Lecture 03 Shocks and Patterns in Hyperbolic and Hyperbolic Parabolic Balance Laws

Yeah, maybe you want to be more in the Yeah, you should be more in the center. guess won't be writing there. So, students. Yes. Yes. These these are my two grad students. there's Dorothia Gallos back in Indiana and Cristiana Galos is somewhere now. don't see her anymore. and it appears that Xiao Yang who is former student. hello. Welcome guys. no. Kevin, nice to meet you. David. Okay, Zachary. Okay. Fico, nice to meet you. am PhD student here at very good. And and are you the program? okay. Nice. In the the shock program or or in the graduate program? wonderful. So, unfortunately, was in Italy. So, cannot attend the first two lectures. don't know if today will understand something. I'm starting at the beginning. The first two lectures were kind of dependently. They were independent. Yeah. kind of like an advertisement of all the wave things that maybe we could do after the course. Maybe. Yeah. So, yeah, maybe we'll see. Yeah. Anyway, so we're going to study what kind of tools would be needed to to do those things. So, send you his regards because told him that. very good. Happy birthday. Yeah. Another one. And Xiao, are you are you there? Xiao Yang maybe he's not there. So So Xiao is in Beijing. That's my my former student of five years ago and he's Yeah, he he was the main collaborator in the in the second talk that gave. Okay. Well, let's see. Two. Okay. I'm going to get get started then. Okay. Okay. Stability. Let's see how that looks. That doesn't look good at all. Can you see it? You can't see it. Wait minute. Let's try again. No. Really? Can you see that, Dorothia? can see that it's little dark on the left side. Like the left two boards are bit dark, but Okay. How about this? shots plus periodic patterns. Periodic waves. That's the course. Let's Yeah. Okay. Is that readable? My goodness. Okay. Good. All right. So, and we're basically going to the course is basically broken into two pieces, one and two in my plan. So, we'll start with the shocks and then we'll we'll in in the second half of the class we'll do periodic waves roughly speaking. So, today we start with shock waves and I'm just going to give little intro motivation and it's kind of It's it really goes there's 65 small note by Peter Lax in Sam just note that was was one of the first things read as student and is still great motiv motivation. So I'm kind of distilling from that what give here maybe combining with some other things but all right so here's the first shock wave equation learned okay oops want to want to write this okay okay so dt of so we have of XT bigger than equal zero. in is fluid height. This is this is the light hill model for shallow water flow of an incompressible fluid. So this you you could think of canal or something coming over dam anyway or flood plane. These are models for that. Okay. So then we're going to write plus dx of equals zero. That's it. And of is Okay. So here's the height is velocity velocity is down the ramp. So is the flux the mass flux. So that is if look at vertical line here, how much fluid mass is is passing through that point? Well, it's just the height times the velocity. That's that's the mass per per time. Okay? And this just reflects conservation of mass which we can see by integrating We're going to take this on the whole line even though dams aren't on the whole they aren't infinite so on but we're going to study it on the whole line. So if the total mass sorry it's the integral of dx. And the derivative of the total mass by the equation is sorry this is integral of ht which by the equation is integral of dx of and that's perfect derivative. So it's just of from minus infinity to plus infinity. What's is is fluid velocity. this is is is velocity. It's fixed. It's No, it's not fixed. It's changing. The velocity is changing with and So but it's depending in in way that I'm going to tell you in moment. Okay. Yeah. Okay. And but so this if if so so in yeah. So we're going to assume that the velocity is function of So equals of That's that's what's going to close this. So is of because it's * of And if we're going to take say in L2 some localized space then okay this is first order hyperbolic scalar equation. So it has finite propagation speed by method of characteristics. So if start with vanishing height at plus and minus infinity will always have vanishing height at plus and minus infinity. So this will be zero. In other words, the total mass of the system will not change. Okay, so that's conservation law. Now, here's what happens. Okay. So okay so of is not constant in in in light hills model which is derived from bigger model of incompressible flow. so you have there are several steps but it's you look at the oiler equation inside under the surface you look at free boundary problem you take small height limit and you take more more care then take some relaxation and eventually you get to this nice model but there are many steps and okay interestingly let's see this looks more like compressible flow. should say that like the fluid model because it's very visual, but it also can be derived from almost any model as an approximate single characteristic behavior behavior along single hyperbolic characteristic. So this is also model for gas dynamics in in in preferred coordinate preferred mode of propagation. So it it's it's still more general. Okay. So, du over dh does not equal zero is called genuine nonlinearity by lax name that is stuck. It's catchy and okay. So what's that going to give us? Well, that's that's very interesting. First of all, mean, it's not at all obvious why this should be, but it turns out that way. In fact, of this is it doesn't matter, but of is actually constant time to the 1/2 turns out in in standard Lighill. Okay. But we could take of equals for example. anything anything that's not constant will lead to the behavior that said but also in fact it's greater than zero so it's increasing with height that's very very telling so that means that's just for for definitess if it turned out it were less than zero there would be similar behavior but different picture so here's my picture what if start with wave And now I'm going to flatten the bottom. Just look at is the direction along the ramp. I'm going to forget about the incline. And suppose have something that looks like this. It's decreasing. Okay. And now I'm going to look at what happens. Well, it's quite interesting by method of characteristics says want to look at if want to see the propagation look at this as re write it out as dt plus okay now I'm going to take prime of dx XH equals Okay. Now prime of Okay. So what what does that tell us? This tells me along characteristic which is often written just dot dot equals zero. Where dot means DT plus prime of DH. to just motion along certain direction in and time and And the next thing we see is because dot is zero, is constant along these characteristic curves. And therefore the characteristic curves are lines in XT. This is standard. But want to look here. This is nice think this is nice way to look at it. So very So let's here we'll plot against height And I'm going to invert. So want to view x= of knot. sorry. Yeah, just of All right. And then the solution will be by this will implicitly give the solution. That will give h= of xt. If this is invertible, if dh of does not equal zero, then we can invert this locally. And that'll give us nice smooth solution. But look what happens. It's it's very simple. dx over dt equals That's the equation. So if want to propagate this graph, it's it's easier to do viewing it as graph over and propagating in the direction. And what see is well because of the monotonicity of the speed with respect to height, the upper parts are going to be moving faster than the lower parts. And they will eventually catch up but at some point they will just get vertical slope which is zero slope from from the point of view of with respect to and after that they will turn over. So you see breaking of the water wave. Okay. And when does that happen? Well, that's yeah, that's pretty easy to see because we want that happens when breaking happens when dhx equals zero. But let's let's differentiate dhx and see how it changes. we have = knot plus of So dx dh of is is just and let's see moreover let's see something's not not I'm sorry this is wait no one it is plus prime right it's prime plus prime of right? Okay. So, is prime of prime prime of Wait minute. right. Something wrong No, that's it. need not over DH. So it's it's DH KN plus prime prime of HTT. And this was assumed less than or equal to less than zero. And this is assumed greater than zero. Sorry, it's it's not it's prime. already made mistake here. We we want this is the propagation speed. We want prime prime does not equals we want the derivative of this this propagation speed. I'm sorry that that led to other issues. And so now here we see it. This is positive. This is negative. At finite time it will be zero. It's not playing the role. No, you is just to understand. yeah, is actually not playing role. It's gone. The fluids the the fluid velocity does not really tell us the shape of the wave. What tells us the shape of the wave is this prime, this propagation speed, which is sound speed roughly. This is sound speed. So the sound speed, not the fluid velocity. So the individual particles are moving along but the the shape of the wave moves distinctly. It's not just if if you were constant these would be identical. Sound speed would be the fluid velocity but it's not. Yeah. Because in that if were constant then prime would be just but yeah and second would be zero right yes right and so this would never happen yeah it would always be but it also would not mean the shape would never change it would just propagate independently of of just at constant so this will vanish at equals hx knot over prime prime of So you can explicitly find for each that is each one at each point we will eventually lose lose invertability but you know in reality there will be first one. So it's so it will first happen the first time will be at the minimum over of dhx knot over prime prime of So you you find in your initial profile there's some minimum value of that ratio and and that will tell you the first time that you become singular and an extreme extreme case might be you know if if they're just tuned you could have all of them meet at point that could happen in which case you would suddenly have what does this look like here. This looks like jump from from minus say to H+. So it would be in terms of and it's kind of catastrophe or or shock. Okay. But generally it looks like this. It's something else. So we had whole beautiful series of lectures on this last week by by Cole Grant on exactly this concept. Okay. So what but going back more in more primitive way how how do we continue this? Because actually in nature these waves continue. So what's the right way to continue it? Well, it's pretty clear we should do an equal area rule here because after all, this is about conservation of mass. And so we want if if mass is to be conserved, we have to chop out an equal positive and negative mass. So what we say is that the wave and you know maybe this is not exactly at this point quite what you see in nature because you do see something like this sometimes, right? But but we're not allowing that. We're and and I'm thinking more at this point of gas dynamics where where we see this and we have the same equations. Okay. So and this actually so this is described by by lax and it's also described for in ontropic gas dynamics equations. So that's two 2x two vector system by lax. The argument is the same as the one of Bernard Reand. He was he was the first to observe this which is very nice. Okay. So yeah and so many things in this in this area bear his name. Reman data, remon invariance, remon problem. Yeah. But there was little okay so but some others who came later were ranking and Yugonio and they introduced the so so there should be okay so Reman conjectured we should continue with the discontinuity somehow which is what you see in gas dynamic. So this is beautiful you know Beautiful analysis really. So continue with the discontinuity of and dot we'll call of And want to figure out so it should continue like this. And now I'm going to abandon this reparameterization and look at as function of again. So this is moving with speed All right. So what is the what is the mass flux across this boundary? Okay. First of all, it's it is going to be of minus on the left. That's how much is is trying to come across. of H+ I'm sorry of H+ minus of minus the mass flux on this side minus the Let's see. no. said it wrong. It It was It's mass coming through. So, it's positive on the left. did it wrong. Okay. So, but then there's another there's correction which is due to So there's simply there's quantity of water that's just passing through this boundary due to the motion of the boundary not the motion of the fluid and that is this and we want this to be zero in other word in order that there be mass conservation. Okay. So that means of jump which is denoted by brackets. So you just take the the value on the right minus the value on the left. That's the definition of the bracket. That should be equal to times jump of And that's the ranking Hugonio condition. through the weak solutions. You can also get it through weak solutions. Yeah. Okay. All right. So, so that's the first observation, first two observations really is that first we need we need to allow discontinuities to continue the solution globally in time because they will lose regularity. And second observation is well. So we allow discontinuities but with the ranking hugonio condition. But of course there's the the problem is so we can look at two canonical types. One is this is the remon problem remon data. So we have on the left of zero we have minus and on the right we have h+ and let's just see what happens. So first example we'll take minus greater than h+. So here we have since we have constant values all characteristics this is and Now that's how this is drawn and all the characteristics are parallel and they enter shock and drew the shock too carelessly. the shock will have some speed and these will all be coming into that shock. That's why it that's why so it forms at time zero because we've built it to form at time zero. This is more of thought experiment. But but what if minus is less than H+? Okay. Well, is unchanged because if we change the direction of of if we change to to minus this becomes negative. This becomes negative. But the if we change time as well, if we reverse time and space, is unchanged. Okay? And that's what we're doing. But now, so we have now h+ minus switched And so they come out oops parallel out of the sh like this. Okay, that may seem okay. But there's another nice solution which looks like this. It's called rarifaction. And it just you just emanate characteristics smoothly from the origin. This one has no discontinuity. So this one's lip shits. People like to choose this one. People like to choose this one for all kinds of reasons. First of all, physically you don't you see this one and that's what we're trying. You know, it's think of it like in gas dynamics, it's like pressure front. It's high pressure here, low pressure here. that makes sense that it should compact and form shock. But if it's you know if it's moving away we should expect that it rarifies it doesn't form pressure front like that but but just almost kind of little place of low density rarified vacuum not quite vacuum but lower pressure area in here. And this this is what we see here. Okay. But there's okay. So now let's think of all kinds of reasons. So so we want to allow one and not the other. And there are all kinds of reasons why this just looks more appealing. And one reason is what are we doing here? We're we're creating information coming out of this discontinuity. Where does it come from? It it it's it doesn't make lot of intuitive sense. Whereas here everything starts from the initial data. We we prescribe some data and it just propagates forward. Now don't have problem here because everything propagates forward. It can terminate at the shock. So lose some information but I'm not making up information out of nowhere. So people think of that like an sort of an entropy kind of thing. So that seems preferable. But okay. So there are many arguments for this and the first one is just consistency. So we want not Okay. So consistency argument. We just want wellposed pre-boundary problem. And clearly this one is not well posed. It it's missing bound. sorry. This one is not well posed. It's missing boundary condition at the at the boundary because actually two boundary conditions for both outgoing signals. It's effectively we're prescribing at random, but we could prescribe differently at any point here. We could put in another one of these rare factions. It's quite non-unique and it's not it's not well. So this is non-unique but but this one's unique right if if we if we don't allow things to come out. So from this we get the lax characteristic condition which says okay I'm going to use his notation with of defined as prime of that's the sound speed or the speed of propagation of of signals that we saw the of characteristics so we'll take minus should be greater than should be greater than plus. In other words, case case is going to be prime. It's the characteristic speed. So we have dot equals dt plus dx Yeah. So I'm just giving it the is giving me trouble because keep for you know is prime kind of entropy condition. Yes. Yeah. And it's also sometimes called it's sometimes called the entropy condition but I'm going to put that in quotes because this is just entropy in some uristic way right but in moment we we we'll do the entropy in more real way. Yes. And so that's the characteristic speed and it's showing that they enter the shock because they're greater the speed is greater on the left than the shock speed and and smaller on the right. So they they pinch together. Okay, so that's one way. But what seems more fundamental is the what's called the vanishing viscosity principle. And the idea here is that for most of these models including the the both the light hill shallow water model and gas dynamics in in kind of single characteristic mode. There really are other effects physical effects that we've left out. So higher order effects give dt plus dx of equ= epsilon dx^2h. This is maybe more precise model and this is well for gas dynamic for compressible gas this is comes from actually viscosity and also other transport phenomena like heat conduction and hence the name vanishing viscosity. It's not viscosity in in this other case in this light hill case, but we could call it an effective viscosity for the lighill. Okay, either way, this is something that belongs here. And here epsilon is the coefficient which we'll just call viscosity coefficient whether or not it's an actual viscosity. This is just what people call it for the second order term. Okay. Okay. So there are there are could be other right like would be epsilon dx cubed or epsilon dx to 4th or other combinations depending on your physical model. But and the behavior will be different. And the limiting behavior we we want to say as epsilon goes to zero limit. So epsilon is seen to be small and that's why it's often neglected. But nonetheless, it presumably plays role. We'll see that it does play role. And we want to see what happens as it goes to zero. And that should enlighten us more about case and That's the idea. So first one the viscous profile condition which as far as know that's first introduced by Gelfund in Orlando. okay. So he just says we look for traveling wave of okay let's call this now one epsilon. So let's with speed some speed because in the invisit case the case with epsilon equals zero that's what our shock was it was at least for the remon data we had so like the remon remon data version which was of xt equals minus for less than ST, H+ for greater than And it was just shock, right? If we look in XH coordinates, it's minus to H+ and it's traveling along with speed X= or well with speed unchanged, just jump between two constant states. So we're going to look for something like that. So we seek of xt equals bar of - stationary sorry with and what does that give us? with solving one epsilon and with goes to bar goes to plus or minus as bar goes to plus or minus as goes to plus or minus infinity. So we want asmtoic limits. So in other words, we're going to blow this up. Here's the invisit one. And we're going to look for smooth smoothed out version because it's going to be smooth. should have pointed out as soon as put an epsilon that's positive. This is now parabolic equation. It does not spontaneously generate singularities. In fact, it continues until mean it might blow up in but it's in fact, we can also see it doesn't, but it does not. It stays smooth. So we expect smooth wave and we ask could there be one. Okay. So let's put this into the equation. Well, we get Okay. So dt of will then be of bar prime time and dx of will be bar prime and okay so what we get here is negative har prime plus of bar prime equals bar prime prime. That is our travel. It's traveling wave equation. And so this looks so funny. Got Chris Jonas starting it looks like nursery rhyme or something. traveling wave OD. It's kind of fun. Okay. And if equals zero, it would be SWO standing wave OD. It's convenient. All right. So, well, that's fine. What do we what do we have? we would write well, one thing we could do is integrate this. It's perfect derivative. That makes it easier. So we get + of bar from to minus infinity equals bar prime. Now we have first order OD and okay first thing we can see is it has an equilibrium at at minus at sorry at bar equals minus. So, this is an autonomous OD. And if stick in maybe shouldn't maybe should write that out. So, let me let me write that little nicer. So bar prime is equal to ne say of bar minus bar minus of minus minus minus writing it out so clearly bar= minus makes the right hand side equal zero so this is an equilibrium of the od so if want to write So and prime is actually derivative with respect to So this is what do have? have minus and h+ and I'm trying to get from here to here. Okay. With with something that asmtoically converges to constant. That means that constant has to be in equilibrium. Okay. So so far so good. minus built it in though. assumed had an equilibrium at minus and got one. But bar= H+ is an equilibrium if and only if of H+ minus H+ equals of minus minus minus. And that's equivalent to the ranking condition. It's jump in equals jump in So that's great. That's good start. So at least at least the the possible pairs of points that could connect match up with the invisid problem. That's no accident though because they're both basically conservative. Well purpose each larger. yeah. do it on purpose because it it might or might not be. It turns out But did because had to pick one and because know that's the one want to turn out to be good. Yeah. Yeah. That this corresponds to Yes. Exactly. case that think will be good. And hope to get case and not case no. we lost connection probably. it looks like you're they're back. We're back. We're back. Can you guys hear? Okay. And if you have any questions? Yeah. Yeah. Speak up. Yeah. Yeah. We didn't lose connection here. you didn't? okay. We just lost it on our our side, guess. Okay. And you still can see the board. Okay. Yeah. Okay. I'll try to stay more toward this side, guess. think it's better. Okay. So, now what do we need next? We need more, right? We need this has got to be repeller and this should be an attractor if we are going to have connection. That's necessary condition. So, let's linearize the traveling wave OD. So linearized is okay let's call it let's just call it again okay is the derivative with respect to of the right hand side at bar sorry at plus or minus * That's how you linearize an OD. And what we get is this is so the right hand side this is constant so it goes away and this one becomes prime of plus or minus minus plus or minus okay let's just write prime of minus plus or minus depending if plus equals plus or minus which which equilibrium we're looking at. But this is minus No, I'm sorry. This is minus Yeah, minus That's didn't do well linearizing my linearizing my linear term. Okay. yeah. Okay. But this is at plus or minus. So if I'm going to be repeller at minus, need that this be positive. So minus min - min - must be positive and plus minus must be negative in order of prime of is that was the definition of that's the characteristic speed that's here so have minus and it says minus must have specific signs at the two end points. So it must be positive on the left. think lost. no, here here. okay. As always the entropy condition. yeah, it's it's here it is. Yeah. Yeah. the ent Yeah, here it is. Here it is. Thank you. Yeah. So minus must be positive on the left and negative on the right. So that's fantastic. So we we recovered this condition. So necessary condition for profile to exist is that we are in case and not case If we were in case then minus the the state at in minus infinity would be an attractor and we would never we could never leave. It would just be stable rest point of the OD. So we could have no heter this is heteroc clinicic connection in very simple scalar OD. Okay. So that's necessary condition. And one last point is that okay now want to look at this let's think of go one more comment. What about sufficiency? So that's the last condition is necessary. But claim lax plus genuine nonlinearity is sufficient. Because all we need is so that we have this right hand side which is of minus SH minus of minus minus SH minus okay we need that that be less than zero between we needed that this should always be it starts off well it's going from repeller here and it ends well it's an attractor over here we need that in the middle it still keeps going keeps going downward but what we can do is just call this phi of Let's plot that against Okay, we know five of is zero at the two end states and in fact five prime has the right that's minus it has the right signs. We need to know what happens on the interior. We want that it stays below the axis. But we notice that okay so pi prime of equals of minus pi prime prime of equals prime of which is positive by assumption. That's the genuine nonlinearity or sorry GNL which you could write genuine nonlinearity or genuine nonlinearity by lex can it be either way. So this is therefore this I'm sorry this is not this is prime prime yeah second derivative is positive is convex and therefore it lies beneath its seeant line so it's always negative or another way to say it is yeah all we need is that there is no intervening rest point. That's another way to say it. And we could see that by differentiating. Well, that's that's what we're doing really, aren't we? So, okay. So, we we get that that's pretty satisfying. Okay, let's see. maybe this is good time to take break. So we'll take five minute break and we'll come back and it's from 3 to 5, right? Yes. And the other is on Thursday. Yeah. And that one is only 3 till 4. Yeah. There's there's another lecture that dove tails after. It's not like real class for your students. Is it true? Yes, they can. yeah, but where are they? don't know. So, but could put also in my AOL this class. That's amazing. Yes, you could put it if could do that. Yes, think so. Yeah, it's cross-listed. You can as course, think. Okay. Through University of Toronto though, not Fields doesn't deal with the credits. was told. Okay. So let's come back at SP44. hello you there there there you are. Hi. Okay. want to get some coffee. hope you have coffee in your room or something. as well. Okay. So, we start we restart and so mentioned entropy before but in very vague way. Now I'm going to mention it in more strong way. So the definition of an entropy entropy flux pair let's see and I've called this ADA and is that data dq equals So that would be yeah. So if pick for for scaler any ADA has sign just the anti-derivative of da dq. So can pick any ADA that like. Okay. And one that like is that ADA should be let's see which one do like squared okay and then well I'll come back to this right now don't care what is ADA but what I'll is now I'll multiply the equation by ada by da and get dta which is da of * dth chain rule plus dx of equals zero because this is dai of of what did do there? the of of sh I'm sorry this is the ada and this is okay and this is data dq DXH and this is DT of So cancellation. So mean it's really da over DH. It's just the the differential. So D8 over maybe should write it out. D8 over DH DQ over DH equals DSI over DH. This is what want. Perfect Perfect derivative. Yeah, this might be easier to read. Well, it it's more things to write though. So, I'll leave it like this with the understanding. That's what it is. Okay. And right, this is actually negative, right? This is dxq, which is negative dt. And so we have cancellation. This term is da * dx So we look for this couple of functions and Yeah. Such is this relation. Yes. And it's always possible. It's always okay. In the scalar case, it's always possible to find here it is. You just take an anti-derivative of this. It's just product of some functions. You can take the anti-derivative. That's But in and in even in with two variables now it becomes much more restrictive. There are many of them but you can't choose them at will. For 3x3 systems in general, generically there would be none. But for gas dynamics, for plasma dynamics, many many many other things, it turns out there is one. There's physical entropy. It actually corresponds to the the thermodynamic entropy. There is one. Yeah. So for the the important physical systems, we have an entropy for nice equations of state. And so this is an important concept for the applications. Okay. So all right this is for smooth solutions. Okay. but now smooth invisides and that's the point is we're going to get something very interesting when we have the vis viscous term. So now let's take one more assumption why this should be like kind of entropy definition like well This is because in in the in the compressible the physical compressible gas dynamics equations if if you put in viscosity as here sorry without viscosity for the oiler equations there is thermodynamic entropy and it is it is conserved for smooth solutions but it's it's not conserved for shocks. This is what we're going to see. So that yeah it's an additional conserved quantity of the invisid equations that will turn out to be different in way for the viscous equations. Okay. So here now let's try the viscous case and take ada convex. So ^2 ada positive. And now let's see what we get. So we take the equation which is dh + dx = epsilon dx^2h and we multiply the whole thing by da. We know what happens to the left hand side. We just did that. We get da of dta of plus of And now the right hand side though is interesting. We're going to have data * dx^2 and we're going to manipulate that bit. So, so I'm going to write that as epsilon times. Okay, this looks like deta of dxh with dx kind of almost if the dx falls on on the the dxh term. But then we have to subtract the term when it falls on data which would be dxh d² data dxh. We subtract this term and notice this is negative this is this is coercive. It's negative quadratic form dxh because this is positive by convexity assumption. Now let's integrate. Take integral of 8 of dx and take the time derivative. Okay, that will be negative integral of dx of And and notice that can put that inside. And so that's negative dh. But then there's another term right then and there's integral of dx this is the the the conservative one the perfect derivative which is dxh that gives zero this gives zero but there's another term this term this is less than or equal to zero and in fact is strictly less than zero unless is constant. If there's ever any any gradient any nonzero gradient dxh then I'm going to get negative contribution. So the entropy will decrease. let's see did do do this right? let's see minus epsilon this thing that wrote here less than or equal to zero. that's dxh d^2a that's this. I'm just taking this term. and there is an epsilon. So, it's small but it's has sign. Okay. Okay. So the entropy with this setup with convex entropy the entropy decreases with time. The entropy the total entropy. Yes. But for the physical system it's actually concave entropy. Entropy increases where is emphasis because for most people Yeah. Yeah. Entropy increase decreases. Yeah. Yeah. Okay. So so this gives this gives another this gives an arrow of time, right? direction as entropy does. And that's kind of what was happening with the and cases. They were time reversed. They were identical, but time reversed and space reversed. So it says one one direction is okay, the other direction is not. Okay, we don't see that yet quite, but we'll we'll see it in minute. So now need to do another computation. And this one find pretty interesting. Something that neglected to say was it's actually traveling wave OD wasn't quite as wrote it. It's got an epsilon actually got an epsilon in front of the derivative and forgot to write that. But what you can do is take Take capital x= epsilon * little sorry capital equals epsilon * actually the other way around. So because have an epsilon here that makes the time derivative faster by one over epsilon. And what it means is I'm going to have very fast, very steep jump here. And what I'm doing is I'm actually stretching the coordinates out. And in these coordinates, it will look like just so by dx of bar. is unchanged. That's the point. If scale and together, the speed stays the same. and just get just remove the epsilon. Okay. So therefore of xt is equal to ep epsilon is equal to bar of - over epsilon where bar is the epsilon= 1 profile. And that means that this picture is roughly there's shock width of size epsilon or or shock layer width where there are features of the layer. And you might ask how can one see these these features in an actual shock wave? Well, if it's water wave, you can see it because they're bit spread out. If it's gas dynamical shock wave, first of all, it's happening usually inside reinforced tube that is not transparent. And so you have to see it with these either either with measurements through through wire like temp measuring temperature or with these shadow pattern, slicking patterns. and it's difficult but surprisingly people can detect the shape of of guest anvil shock wave and it's pretty close to to what's predicted by the viscous profile at least for decently low mock numbers that's so find that very interesting okay now want to compute what's the entropy flux maybe it's small is it going away as epsilon goes to zero. This is what what want to compute. But let's just take for fixed epsilon. So the entropy production will be integral of ^2 dx dx dx with an epsilon. Yeah, but this is an epsilon. epsilon times da of d^2 of epsilon. So I'm trying to reproduce. Let me let me make it bigger. That's way too small for the That's terrible. Okay, let me try it again. The entropy production is epsilon integral of dxh harp epsilon d^ 2 of bar epsilon dx epsilon dx and that's from minus infinity to plus infinity. What was the turn that was had special sign there up there? Yes. That was the turn that has special sign and Okay. Right. Okay. Now want to eval can evaluate this. It turns out because I'll change first of all I'll just use this to change variables to DH. DH bar. yes. So this should be bar dhar and this one should now be that's epsilon is bar of - / epsilon - over epsilon. So that will bring 1 / epsilon. It's 1 / epsilon dx bar. And should have done that. should have done that at the start because there's one there's can put there's an epsilon from this and an epsilon from this and can change them all to bars. There we go. And there's one over epsilon squared coming from the derivatives. And now there's no more epsilon. Okay. But now use the equation or dxh dxh is here it is it's the traveling wave od so it's going to be of bar minus bar ^2 and then minus constant ^2 8 of DH and with one. now it's one over epsilon here. And let's see. Yes, it there will be let's see there should be this should have been an epsilon. Yes, it should have. There was an epsilon. Yep. So, at this point, yeah, no more epsilons. Yeah, now the epsilon's cancelceled. So we see already it's constant. It's independent of epsilon. But not only that, but the there's also the the xh at the end. So you pay one epsilon for the first and one epsilon for the other. That's yeah, that's how got this. So changed from epsilons to bars. Yes. And then that's where the one over epsilon squared came from. had an epsilon because it it comes in front of the term. And then yes, when when when used just moment now. did then they then think that actually think that by my pants by sorry. It should have already been should have just been this. no, no, no, wait, wait. feel like I've still got one over epsilon, but something what happened here. This is the best computation. let's see. no, no, no. used let's see of equation for bar it's written in terms of capital right yes so it doesn't you have to change with respect to with the capital Yes, that was it. This was demo. Let's see. This one is let's see. See, maybe should have just and then when you switch from yes. Yes. Yes. You're right. because small and so yeah if keep small which is maybe what want if keep small and this one just becomes an there's no epsilon there from that and then this one just trade in trade in for one over yes trade in for boy Yeah, I'm going to get Okay, I'm not sure why it's not coming out. I'm going to get something the epsilons cancel and get something constant. And not only that, can figure out what the constant is. Yes. Yes, see. Now, because you write dxh dx as dh. Mhm. And so you pay one epsilon for dxh. So there was an epsilon still. Yeah. Yeah. Yeah. So now this is So now it's going to balance. Yeah. Yeah. Yeah. Now it's going to balance. Okay. Yes. There we go. There we go. Yes. And so it's constant independent of epsilon. And moreover, can integrate by parts to find out what it is. Yeah. Because if integrate by parts. Okay. So it'll be this times DH. So it'll be minus sh deta from minus infinity to plus infinity. Okay. But this is the derivative. This this vanishes at minus infinity or or I'm sorry it is let's see this is yeah this vanishes because it's equal at plus infinity and minus infinity. That's that's the condition to be rest point. That's the ranking Hugo. when you because this had it wasn't just I'm sorry. It wasn't just It was minus the value at infinity. And now I'm taking this time data from minus infinity to plus infinity. this vanishes at both ends. It's the condition. It's the derivative and it's the condition to be rest point. So it contributes to nothing. Then the other term integration by parts. Now get negative integral of dq da minus da dh from minus infinity to plus infinity. But this is So they're both perfect derivatives. And get jump in minus jump in which is the invisid entropy flux but it's not zero. So this is like visc sometimes people talk about the viscous anomaly in mixing and things that there's some effect left over from the viscosity even when the viscosity goes to zero and this is one of those examples because the entropy dissipation is independent of epsilon for these viscous shock profiles and you can take epsilon down to zero and you get the normal entropy production And just from looking at flux across the shock and we know because of this because of of the original formulation this is not zero. So entropy is not conserved. It has sign. So it's there should have been negative in front of everything. Okay. So this shows us that that the entropy indeed is going to give us preferred direction. Let's find out what it is. Okay, let's take I'm going to take = ^2 over 2. I'm going to take = ^2 over 2, which by the way makes this burgers equation. So, so data is = ^2 ADA = 1. And okay, don't need ^2 just need that dq a= dq = that's h2. And so is the integral of dta dq and that's cubed over 3. That's my this is my entropy and there's my entropy flux. Now let's compute jump in minus jump in theta. But first let's compute which is jump in over jump in And that's +^2 / 2 - - 2 / 2 / + - hus. And that is the average h+ + minus / 2. So that's Okay. So here this is the jump in cubed over 3. So that's + cub - - cub over 3 minus + - 2 that's the and then ada is + 2 - minus 2 over 2. That's the jump in ada. Okay. And I'm not sure can make this come out right, but it turns out this is + - minus cubed over 12. That's what it turns out to be. So it's in in in the shock strength jump in This is the shock. This is kind of the shock strength the difference in heights. So it's cubic very small in the shock strength. So if the shock were small, you you want to neglect it almost and the same thing would be true for any convex entropy. mean it wouldn't be nice jump in cubed over 12, but it would be some something positive. And we know that by the derivation. We could do the same argument in reverse. Here we started with the viscous entropy production and ended with the invisid production. But we could follow all the steps in reverse to find out that in very funny way that the production is positive. Okay. And that yeah when you choose the entropy and entropy first you have you have choice right you can choose yes yeah and any any convex one would kind of mimic the properties of the physical entropy in in the oiler equation yeah you have lots of yeah but you for the physical equations you have no choice there's only one essentially for oiler and even fort on gas believe there's only one convex entropy up to you could add of course any linear function linear functions are automatically entropies linear function of it's just multiple of the original equation because it would just the you know + * and the gives zero in the equation which is perfect derivative and the ch gives times the the previous equation which is then zero. So any yeah any any linear linear affine function of will satisfy the original you know the same kind of equation it will still be conservation law and that's what it means to be an entropy. that it satisfied conservation law with some flux but but notice should have said this is odd that's important so the sign will change if flip minus and H+ this sign will change so again it's giving us directionality and it tells us case is good case is bad and and This would be true for yeah this is more of an invisid limit computation just want to say kind of one one more thing here it is all right is but maybe let's stop here so we've had few things what we did the consistency which is just wellposeness of this problem we we made up this free boundary problem that we supposed but then we use Gon's criterion that if we're looking for vanishing viscosity well surely there should be some viscous profile near the original shock that's pretty nice it's kind of leap but it it makes sense and using that you get more more convincing argument think this is maybe even more convincing with the entropy and then now I'm going to say I'm going to lead into the time asmtoic stability question from here. But did you Reno, did you have question? No. Okay. So, the last topic for today is the setup for what we're going to do in the rest of the time. We've already more or less done it. Okay. Okay, the last point is the large time versus small viscosity limits. We've in way already seen this. Let's take capital x= little over epsilon. Capital t= little over epsilon in the viscous problem. And that takes us to dt plus capital of equals just dx² capital x^2 the epsilon= 1 problem. But what we what we've really done is we've taken this kind of microscopic shock layer of of width epsilon and we've blown it up to be width one. So this was this is in the capital coordinate. Okay. So we would like you know what do we want? by the way maybe comment that missed that should have said what about that consistency problem the wellposedness wellposedness in what space? It would not be in C1 say because we've already seen singularities can form so if you have an invisid shock and and you have some other things around in the solution probably they'll form another shock out here. So it's going to be short time well postedness in in H1 or C1 or any smooth space. That's all one can hope. Okay. And and that is all all that's been done so far. Short time. the famous works of of of Maida and Christ and Mativier these are all just to establish bounded well posess for some small time local theory now with with the viscous problem we'd like at least that much so we'd like small time we'd like capital less than or equal to say one or maybe some other small con constant but that's what we would like this is equivalent to I'm sorry no little capital would be less than equal to 1 / epsilon it's large time existence it's it's basically stability type problem we would like bounded some kind of bounded stability you well posess up to that time and from there it's maybe small small leap to what we're going to examine. We're going to examine time asmtoic stability for the epsilon equals 1 problem. So the fully viscous problem not vanishing viscosity. So we want to we want to show that not only yeah well we'd like to show at least that it remains perturbed viscous profile remains somewhere near perturbed viscous profile for all time. And that's what we would need if we were if we were to get you know epsilon is going to infinity. So that's what we would need in order to make sense of kind of vanishing viscosity problem. Are they equivalent? No, they're not. And they're not because didn't mention what's the data. The data are different. It's because if we take perturbation the L2 norm of tilda of the perturbation what if we write that now in the capital coordinate the stretched out coordinate that becomes 1 over epsilon times oops each. It's much bigger. So if because want I'd like this to just to be right. I'm thinking of very narrow shock. But the perturbation I'm thinking should be just maybe L2 or something in the little coordinate. But when blow things up, that stretches everything. It makes the L2 norm. that's one over square root of epsilon. say 1 over square root of epsilon because now the integral increases the integral of tilda squ increases by factor 1 over epsilon take the square root 1 over square of epsilon and that's why those problems are not exactly the same but they're related and for work on this allowing well it actually the only know one work allowing data like this and that's my academic Brother Shashen, you did some work on this. There are related works by Goodman and Shin, by Grenier and Ruse. And there are several works by Olivier Guess Givier, Mark Williams and me. There are maybe six or seven papers about that that that problem but not allowing but not allowing the one over square root of epsilon or like just showing existence of nearby solutions. It's different problem but I'm not going to do that in this class. That's lot of extra machinery that's in different direction than we're going. But they might it might be interesting to just look at what the results are that are known. Okay. there but it's not the only motivation and it's actually not my real motivation. My real motivation is that the physical equations do have some viscosity and we do see shocks propagate over long distances unchanged. They're coherent structures that are important in compressible Navia Stokes. So we would like to understand their stability and that was mentioned at at at one point as very important open problem and think the first the first big result on that was Jonathan Goodman and think it's 80 six and then Lou no 84 86 typing Lou had partial revol this result was for perturbations without mass but it introduced many of the ideas that were important. this one introduced some further ideas. The first full result was sorry was was and Shen feel like that's 92 or so and then on and on whole lot of results. Most of these were for for small shocks. So what my point of view is don't want to restrict small shocks. They were small shocks and they proceeded by careful understanding of the small shock structure and building from there. sort of like in the description of singularity formation that we saw in the lectures of the previous week. They used there there were used closeness to hop core and special special techniques that came from knowing you were close to burger shock. We want to leave that and just see this in terms of spectra alone because we don't know the structure of large shock is pretty random. depends very much on the equation of state. We don't know what it is and we don't have any nice exact solutions to build from. So, we're going to try to do nonlinear stability independent of shock strength or structure. Meaning, don't don't don't know what the wave looks like whatsoever. only know the two end states. I'll write minus and plus now. and maybe little vector sign to indicate that I'm no longer talking about this scalar model which one could conceivably solve to find the structure. But for more complicated vector models, the structure can be well it can be variable quite variable and it's it's not small. I'm going to put viscosity one. Yeah, there is no small parameter there. There's no small parameter but there's going to be an assumption. So you have to pay somewhere right and the only assumptions are going to be assuming spectral stability condition of the spe of the spectrum of the lines. Yes. And this is point spectrum of the linearized operator which we're going to look at next week. or on Thursday. On Thursday, we're going to look at the linearized operator, what it looks like, and talk about what the spectra look like or begin to talk about that. We're going to assume that it's good. Now, this is something that could be checkable numerically. So, that's the idea is if it's big shock, well, you would pretty much have to find the profile numerically to begin with. You're not going to find an nice expression for it. And then if you wanted to know the linearized the the point spectrum you could find that by also by numerical methods respect small perturbation. So line then so we'll think of small perturbation added to this. This is given shock and we're going to perturb we're going to give as data this shock plus small perturbation and we want to know what happens to that. We'd like to show that it converges back to this shape. Yeah. For long time. So as time goes to infinity, we want to show it converges to stay well, first we'd like to know it stays near this shape, but actually we're going to show it converges to this shape. Sign is sign. Yeah. As for OD, we would like the the real part of the spectra to be negative. But this is what I'm going to I'm going to talk about, you know, start from OD on Thursday. Yeah. So, so we'll we'll we'll really discuss this, but this is the overview. We want to just assume things about the the spectra which could could be checked especially these days one would hope it could be checked by numerically assisted proof rigorous numerical proof it certainly can be checked by very accurate and convincing numeric non-rigorous we've done that we even have one Blake Barker and even have one result for fort onropic gas dynamics of numerically assisted proof about the spectra but it turned was very hard at the time that was in 2015. The field has advanced tremendously since then. So think for instance think Gio's group or Blake Barker's group could do this. Yeah. yeah with Javier Gomerano and and all the people attacking really tough problems with numerically assisted proof. feel that yeah if not tomorrow will pretty soon people could do that it's anyway this is starting point it's way you can well just assuming this it turns out you can find all kinds of behavior so that's surprising and you don't need to know all the details but we'll see why and how in the coming lectures and think you know it's it's actually very fun just from the point of view of spectral theory of unbounded operators, how how do you show nonlinear stability of of you know PTE and this one is just little bit different from what think you might be used to even if you do it. If you haven't done it at all, it's interesting. But if you've done different things with reaction diffusion fronts or something, this will look different the the way that we do it. little bit different, but recognizable and usually that's fun. Okay, so I'm going to stop there. Next time we're going to start in on the analysis. This would all you know in way digression, but think these are important motivations and actually these ideas will recur. the ranking hugonio the lacks characteristic condition and and much later entropy in certain examples will show up. Okay, thanks. Stop there. perfect. Condition is little bit point spectral because you know already know that the continuous one is turns out. Yeah, this is something you can compute the the essential the continuous or essential spectrum can be computed. you know basically mean no even though drew this as very strange still know the strange part is confined to bounded domain and then it's exponentially approaching some limit so out here it behaves very much like constant solution and it turns out all basically the point spectrum encodes near field dynamics things on the bounded region and the essential spectra encode information about far field and also interactions across this layer. But let's see no it does encode that but no in way but first of all it encodes we we can compute the essential spectrum by linear algebraic study by just looking at stability of the constant states and you know spectra of the constant states those basically control the continuous spectra of the whole thing kind of like the union of the two spectra. because they're two states. Yeah. You Yeah. Yeah. So there should be discrete spectrum associated with this part and continuous spectrum associated with the outside and that which is computable. Yes, you do. Yeah. That's Yeah. But you'll see that we don't exactly do matched asmtoics. This is more just uristics to help us understand what's going on and it's not really what we do. The analysis is actually very well yeah it doesn't look like asmtoic development. It just looks like some some very sort of almost soft estimate but you're getting you're getting very good estimates. But the the method by which you do it is is is not complicated is not there's not let's see that's not completely for the scalar case it's not complicated. There is some bookkeeping when you have vector case but we know that even from the invisit case right you have many characteristics the characteristics the flow along characteristics inter they they are coupled. So this is complicated and we're going to see an echo of that in the viscous problem. We can't avoid it. But I'm not going to be matching gluing across boundary or anything like that. It's something easier. Basically I'm going to convert it into two constant coefficient problems to the right of zero and to the left of zero. Almost like the invisit case. But I'm getting way ahead. probably shouldn't even mention it, but but thank you on Thursday. Thanks for coming, guys. Thursday. Thank you. Do you have snow back home? no. Well, in Bloomington there's no snow right now. don't know about Michigan. think there's another winter storm something predicted for them but don't really know. Yeah, think Michigan would be better bet. Okay. So, the next lecture is on Thursday at 3 3 to 4 but before that we have lecture by by in our reading group. So, see you tomorrow. Bye guys. Thanks Kevin. Thanks, Kevin. Professor