On the behavior of approximate solutions to conservation laws near singularities

Okay, so welcome back for the final session of the morning. It's my pleasure to introduce the Italian grandmaster, Alberto Bressan. Who will speak on the local asymptotic behavior of viscous approximations of conservation laws. Okay. Well, so thanks for the introduction and to the organizer for the invitation. Can you hear this? Okay, looks good. Okay, so in this talk want to discuss family of problems related to how various type of approximations behave in neighborhood of singularities. And singularities mean singularity of solution to either scalar conservation law or system of conservation law in one space dimensions. Okay, so let's see. So let me first talk about what is generic singularity. That's something well known. So let's consider scalar conservation law. Let's see UT plus of UX where is smooth convex flux. And we say that well, one result which is well known is that for generic initial data, the solution is smooth outside locally finite number of shocks, shock waves, and singular points. so, what does it mean generic? Well, we say that property is generic if it holds for all initial data, let's say in this case would be in C3. where is intersection of countably many open dense set in C3. So, it's it holds on topologically large set of initial data. Okay. by Baire category theorem. And the main result in this direction, which is often quoted, is this by Scheffer, regularity theorem for conservation law. It was back in '73. Okay, so if you take almost any C3 initial data for scalar conservation law, the solution will have locally finitely many shocks. Okay. since I'm talking about this, you might wonder what happens this for for systems of conservation law. it turns out that for 3 by 3 systems, or by system, it doesn't work. And well, you can find counterexample in this paper by Caravenna and Spinolo. so, saying that it cannot work for 3 by 3 system, and it's fairly simple to understand why cannot work, because 3 by 3 system typically, when you have two jumps that come together, then you have three jumps that get out, right? So, the number of jumps in solution can become unbounded in finite time. And if you have diagram like this, there is no way to change little bit the initial data to eliminate this number of shocks. Okay. So, it's hopeless to have genetic regulatory result for 3 by 3 system. difficult open problem is the borderline case, 2 by 2. For 2 by 2 systems, well, in principle, you have two shocks coming in, two shocks going out. So, diagram like this, of course, is not would not be possible. So, new shocks can only be born by when compression waves actually break. And depending on the initial data, this hopefully will happen only finitely many times, but so far no one has been able to prove this, okay? So, this is in fact hard open problem to extend Scheffer result to 2 by 2 systems or prove it doesn't hold, okay? Okay. So, generic singularity for solution to scalar conservation law of convex flux are well known. So, these are, let's say, points along shock, point where new shock two shocks sorry, point where new shocks forms, or point where two shocks merge together. So, what want to discuss is essentially describe the local behavior of the solution near these singularity points. And more deeper, actually, more challenging is understand how various type of approximation behave in neighborhood of these points, okay? Well, the fact that this generic regularity result does not hold for by system doesn't mean that you cannot consider these problems also for by system. So, instead of thinking of generic singularity, you can think of structurally stable singularity, so singularity that arise from an open set of initial data. And these are sufficiently regular or you There is hope to classify this singularity within finite number of parameters. Okay, so let's see, how do we classify this different type of singularity? Well, the technique is also very well known. So, one thing you can do is do local rescaling, which essentially amounts to take microscope and enlarge the solution in neighborhood of this point. So, for example, the simplest type is along shock curve in x-t space. You take point on the shock curve. And what you do, you rescale time, you rescale space. And as epsilon goes to zero, it means you take this microscope and enlarge more and more. What do you see? Well, you take you see essentially single shock. So, you see straight line and two constant state, minus plus. And the speed of this, of course, is the Rankine-Hugoniot speed of the shock. Okay. Similar thing you can do for point where two shocks meet. Let's say you have left, middle, and right state. left, middle, right. And then you do exactly the same type of rescaling. And in this case, this the rescale function after you take this microscope and keep enlarging, what do you see? Well, you you see diagram like this. So, two straight lines and they join together in other straight lines. And here you have three constant state. Okay? and of course this the speed of these the slope of these lines are the corresponding ranking audio speeds. Okay? And here stress the fact that in both cases the set of limits that you'd get after rescaling depends on finitely many parameters. Right? in order to discuss the case of point where new shock is formed, let me first remark that if you have kind of generalized Burgers equation, so here instead of having just squared over two, have AU squared over two plus BU as flux. And if have an initial data of the form minus cubic root of can just make change of variables so that out of here get exactly the standard Burgers equation with initial data minus cubic root of Okay? So, this is just rescaling change of variables that trans essentially we get rid of these three constant and by suitable change of variables. Okay. So, point where new shock is formed, well, in this case we can yeah, as said before, by this affine variable change, you can assume for example that the point where new shock is formed is the origin and of zero is equal prime of zero is zero and the second derivative here is one and at the time where at the time where the shock is formed the function of as function of is all locally invertible and it has this type of inverse like that okay, so that's all standard and then after you do this rescaling and here the rescaling is definitely different from the previous one. So you could take you rescale by epsilon minus two by epsilon minus three by epsilon minus one and then out of this you get solution to the standard standard Burgers equation, okay. So after this time you again you take your microscope but you enlarge in different ways the different components and then after you do this you this inviscid solution converges to this standard solution of Burgers equation okay now as mentioned before in all above cases the limits of these rescaled solution lie in finite dimensional manifold that is they depend on finitely many parameters okay now the point is how do various type of approximation behave in neighborhood of these points so we have many different types of approximation that have been extensively studied in literature of how do you approximate this either scalar conservation law or system of conservation law. That doesn't mean we know everything about this. In fact, more many of these approximation we don't even know if they converge in the case where we have baby solution, small baby solution, but anyway. So, in for the scalar case, one is the vanishing viscosity approximation, of course. another is relaxation approximation. Here there is large set of relaxation approximations like I'm not mentioning all of them, but just one. For example, here what is the idea? The idea is that you want to reduce or to approximate your initial equation with an equation of the form well, on the left-hand side is system, hyperbolic system with constant coefficients, so very easy to solve, but on the right-hand side we have something that forces here, for example, it forces to be equal to Okay. so, for example, if want to approximate this equation here by an equation of this form, how can do that? Well, should take prime, well, such that prime is positive, and want is equal of to be the inverse of is equal plus of If can do this, then it works. let me check. It works because let's see. If sum these two, the left-hand side, well, the right-hand side is zero, so you see immediately. So, the this becomes VT plus VX plus WT is equals zero. And because of this one, you see VT is pushed very close to of So, formally we expect that should be almost equal to of So, what happens to UT plus of UX? Well, is plus GV by construction, so this is plus of sub plus sub yeah. of of is so this is sub of sub is sub And this would be sub plus of is almost so WT plus of sub and this is equal to zero. Okay, so formally it should converge to solution of the original equation. And in fact, this is known for smooth solution. tough of problem is to prove that actually still converges for general BV solutions or the small BV solutions. In fact, there are some results of Stefanov and Yang Kimi using center manifold theory in infinite dimensional space that generalize the convergence of vanishing viscosity actually not with with constant. With constant, this is known and for in system with this type of like Shin engine relaxation problem, this is known in general it's not known if they converge. Okay. another type of approximation which has become very popular in recent years, it's related to model of traffic flow. So, model traffic flow here if row is the density of cars and is the velocity of cars, the standard conservation law says that row plus row times of row sub is equal to zero, okay? But people have been looking at non-local models. So, here the velocity does not depend on the density at that same point, but depends on some average value of the density in the forward direction, in front of the car. Okay. So, for example, you could take row sub epsilon here some average value of row at points ahead in the okay? And it's clear that when epsilon goes to zero, this approaches row of right? It's like kernel that converges to Dirac delta. Okay. And we have some results on convergences of these. and also of course there are lots of discrete and semi-discrete numerical schemes. So, for all these approximation as epsilon goes to zero, it's well known that they converge provided that the exact solution is smooth. That's my mostly Taylor expansion. for general BV solution, even in some cases also for by hyperbolic systems, we know that this approximation converge in L1 lock. the question want to discuss here is how do these approximation behave in neighborhood of singularity? Of course, know that the vanishing viscosity approximation converge in an L1 sense everywhere. So, if you have jump, of course, near the jump, they will be close. Okay? But, can we describe the behavior of these jumps more carefully in neighborhood of point? okay. So, here should mention some related work. So, given piecewise smooth solution of this conservation law or system of conservation law, we can construct this family of viscous approximation, and then there are works by first by Goodman and Shin, and more recently by Anderson, Chaturvedi, and Graham that they construct solution to the sorry. Construct solution to the viscous equation that converge to the solution of the inviscid one. Okay? here, the point of view is somewhat different because we do not construct anything. Okay? So, what we want to do is just take any family of approximate solution, look at one single point, and rescale, and see how they behave in neighborhood of that point. Okay? Typically, point of singularity. Now, one thing which should be obvious, but let me stress this, that if have solution of the non-viscous equation, for the when do this rescaling, always rescale the same inviscid solution. On the other hand, when have different various res- approximation algorithms, their rescaling parameter must correspond with the epsilon of the approximation, right? For example, here, one This is the inviscid equation, and this is the one with viscosity. And then, of course, to get the non-trivial, non-singular limit, the way rescale, so when take my microscope, and enlarge, the amount by which enlarge should depend on this epsilon here of the approximation. So, they they cannot be independent. So, in this case, for example, it had it should be exactly the same, because what want is that as epsilon goes to zero, the solution of this should converge to solution of this with unit viscosity. Right? Okay, so what could be rough guess that works for possibly all sorts of approximate solution, okay? mean, making conjecture is easy, then solving proving it's different thing. But, we can make conjecture that is if we have solution to the Cauchy problem in non-viscous, then for any of these approximation methods, consider family of approximations of approximate solution with initial data that converge in C3. So I'm taking smooth initial data for example and want convergence of the initial data. Then for every singular point of type 1, 2, or 3 what do expect? expect that if take my microscope and enlarge in suitable rate, the rate should depend on epsilon. So for example, epsilon should be some epsilon to the minus alpha. epsilon could be epsilon minus beta. So can have to rescale different components in different by different exponents. And then possibly like slightly shifting my solution. Here epsilon and psi epsilon would be way to center my rescaled function at the right point because want to capture the point of singularity. If start enlarging at the wrong place right? get something which is constant and don't see anything, right? So it's important to rescale at the right place and this can change as function of epsilon. So the conjecture is that this rescaled function should converge to solution of the approximation of approximated equation with parameter epsilon is equal 1. So for example, vanishing viscosity should converge to solution of the viscous equation with unit viscosity, okay? And the same for the previous one. For example let's see. Here expect it will converge to solution of this, but with epsilon is equal one. Okay. When do this rescaling, if can do the rescaling in appropriate smart way. Okay. So, that is kind of the general expectation. So, we can prove this in the case of vanishing viscosity. And must say, at first thought it would be kind of straightforward because vanishing viscosity it's out of all these approximation is the easiest one because you have comparison arguments, upper and lower solution, you have lot of tools which you don't have the other case, but it turned out that even proving this for vanishing viscosity, it takes some work. Okay. It's So, So, for one, we take we consider very rescaling of the same inviscid solution. For two, we consider family of solution epsilon. And So, this rescaling parameter, as said, should depend on epsilon in suitable way, and in all cases, we should get some as limit something which is solution with unit viscosity. Okay. So, basic assumption So, here are the three types of singularity. New shock forms, point along the shock, point where two shocks meet. Simplifying assumption is is smooth and strictly convex. The initial datum is generic, so that the the has finitely many shocks, singular point, and we have the convergence of the initial data in C3. And the question is, can we rescale this viscous solution in neighborhood of these points so to obtain well-defined limit? Okay, so point along the shock, this is quite expected in way. So, what do we say? say that if we have point where the initial solution has shock with left and right states U- minus U+ then, given family of vanishing viscosity solution, then we can take rescaling of this point so that this converge to viscous shock profile joining the state U- U- U+ uniformly on bounded sets. Okay. So, And of course, the viscous shock profiles is solution to this first-order ODE where is the of U+ minus lambda U- minus lambda U+ and so on. And Of course, viscous profile is defined up to shift. To determine it uniquely, then we can assume that at the origin is U- + U+ over two, just to fix what Okay. how about two shocks merging? Well, first of all, we have to define what limit solution do we expect to obtain after we make these rescalings like with the with the so, to describe the limit of vanishing viscosity solution near point we construct an eternal solution to the viscous equation with viscosity one such that, well, at minus infinity is essentially the superposition of two viscous waves, viscous profiles, and at plus infinity it is, well, whatever it is. mean, they they come together and they join. So, how do we construct this? Well, it's fairly straightforward in the sense that given these two viscous shock profiles, you have to glue them together. So, you take one, let's say So, we obtain this as limit of in of of viscous solution with initial data given at minus Okay, so you go backward in time. So, at time is equal minus you put two viscous profiles, and you glue them together in the middle, which is not difficult to do because they converge exponentially to the middle value, okay? so, then we take this is the limit of this of and it's not difficult to prove that in this case the limit exists uniformly on compact subset of 2. So, there is this unique solution. and then in this case the result would be that yeah. Under this assumption, let be point where the inviscid solution has is piecewise Lipschitz with two interacting shocks, let's say left, middle, and right state, minus, star, you plus. and then given family of this bench of viscosity solution, after you make this small shift, right? So, you take this small shift. these are points that converge to dark side. then these rescaled functions converge uniformly unbounded set to this eternal viscous solution that at minus infinity is the superposition of two independent shock viscous profiles. And the plus infinity is, well, it has to converge to single viscous profiles, of course. and this is approached uniformly on bounded sets. finally, point where new shock forms. let's see. In the inviscid case, well, we know that after an fine change of variables, there these rescaled functions converge to the globally defined solution of And here can take just Burgers' equation with initial data, well, data at time is equal zero equal to minus cubic root of Okay, so just eliminate these parameters by just some straightforward changes of variables. And What is the limit of rescaled viscous solutions? Well, it turns out that you have to rescale in the form of epsilon is epsilon to minus 1/4, tau epsilon plus epsilon 1/2 and epsilon to the 3/4 mean, compared to the previous one, have to take minus 1/4, 1/2, and 3/4 instead of minus one, one, and three. Sorry, minus one, two, and three. But, just in order that in here get coefficient one. So, want to find do the rescaling to get viscous conservation law with the viscosity equal one. Okay. So, that's why mean that. so, what is the limit of this? Well, the limit is it converges to what we call the eternal solution of this viscous Burgers equation such that at minus infinity converges to the solution of the inviscid Burgers equation with terminal value Well, with value at time is equal zero given by minus cubic root of Okay. So, that's in fact, it it's known in the literature. If there is any limit, it has to be this one. So, mean, this Cauchy problem has unique in fact is as smooth solution backward in time. Forward in time it has shock, but backward in time has smooth solution. So, what do you do? yeah, you take the solution of this one backward in time at time and then when you find the solution at time minus you take the solution of the viscous Burgers equation from minus to plus infinity that convert that coincides with the inviscid one at time minus Okay. So, in in essence, we want solution to the viscous Burgers' equation. So, which is this one, actually. But, that at minus infinity converge is the same as the inviscid one. Okay. So, that's what you get. And stating the result is the following. So, let be tau psi point where the new shock forms in the inviscid solution. and then, of course, without loss of generality, we can make the coefficient to be one. so, after this rescaling change of variables, given family of vanishing viscosity solutions, let's say of this form. with epsilon converges to Well, actually, we prove we ask the convergence to be in C4. believe it still works in C3, but we were kind of lazy to work out the whole this estimate. The estimate are much easier if you assume convergence in C4, so that's not the major point. so, we assume that the initial data converge to bar in C4. Then, up to some small shifts, this rescaled function converge to this eternal viscous solution uniformly on bounded sets. Okay. Okay, so these are the main result. Let me say, we expect that similar results should hold also for the other types of approximation, like relaxation approximation, non-local approximation of the traffic flow equations and so on. However, none of this is easy, will be easy to prove. They're all open problems at this point. Okay, so so in the remainder of this talk, let me describe some techniques. Okay, so what tools do does one need to prove results of this type? Well, first let me recall some result with Carlotta Donadello, 2007. which is about the formation of disco shock. So the setting is this. Okay, have conservation law with this unit viscosity here. And start with an initial data which has this form. So there is an interval AB where it does whatever it does. Okay, anything in the middle. And then for less than is very close to some value minus. And for bigger than it's very close to some other value plus. Okay. And between and is anything. So it's between little and big Okay. So the question is we expect that as time goes by, the solution will be close to traveling wave profile, right? How long does it take to become close to this traveling wave profile? Okay, notice that this is not an asymptotic result. In fact for this initial data, the solution does not converge to anything as goes to infinity because we are not assuming that at plus infinity it converges to plus, or at minus infinity, it converges to minus, right? In fact, it could keep oscillating inside this tiny tube lot. Okay. So, it will not converge to anything as goes to infinity. But, still we expect that after some time, it should be at least close, not identical, not asymptotically converging, but So, how long does it take? Okay. So, what are the tools to prove this? Well, the basic tool is to look at in the same equation, but in transformed variables. So, given solution to Ut plus of Ux is Uxx, consider the variable and this would be the the corresponding flux. mean, here you think that the flux is of minus sub right? If you include the viscosity in the flux, right? Okay. So, now can plot my function as here plot and here plot the flux of minus Ux. Okay? So, what happens? For example, if you have is this function here, what is the corresponding curve? Well, you see, when is zero, am about I'm here. So, am on the graph of When is negative, am above I'm higher than the graph of right? So, am over here. And then again, down here the is zero, so am on the graph of Then is positive, so am below the graph of Another point where is zero. At this point here, am here, and then eventually the solution goes to this limit here. And this would be the corresponding curve. Okay? So, as you have solution of your conservation law, you this curve also moves in time. So, how does it move? Well, it's again well known. After some manipulation, you get is this square times And the nice thing is that it evolves in the direction of curvature. And in fact, this is tool which has been widely used to prove stability of viscous traveling wave. Okay? Because well, because in the transformed variables, traveling profile corresponds precisely to segment. So, if you have segment in these new variables, you know that this correspond to traveling profile. Of course, the traveling profile is determined up to shift. Okay? But and one thing we proved with Carlotta Donatella in that paper is quantitative converse. So, assume that have profile and tell you that the corresponding curve gamma is very close to straight segment. Okay? And then you want to say that the corresponding function is very close to traveling profile. Okay? This is very useful to say in particular stability of traveling profile and Okay? So, So, that means that for every small delta, let be smooth profiles which satisfies this. So, it means that of minus this is point on the curve gamma. And we are saying that this is very close to the segment capital to the right segment capital gamma. Okay. So, if it is close, then the corresponding profile is very close to some traveling wave profile. If you choose carefully the shift. Okay. and the difference, of course, if this is less than delta, this is less than, let's say, times delta. Okay. So, that That's useful tool. so, What is the main result in this direction? Well, We want to estimate the time it takes for an initial data which is crazy like this, but here is almost constant, and here is almost constant, to approach traveling profile. And so, assume that initially it's close to minus at the distance delta naught, and here is delta naught close to plus. Okay. And then, if you give me some any delta, but delta, of course, cannot be smaller than delta naught. It will never get small short closer than delta naught. Okay. So, let's say delta bigger than four delta naught. Okay. So, not too close. And how long does it take in order that your profile differs from the traveling profile by less than constant 01 times delta. Okay? And the time, of course, depends on delta. If you want to be very close, you have to wait long time. How long? Well, 01 times 1 plus minus which is this, times 1 over delta delta to the minus two. Okay, so that's That's the main estimate. So, how does one prove something like this? well, said before, it's the trick is to look at the equation in this transformed variable. So, look at what happens to the curve gamma that moves in the plane. So, let's say initially have curve gamma like this, which may take value much smaller than plus and much bigger than minus initially. So, if wait time T1, then can say that the curve will take values almost very close to plus and very close to minus. So, the the domain shrinks to small neighborhood of plus and minus. If wait another time T2, can say that my curve gamma is within small neighborhood of the convex closure of this arc. So, you take the graph of which is this, you take small neighborhood of this convex closure, and second step of the proof with Carathéodory, essentially is to prove that after some time T2, it's in small neighborhood this shaded area around the graph of And the most tricky part is if you wait another time T3, then it really lies in small neighborhood of this segment. Okay, so this part really moves up and the whole curve lies in neighborhood of that segment. Okay, that is the most tricky part of the whole argument. Okay, but you can do that and once you know this, you know that the corresponding is very close to viscous pro- profile. Okay. okay, so next another thing to which is very useful is about the domain of dependence. So, the idea is that for example, if you want to consider singularity at this point here. for the inviscid solution, this singularity lies in the domain of dependency of some interval AB. Okay. So, if you if change the initial data in this region here or in this region here, of course the inviscid solution does not change at all in this region in here. Just by the method of characteristic, if assume that all the characteristic move out. Okay. And this is quite useful because could have an initial data like bar like this very complicated and would like to appro- to replace it by something much simpler like here, global bounds everywhere. And one thing that can be proved actually is that well, if have solutions of the viscous equation with small epsilon, of course, if change here and here, even if have small viscosity, the solution will change everywhere, even in this region here, of course. But, what's useful is that the change is really exponentially small with respect to epsilon. So, really nothing changes once you do this rescaling. When you take your microscope and large the change that you see by changing the viscous solution, let's say here or here, is higher order with respect to epsilon to whatever power you like. So, in the the limit of this rescaling does not depend on what you do here or here. Okay, next. Yeah, how do we prove convergence of the scale solution uniformly on bounded sets? For example, let's take this case where two shocks interact. Okay, so we want to prove that this rescale viscous solution will converge to this eternal solution here of uniformly on compact sets. so, the idea is first to determine family of intervals so that if take the L1 distance of my rescale solution with this eternal solution if take integrate this distance on this interval here, this goes to zero. That's the L1 norm goes to zero. And the point is that let's see. want to choose alpha and beta like this so that well, of course, first of all, need that this L1 norm goes to zero, but want alpha to be smaller than beta. So, it means that this thing this distance here is much smaller than this distance here. And so that essentially if if it converges here, each the L1 norm should be go go to zero on the domain of dependency. And by choosing this much bigger beta much bigger than alpha, can make sure that the this domain of dependency contains the origin in fact when epsilon goes to zero, they invade the whole plane. Okay. So, well, one way to do this actually is to observe for example that this equation here generates contractive semigroup in L1. So, one thing could do is the following. could change my viscous equation. change the data at time epsilon. Okay. So, what do, take another function hat epsilon which coincides with epsilon here and over here essentially coincides with with Okay. mean, know that the difference between epsilon and goes to zero. So, it's very small. If take the the integral here, of course, if take the integral over the whole real line, that might be huge. Okay. So, it's not going to work. So, what do, take change my function epsilon outside this, so that now the difference between hat and is still small and goes to zero. Okay. And now can say that hat epsilon minus at any time at any time here still goes to zero on the whole real line. Okay. But then can can say that if look at the domain of dependency, so if look at just this interval epsilon of is in the domain of dependency, so if and hat are the same on this interval, the red one here, then since this is in the domain of dependency, they're almost the same also on epsilon of Okay. So, can say yeah. So, the difference between epsilon and the limit on this interval epsilon of is almost the same or smaller equal than the difference sorry. It's It's almost the same as the difference between hat and in this interval here. Right? Because it's domain of the kindness. Right? If the two solutions are almost are the same on the this red one are almost the same also on the blue one. Right? mean, this would be obvious if there was no diffusion. But even with some diffusion you have this characteristic are strictly flowing out and the size of this interval the thickness of that goes to infinity. Okay. So, even with any diffusion the amount of mass that percolates inside goes to zero exponentially fast essentially. That's the bottom line. So, and finally so can conclude since epsilon and satisfy conservation law with unit viscosity know that they are uniformly Lipschitz continuous on compact sets. Okay. So, if know that they converge in L1 and they are uniformly Lipschitz continuous, it means that they converge uniformly on bounded sets. And that is concludes the the proof in the vanishing viscosity case. Okay. don't expect that the proof will be easy in the other cases like relaxation or non-local approximation because here we are extensively using comparison principle upper and lower solution. believe that similar result would be true but think the proof would be considerably be deeper. Okay. so think with this can conclude and thanks for the attention.