will show you so first of all we instead of velocity we use we use momentum here but this is not really a big big issue because all the way through through the through the paper we always assume that the density is bounded from below by some constant so this is not really an issue and we introduce first of all something which is called a generalized fun partition of our space time so the generalized fun partition consists of three sets so essentially what we do is we split our space time starting from the time where we have the discontinuity into three into three regions so one of the region is P minus and the other one is P plus which are here and they are separated by curves which okay which are functions of time here so this new minus tilde and new plus tilde they define the generalized fun partition and then using this fun partition we introduce something which is called generalized fun subsolution and I will not go through the details here but it is a piecewise continuous object or piecewise differentiable object which has jumps across these curves so actually what happens is that the fun the fun subsolution actually coincides with the solution to the Euler equation in this P minus region and in this P plus region and it has a jump discontinuity across these curves and here in this region P1 it is some object which which we called a strict subsolution which satisfies a relaxed version of the Euler equations which is written here together with some point wise inequality between these functions Mu and rho and C which are all parts of the generalized fun subsolution and then the rest of the story is is simple yeah and moreover of course you need to you need to define the subsolution to be admissible if you want to if you want to end up with admissible solutions to Euler and then the rest of the story is simply to find the admissible fun subsolution because the work of Delalis and Sekalihidi tells you that if you find just one subsolution then you have infinitely many solutions and yeah after a lot of technical stuff you can indeed do that and find the admissible fun subsolution with the initial data which I which I told you before which are generated by the smooth compression way and you end up with the result and that is actually the end of my talk so thank you for your attention. Thanks a lot Andrei.
be imposed in order to select a unique solution. that you have infinitely many bounded admissible admissible weak solutions to the Euler system. time interval between 0 and some capital T plus existence of the unique smooth solutions which the time of the discontinuity and therefore you Czecklihidi to show that to prove the existence construct the compression wave which is C relation between the Euler equation and the the Burgers equation and the point is that if you smooth compression wave and we end up at the time