The Global Solutions Of Some Partial Differential Equations In Fluid Dynamics

We mainly study two kinds of partial differential equations in fluid dynamics. One is the compressible Euler equations with damping for an ideal gas, another is Camassa-Holm equation. It contains the following five parts. 1. We study the compressible Euler equations with damping (CEED) for an ideal gas in Chapter 2. The damping coefficient is a positive constant. When the initial density has compact support and the total mass of the gas isn't zero, then the regular solution of the Cauchy problem (CP) of CEED must blow up in finite time. This regular solution of CP of CEED is a classical solution while the velocity satisfies a transport equations with damping in the vacuum region, this transport equations with damping is just obtained by the momentum equations removing the density. The blowup result is proved by estimating a functional of mass with power and simultaneity using the method of characteristic to determine the region of the gas. But the previous research of Euler equations (that is, with no damping) pointed out that when the initial density has compact support, the regular solution could exist globally if the initial data satisfy some hypotheses. So we turn to investigate the compressible Euler equations with degenerate damping.2. In Chapter 3, we investigate CP of the compressible isentropic Euler equations with degenerate linear damping (CIEEDLD) for an ideal gas. When time tends to infinity, the coefficient of degenerate linear damping goes to zero, the power of reciprocal of time in it is called the degenerate order of damping coefficient in this paper. We find a critically degenerate order of damping coefficient for this problem, actually it is the number one. When the degenerate order of damping coefficient is less than this critically degenerate order, if the initial density has compact support and the total mass of the gas isn't zero, then the life span of the regular solution of CP of CIEEDLD must be also finite.While the degenerate order of damping coefficient is greater than this critically degenerate order, if the initial density properly smooth and is smallenough, the initial density has compact support, and if the smooth enough initial velocity has the property of dispersing the gas out, then the classical solution of CP of CIEEDLD exists globally, in fact this classical solution is a regular solution. When the density identically equal zero, the velocity in regular solution satisfies a CP of the transport equations with degenerate damping, this problem is called approximate problem. The initial velocity is so given that the approximate problem has global classical solution. Furthermore the effect of the initial velocity and the degenerate damping is that the global solution of the approximate problem decays when time tends to infinity, and the higher partial derivative with respect to (w.p.t.) the space variable of velocity is, the higher decay rate it has. In fact, utilizing the state equation of the ideal gas, we transform the density to a new variable, then CIEEDLD can be transformed into a symmetric hyperbolic system (SHS). In the case that no vacuum occurs, these two systems are equivalent, even if there exists vacuum, the solution of this SHS is also a solution of CIEEDLD, but the uniqueness of the solution of CIEEDLD is lost. We actually study this SHS, and prove that the solution of CP of this SHS exists globally, and so does CIEEDLD. Since the initial velocity don't belong to Sobolev space, we first construct the local solution of the SHS, then we use the method of energy estimate to prove this solution actually exists globally under the condition that the initial density is small enough. We define an equivalent norm of Sobolev space, and estimate the difference between the solution of the SHS and the approximate solution in this new norm, then prove its norm doesn't blow up under the condition that the initial density is small enough, in the course of the proof we use the property of the approximate solution. Furthermore, the solution of the SHS tends to the approximate solution when time goes to infinity, and their difference also decays in some rate, the decay rates are some like that of the approximate solution.In the case of the critical degenerate damping coefficient, we also has a detailed discuss. The conclusions are similar: when the coefficient of degenerate damping is larger, if the initial density has compact support and thetotal mass of the gas isn't zero, then the regular solution of CP of CIEEDLD must blow up in finite time; while the coefficient of degenerate damping is smaller, under some condition on the initial data we can also prove the regular solution of CP of CIEEDLD exists globally.Removing the condition that the initial density has compact support and preserving the other hypotheses, finally we also obtain the global existence of the solution of CP of CIEEDLD when the coefficient of degenerate damping is smaller.3. In Chapter 4, we generalize the conclusions obtained in Chapter 3 to CP of the compressible Euler equations with degenerate linear damping (CEEDLD) for an ideal gas . When the infimum of the initial entropy is finite, we extend the blowup result in Chapter 3 to this general Euler equations. While the initial entropy is small enough in a proper Sobolev space, we extend the global existence results of the regular solution of the isentropic Euler equations to general Euler equations. Since we can also define another new variable by the state equation of the ideal gas to substitute the density, and transform the general Euler equations into a new SHS, except that the entropy satisfy a transport equation, the structure of the remainder of this system is the same like that of the isentropic case. So if the initial density and entropy properly smooth and small enough, and if the initial velocity has the property of dispersing the gas out, then we can also prove that the regular solution of CP of CEEDLD exists globally while the degenerate order of damping coefficient is greater than the critically degenerate order.Usually the damping plays a positive role in the existence of global solution and in the property of the solution, this point comes into being by the influence of the case that the density is positive, that is, no vacuum occurs. But if there are vacuums, the damping plays a negative role in the existence of the global solution; this point can be viewed from Chapter 2 to Chapter 4. So we should draw some attention to this phenomenon.4. In Chapter 5, the initial boundary value problem (IBVP) of compressible isentropic Euler equation with damping on half axis is considered.The coefficient of damping is a constant. The boundary condition is that the velocity is zero on the boundary. We deduce some higher partial derivatives w.r.t. the space variable of the smooth solution of the Euler equations on the boundary, concretely, the odd order derivatives of the density are zero on the boundary, and so are the even order derivatives of the velocity. If the initial data is a small perturbation around a constant equilibrium solution and satisfies the compatible condition, then we prove that the classical solution of this IBVP exists uniquely and globally, and when time tends to infinity, this solution tends to the equilibrium solution. The proof is to prove an a prior estimate to extend the local solution.5. We study a kind of shallow water equation, Camassa-Holm equation (CHE) in Chapter 6. We first prove that the nonzero classical solution of an IBVP of CHE on finite interval must blow up in finite time. Next we study CP of CHE with an odd initial datum. We prove that if the initial velocity belongs to a proper Sobolev space, the corresponding potential (the difference between the velocity and its second order derivative) is nonnegative on the positive half axis and is integrable, then the solution of this CP exists globally, and the solution is an odd function w.r.t. the space variable forever; if the initial velocity does not identically equal to zero and it derivative at the origin point isn't positive, then the life span of the solution must be finite. Finally we apply the second part to an IBVP of CHE on half axis, the velocity and its second derivative are zero on the boundary. Since the given initial velocity can be uniquely extended to an odd function on the real axis, so we solve CHE with this odd initial velocity to obtain a solution, then limit this solution to the half axis to obtain the needed IBVP of CHE, and gain correspondingly the global existence and blowup results. Comparing the two kinds of IBVP of CHE, one is on finite interval, the other is on half axis, we find that the effects of these two boundary conditions are different.