Blowup Phenomenon of Solutions for the IBVP of the Compressible Euler Equations in Spherical Symmetry

The blowup phenomenon of solutions is investigated for the initial-boundary value problem (IBVP) of the N-dimensional Euler equations with spherical symmetry. We first show that there are only trivial solutions when the velocity is of the form c(t)|x|α−1 x + b(t)(x/|x|) for any value of α ≠ 1 or any positive integer N ≠ 1. Then, we show that blowup phenomenon occurs when α = N = 1 and c2(0)+c˙(0)<0. As a corollary, the blowup properties of solutions with velocity of the form (a˙t/at)x+b(t)(x/x) are obtained. Our analysis includes both the isentropic case (γ > 1) and the isothermal case (γ = 1).


Introduction and Main Results
In this paper, we consider the -dimensional Euler equations for compressible fluid: with boundary condition where , u, and represent the density, velocity, and pressure of the fluid, respectively. n is the unit normal vector on the unit sphere. The -law for is given by (1) 3 . The fluid is called isentropic if > 1 and is called isothermal if = 1.
Euler equation (1) is one of the most important fundamental equations in inviscid fluid dynamics. Many interesting fluid dynamic phenomena can be described by system (1) [1,2]. The Euler equations are also the special case of the noted Navier-Stokes equations, whose problem of whether there is a formation of singularity is still open and long-standing. Thus, the singularity formation in fluid mechanics has been attracting the attention of a number of researchers [3][4][5][6][7][8][9][10][11].
In particular, in [3,4], the authors obtain blowup results for the IBVP of the Euler equations, namely, system (1) with boundary condition (2). By making use of the finite propagation speed property [5,6], they are able to apply the integration method to derive differential inequalities and show that if the initial weighted functionals of velocity or momentum are large enough, then blowup occurs.
In [10], the authors consider the solutions of (1) with velocity of the form and show that, by using the standard argument of phase diagram, the solutions will be expanding if (0) and(0) satisfy some inequalities. It is natural to consider the more general velocity form: The Scientific World Journal For solutions in spherical symmetry, namely, ( , ) = ( , ) and u( , ) = ( , )(x/ ), system (1) together with (2) is transformed to where = |x| is the length of the spatial variable x.
Our main contributions in this paper are stated as follows.   As a corollary, we also obtain the following. for some constant ∈ R. Furthermore, one has the following five cases.

Lemmas
It is well-known that is always positive if the initial datum (0, ) is set to be positive. Thus, we suppose (0, ) > 0 in the following to avoid the trivial solutions ≡ 0.
Proof. From (5) 2 , one has and the results follow.
Similarly, we have the following two lemmas for = 1.
Lastly, one has the following lemma that will be used to prove that there are only trivial solutions when ( , ) = ( ) + ( ) and > 1.

Lemma 7.
Consider the following dynamical system The Scientific World Journal 3 then (17) is equivalent to where Otherwise, the solution to (17) is trivial.
Next, we consider the case = −1. For = −1, the corresponding equation of (31) is for all ≥ 1, where are functions of only. As ln is not a rational function, one has that all = 0. In particular, one has As −4 + ( − 1)(3 − 5) and 2 − ( − 2)( − 1) cannot be both zero for ̸ = 1, we conclude that = 0 and the solutions are trivial. For = 1 and ̸ = 1, one can show that there are only trivial solutions with similar procedures. The proof is complete.
Proof. For = = 1, the corresponding system of (41) is Note that (49) 1 is a special case of equation (7) in [10] when we set the parameter in (7) to be zero. Thus, by Theorem 2.1 in [10], the results (1) and (2) in the proposition follow.
Finally, we are ready to present the proof of Corollary 2.