On a body with a cavity filled with compressible fluid
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Macha, Vaclav
Galdi, G. P.
Nečasová, S.
We concern the system consisting of a moving body filled with a compressible fluid. We present several existence proofs, however, our main aim is to deal with the long-time behavior of the whole system.
 for invitation here. So it's first time I'm in Banff and it's marvelous place of course. Well, so actually my presentation or there will be several results presented in the next few minutes. And these results are done or were done in collaboration with Professor Galdi, Professor Nechasova and also Bangue Shea. And also I'm very proud that there will be two pictures in my presentation, you know, because usually there are no pictures in my presentation at all. Okay, so well, I'll talk about the body with cavity filled with compressible fluid and okay, so let me switch to my first picture. Yeah, so that's it. I know it's the resolution of this picture is not so high. But I would just like to introduce say my system. Yes, so the system consists the system is called s and it's consist of a body and cavity, as you can see here. So the body can freely move in space and contains some cavity and the cavity is assumed to be filled with some kind of fluid. Yes. And actually, I will present here several results, mainly concerning the long time behavior of such system. Okay, so let me start briefly with history of such such research. Actually, the study of this motion traces back to the pioneering contributions of Stokes, Jukovsky, Poincaré and also Sobolev. And actually, Jukovsky claimed that the motion of such system will be stabilized after some time actually will eventually be rigid motioned and precisely permanent rotations. Well, oh, sorry, sorry. There are many, many works, but say the full generality or for full general system. It was shown by a dessert, Galdim, and in 2016, that actually the Shacoj, Jukovsky conjecture holds holds and they treat the incompressible fluid inside the body. And our aim is to look into the role of the compressibility of that fluid. Yes. So we just put a compressible fluid inside the body, inside the free moving body. Okay, so let me introduce the governing equations. Well, these are usual, the first two equations are usual compressible Navier-Stokes equations. Well, the first equation is momentum equation. Actually, you can see the stress tensor contains some pressure. Yes, so the pressure is given by some barotropic law, a times r power to gamma. This a and gamma are supposed to be positive constants. I will later on show the restriction on them. Then the second equation is a continuity equation or conservation of mass. So the first two equations, as I said, these are usual compressible Navier-Stokes. And then the third equation is a full slip on the boundary. Yes, so you can see here, actually this w is the velocity of the fluid. And on the boundary, the velocity actually is just a rigid motion. Yes, so the rigid motion of the body here is here represented by this omega and this eta. Yes, so this omega is a rotation and this eta is transition. And next we have equations for the rigid motion. Yes, so the first or the fourth equation actually here is the conservation of angular momentum. And the last equation is a conservation of linear momentum. Yes, so you can see actually the fourth equation is a equation for this omega for this rotation. And the fifth equation is equation for transition. Yeah. Okay, so here maybe it's a bit unusual. Here this r is density and w is velocity. Say the reason for this notion is that these two unknowns are on the moving domain. We can switch to the rigid motion. Sorry, we can switch to the non-moving domain. Yes, to the constant in time domain by certain transformation. And we can deduce actually this system. Yes, so this time the domain is fixed. Yes, so you can see it's more convenient. Here just there is two velocity fields, actually velocity v and velocity u. Velocity u is a velocity of the fluid. Together with the rigid motion velocity v is the velocity of the fluid, which is just without the rigid motion. Yes, so this velocity is with respect to the body. Okay, so now let me briefly just mention some of our prior estimates. So we have conservation of mass, we have energy inequality. The energy contains not only the kinetic energy of the fluid and the pressure energy of the fluid, but there is also kinetic energy of the body. These are these two terms. And of course once we have this energy estimate, we are also able to deduce the existence of a big solutions by say standard, theorized Lyon's theory. Okay, so however, because of the boundary condition we have in mind, yes, because the body may possess any free motion, we are also able to deduce the conservation of the modulus of the angular momentum. And this is something that will be presented here without any further technical details. But actually if I introduce the quantity, which is really called m, yes, so this is a tensor of inertia of the body times omega plus integral over the cavity of rho x times u. Then actually this particular quantity fulfills this ODE, yes, so the time derivative of m plus omega times m equals zero. And of course we can multiply the equation by m in order to deduce that the total angular momentum of the whole system is equal to constant, yes. So that's quite nice because actually the system cannot tend to stop, yes. So say the trivial solution is not the correct candidate for the long-time behavior. Okay, so in order to find some candidates, let me consider the steady state system, which of course looks like this, these are the equation without time derivatives. And actually there is one issue, there is one non-uniqueness which might appear due to say some vacuum regions. Actually this non-uniqueness was described in papers by Fiery-Reisel and Petzeltova from 1998 and 1999. And actually this non-uniqueness is quite a bad issue because we would like to have a unique candidate for the long-time limit, yes. So once there will be a unique candidate for the long-time limit, I can say that we are done. Okay, so in order to prevent this non-uniqueness, we would like to assume that there are no vacuum regions and yes, okay, so that will be our assumptions from now on. And we can actually deduce say several lemmas observations. So first of all, a weak solution to the steady state system, if there is any, then it must hold that the relative motion Vs is identically equal to zero. And actually, if there is any weak solution to the steady state system, then we have this particular set of equations and these equations are algebraic equations and not longer PDEs. Yes, so that's quite important for us. Okay, and now I would like to talk also about one existence that is out to the non-steady case. Actually, we are able to deduce that there exists a strong solution. Yes, once the initial data is small enough, yes. So I don't want to comment on this on, but we would like to show that actually every strong solution which is constructed for small initial data converge to some, say, long time limit. Yes. Okay, so in order to show that there is just one candidate, we use the omega limit set. Yes, so just let me recall that omega limit set is a set of all quadruples here denoted by V hat, omega hat, psi hat and rho hat, for which there exists an increasing unbounded sequence of times such that V converges to V hat, omega converges to omega hat, psi converges to psi hat and rho converges to rho hat. And actually, the omega limit set, it is compact, it is connected and it is non-empty. So that's quite standard properties. Secondly, since the density of the strong solutions is bounded from below and from above, then the point in the omega limit set does not contain a vacuum region. Yes, so we can see that the density is always bounded from below and from above. And actually, the omega limit set is invariant under solutions constructed in the previous theorem. Yes, so once there is a strong solution emanating from the point of omega limit set, then such solution remains in that omega limit set. Okay, and now I would like to show or I'm going to show that, say the omega limit set consists of isolated points, because then of course, because the omega limit set is connected, then there is just one point in the possible candidate for the long time limit. And that's our aim here. Yes, so actually, every element of the omega limit set solves the steady state system. And moreover, we have still this conservation of mass and conservation of the magnitude of angular momentum. Yes, so these two equation has to be added to the steady state system. And okay, I'll go a bit quickly through this. Actually, we end up with this set of equations. This is system of eight equations, because actually the first one and the third one, these are system of three equations. Yes, so we can rewrite it as a system of eight unknowns and say eight equations for appropriately defined nonlinear function f, it can be seen as this. Yes, and our aim or our goal was to show that actually the gradient of f under certain condition is regular, because of course, once the gradient is regular, then there are not finitely many, but there are only isolated points can be a solution. So that is our aim. Okay, so sorry, so let me go quickly through that. Actually, this is the gradient of f. You can see it consists of some main part and something what should be negligible, hopefully. Yes, so it consists of some derivatives here. And actually, once we assume that the inertia tensor of the whole system has three distinct value values, and that this rest is small, then actually, the gradient of f is regular. Yes, so this can be of course, claimed under certain conditions. So we end up with this theorem. So actually, if we have a cavity which is of class C4, this is to have this strong, strong existence result. And if we more or less assume that the three eigenvalues of the tensor of the inertia tensor of the whole system are distinct, then there exists a zero such that for all a bigger than a zero, the terminal motion of the coupled system reduces to uniform rotation around an axis parallel to the constant angular momentum. M zero of s passing through its center of mass g. Okay, so here I would like to comment this bit, because here we have a constant a, and I would like to go back to my third slide, I guess. Yes, so okay, fourth slide. Actually, this constant a, it's important, is here in the definition of the pressure. And actually, in order to obtain our main claim, we need to have this constant a sufficiently high. So this means that the main claim is true only for sufficiently incompressible fluid. Yes. Okay, so such theorem is of course, not very suitable for some dependence between the decay rate and the compressibility of the fluid. So that was our main aim. So that's why we treated another case. So this time, we take into account just a system concerning pendulum, not freely moving body, but the pendulum. And in our case, the pendulum is a body which might move, which has just one degree of freedom. Yes, so it can move freely around one axis. Yes. So you can see once again, the system is very similar. However, this time the rigid motion is just described by a scalar omega, and it is expressed as omega times E3 cross x. Yes. So actually, the whole body might move around the E3 axis. Yes. And here, these last two equations are equations for the mechanical oscillator. Yes. Here, this g is actually a direction of gravity. Okay. Of course, once again, we have energy estimate. You can see here the last term or say the one term on the right hand side is actually potential force. And that's why once again, we can deduce the existence of a weak solution. But when we turn our attention to the steady state system, it's much more interesting now because it turns out that we are able to prove some kind of uniqueness. Yes. So that's the stationary system. I would like to comment this on a bit. So first of all, this gamma minus one root of this bracket of this gamma minus one divided by a times gamma times gx plus c, this is actually a density, yes, which is a function of g and c. So actually, the first equation here is the conservation of mass. Then we have here this second equation. Actually, this L, this can be seen as a center of gravity of the whole fluid. This P is just a projection to the first two variables. Yes. So actually, this P of this bracket is just a center of gravity of the fluid. And here you can see there is a scalar times a times some direction of gravity. Yes. And then we have, okay, so the gravity is always the same, or it has always the same magnitude. Actually, we can show that if some density and some direction of gravity is a minimizer of energy functional, then say these two values solve the above system. And we can use the direct method of the calculus of variation to show that there exists a minimizer. So this means that we have an existence of solution to this system. And actually, under certain condition, we are also able to show uniqueness, namely, okay, so since I don't have so much time, I will skip these assumptions here. Just tell me that there are at most two solutions under certain assumptions to the steady state system. One is with D less than zero and one is with D greater than zero. So let me go back. If you take a look here on the second equation, then actually D is something what multiplies the direction of gravity as, and actually, if D is greater than zero, then it means that the direction or the gravity has the same direction as the center of, as the body center of the whole system has. So actually, if D is greater than zero, then you can, you take this particular steady state. If D is less than zero, then you get, say, this upside down steady state. Of course, the second one might be ruled out somehow. For example, we can start with energy, which does not allow to obtain, say, this upside down steady case. Yes. So, and once we, of course, rule out this second steady case, then we are able to claim that actually every renormalized weak solution tends to just one steady case. Yes. So that's our main theorem here. And this pendulum case is for us much more convenient because we are now able to look into the problem of decay rate and how the decay rate of the solution depends on the, on the compressibility of the fluid. And here comes my second picture. Yes. So this is my second picture of this talk. So you can see this is a numerical simulation. And if we take, okay, so this A, this is the constant in the pressure law, actually, if he take A to be small, A equals 0.1, then you can see this is dark line. And actually, the decay is quite fast. And if A is equal to 100, so this is the, say, light line. And you can see that there is still some decay, but it's not so fast. So from this numerical simulation, it follows that actually compressible fluids provides better decay than incompressible fluids. Okay. So that was all. And I would like to thank you for your attention.
 for invitation here. So it's first time I'm in Banff and it's marvelous place of course.  Well, so actually my presentation or there will be several results presented in the next  few minutes. And these results are done or were done in collaboration with Professor  Galdi, Professor Nechasova and also Bangue Shea. And also I'm very proud that there will  be two pictures in my presentation, you know, because usually there are no pictures in my  presentation at all. Okay, so well, I'll talk about the body with cavity filled with compressible  fluid and okay, so let me switch to my first picture. Yeah, so that's it. I know it's the  resolution of this picture is not so high. But I would just like to introduce say my system.  Yes, so the system consists the system is called s and it's consist of a body and cavity, as  you can see here. So the body can freely move in space and contains some cavity and the  cavity is assumed to be filled with some kind of fluid. Yes. And actually, I will present  here several results, mainly concerning the long time behavior of such system. Okay, so  let me start briefly with history of such such research. Actually, the study of this  motion traces back to the pioneering contributions of Stokes, Jukovsky, Poincaré and also Sobolev.  And actually, Jukovsky claimed that the motion of such system will be stabilized after some  time actually will eventually be rigid motioned and precisely permanent rotations. Well, oh,  sorry, sorry. There are many, many works, but say the full generality or for full general  system. It was shown by a dessert, Galdim, and in 2016, that actually the Shacoj, Jukovsky  conjecture holds holds and they treat the incompressible fluid inside the body. And our  aim is to look into the role of the compressibility of that fluid. Yes. So we just put a compressible  fluid inside the body, inside the free moving body. Okay, so let me introduce the governing  equations. Well, these are usual, the first two equations are usual compressible Navier-Stokes  equations. Well, the first equation is momentum equation. Actually, you can see the stress  tensor contains some pressure. Yes, so the pressure is given by some barotropic law,  a times r power to gamma. This a and gamma are supposed to be positive constants. I  will later on show the restriction on them. Then the second equation is a continuity equation  or conservation of mass. So the first two equations, as I said, these are usual compressible Navier-Stokes.  And then the third equation is a full slip on the boundary. Yes, so you can see here,  actually this w is the velocity of the fluid. And on the boundary, the velocity actually  is just a rigid motion. Yes, so the rigid motion of the body here is here represented by this  omega and this eta. Yes, so this omega is a rotation and this eta is transition. And  next we have equations for the rigid motion. Yes, so the first or the fourth equation actually  here is the conservation of angular momentum. And the last equation is a conservation of  linear momentum. Yes, so you can see actually the fourth equation is a equation for this  omega for this rotation. And the fifth equation is equation for transition. Yeah. Okay, so  here maybe it's a bit unusual. Here this r is density and w is velocity. Say the reason  for this notion is that these two unknowns are on the moving domain. We can switch to  the rigid motion. Sorry, we can switch to the non-moving domain. Yes, to the constant  in time domain by certain transformation. And we can deduce actually this system. Yes,  so this time the domain is fixed. Yes, so you can see it's more convenient. Here just  there is two velocity fields, actually velocity v and velocity u. Velocity u is a velocity  of the fluid. Together with the rigid motion velocity v is the velocity of the fluid, which  is just without the rigid motion. Yes, so this velocity is with respect to the body.  Okay, so now let me briefly just mention some of our prior estimates. So we have conservation  of mass, we have energy inequality. The energy contains not only the kinetic energy of the  fluid and the pressure energy of the fluid, but there is also kinetic energy of the body.  These are these two terms. And of course once we have this energy estimate, we are also  able to deduce the existence of a big solutions by say standard, theorized Lyon's theory.  Okay, so however, because of the boundary condition we have in mind, yes, because the body may  possess any free motion, we are also able to deduce the conservation of the modulus  of the angular momentum. And this is something that will be presented here without any further  technical details. But actually if I introduce the quantity, which is really called m, yes,  so this is a tensor of inertia of the body times omega plus integral over the cavity  of rho x times u. Then actually this particular quantity fulfills this ODE, yes, so the time  derivative of m plus omega times m equals zero. And of course we can multiply the equation  by m in order to deduce that the total angular momentum of the whole system is equal to constant,  yes. So that's quite nice because actually the system cannot tend to stop, yes. So say  the trivial solution is not the correct candidate for the long-time behavior. Okay, so in order  to find some candidates, let me consider the steady state system, which of course looks  like this, these are the equation without time derivatives. And actually there is one issue,  there is one non-uniqueness which might appear due to say some vacuum regions. Actually this  non-uniqueness was described in papers by Fiery-Reisel and Petzeltova from 1998 and 1999. And actually  this non-uniqueness is quite a bad issue because we would like to have a unique candidate for  the long-time limit, yes. So once there will be a unique candidate for the long-time limit,  I can say that we are done. Okay, so in order to prevent this non-uniqueness, we would like  to assume that there are no vacuum regions and yes, okay, so that will be our assumptions  from now on. And we can actually deduce say several lemmas observations. So first of all,  a weak solution to the steady state system, if there is any, then it must hold that the relative  motion Vs is identically equal to zero. And actually, if there is any weak solution to the  steady state system, then we have this particular set of equations and these equations are algebraic  equations and not longer PDEs. Yes, so that's quite important for us. Okay, and now I would like to  talk also about one existence that is out to the non-steady case. Actually, we are able to deduce  that there exists a strong solution. Yes, once the initial data is small enough, yes. So I don't  want to comment on this on, but we would like to show that actually every strong solution which  is constructed for small initial data converge to some, say, long time limit. Yes. Okay, so in  order to show that there is just one candidate, we use the omega limit set. Yes, so just let me  recall that omega limit set is a set of all quadruples here denoted by V hat, omega hat,  psi hat and rho hat, for which there exists an increasing unbounded sequence of times such  that V converges to V hat, omega converges to omega hat, psi converges to psi hat and rho converges  to rho hat. And actually, the omega limit set, it is compact, it is connected and it is non-empty.  So that's quite standard properties. Secondly, since the density of the strong solutions is  bounded from below and from above, then the point in the omega limit set does not contain a vacuum  region. Yes, so we can see that the density is always bounded from below and from above. And  actually, the omega limit set is invariant under solutions constructed in the previous theorem.  Yes, so once there is a strong solution emanating from the point of omega limit set, then such  solution remains in that omega limit set. Okay, and now I would like to show or I'm going to show  that, say the omega limit set consists of isolated points, because then of course, because the omega  limit set is connected, then there is just one point in the possible candidate for the long time  limit. And that's our aim here. Yes, so actually, every element of the omega limit set solves the  steady state system. And moreover, we have still this conservation of mass and conservation of the  magnitude of angular momentum. Yes, so these two equation has to be added to the steady state  system. And okay, I'll go a bit quickly through this. Actually, we end up with this set of equations.  This is system of eight equations, because actually the first one and the third one,  these are system of three equations. Yes, so we can rewrite it as a system of eight unknowns and  say eight equations for appropriately defined nonlinear function f, it can be seen as this. Yes,  and our aim or our goal was to show that actually the gradient of f under certain condition is  regular, because of course, once the gradient is regular, then there are not finitely many,  but there are only isolated points can be a solution. So that is our aim. Okay, so sorry,  so let me go quickly through that. Actually, this is the gradient of f. You can see it  consists of some main part and something what should be negligible, hopefully. Yes,  so it consists of some derivatives here. And actually, once we assume that the inertia tensor  of the whole system has three distinct value values, and that this rest is small, then actually,  the gradient of f is regular. Yes, so this can be of course,  claimed under certain conditions. So we end up with this theorem. So actually, if we have a cavity  which is of class C4, this is to have this strong, strong existence result. And if we  more or less assume that the three eigenvalues of the tensor of the inertia tensor of the whole  system are distinct, then there exists a zero such that for all a bigger than a zero, the  terminal motion of the coupled system reduces to uniform rotation around an axis parallel to  the constant angular momentum. M zero of s passing through its center of mass g. Okay,  so here I would like to comment this bit, because here we have a constant a, and I would like to  go back to my third slide, I guess. Yes, so okay, fourth slide. Actually, this constant a,  it's important, is here in the definition of the pressure. And actually, in order to obtain our  main claim, we need to have this constant a sufficiently high. So this means that the main  claim is true only for sufficiently incompressible fluid. Yes. Okay, so such theorem is of course,  not very suitable for some dependence between the decay rate and the compressibility of the fluid.  So that was our main aim. So that's why we treated another case. So this time, we take into account  just a system concerning pendulum, not freely moving body, but the pendulum. And in our case,  the pendulum is a body which might move, which has just one degree of freedom. Yes, so it can  move freely around one axis. Yes. So you can see once again, the system is very similar. However,  this time the rigid motion is just described by a scalar omega, and it is expressed as omega times  E3 cross x. Yes. So actually, the whole body might move around the E3 axis. Yes. And here,  these last two equations are equations for the mechanical oscillator. Yes. Here, this g is  actually a direction of gravity. Okay. Of course, once again, we have energy estimate. You can see  here the last term or say the one term on the right hand side is actually potential force. And  that's why once again, we can deduce the existence of a weak solution. But when we turn our attention  to the steady state system, it's much more interesting now because it turns out that we  are able to prove some kind of uniqueness. Yes. So that's the stationary system. I would like to  comment this on a bit. So first of all, this gamma minus one root of this bracket of this gamma  minus one divided by a times gamma times gx plus c, this is actually a density, yes, which is a  function of g and c. So actually, the first equation here is the conservation of mass. Then we  have here this second equation. Actually, this L, this can be seen as a center of gravity of the  whole fluid. This P is just a projection to the first two variables. Yes. So actually, this P of  this bracket is just a center of gravity of the fluid. And here you can see there is a scalar times  a times some direction of gravity. Yes. And then we have, okay, so the gravity is always the same,  or it has always the same magnitude. Actually, we can show that if some density and some direction  of gravity is a minimizer of energy functional, then say these two values solve the above system.  And we can use the direct method of the calculus of variation to show that there exists a minimizer.  So this means that we have an existence of solution to this system. And actually, under  certain condition, we are also able to show uniqueness, namely, okay, so since I don't have  so much time, I will skip these assumptions here. Just tell me that there are at most two solutions  under certain assumptions to the steady state system. One is with D less than zero and one is  with D greater than zero. So let me go back. If you take a look here on the second equation,  then actually D is something what multiplies the direction of gravity as, and actually,  if D is greater than zero, then it means that the direction or the gravity has the same direction as  the center of, as the body center of the whole system has. So actually, if D is greater than  zero, then you can, you take this particular steady state. If D is less than zero, then you get, say,  this upside down steady state. Of course, the second one might be ruled out somehow. For example,  we can start with energy, which does not allow to obtain, say, this upside down steady case. Yes.  So, and once we, of course, rule out this second steady case, then we are able to claim that actually  every renormalized weak solution tends to just one steady case. Yes. So that's our main theorem  here. And this pendulum case is for us much more convenient because we are now able to look into  the problem of decay rate and how the decay rate of the solution depends on the, on the  compressibility of the fluid. And here comes my second picture. Yes. So this is my second picture  of this talk. So you can see this is a numerical simulation. And if we take, okay, so this A,  this is the constant in the pressure law, actually, if he take A to be small, A equals 0.1, then you  can see this is dark line. And actually, the decay is quite fast. And if A is equal to 100, so this  is the, say, light line. And you can see that there is still some decay, but it's not so fast. So  from this numerical simulation, it follows that actually compressible fluids provides better  decay than incompressible fluids. Okay. So that was all. And I would like to thank you for your  attention.
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