Global existence of the two-dimensional axisymmetric Euler equations for the Chaplygin gas with large angular velocities

1.1 Background

In this article, we consider the two-dimensional (2D) compressible Euler equations for the Chaplygin gas:

where v = 1 ρ is the specific volume, ρ is the density, and u = ( u 1 , u 2 ) is the velocity of the flow. The pressure p is given by the state equation:

with constants A > 0 and B ∈ R . In the following, we shall take A = B = 1 for simplicity. The Chaplygin gas (also known as the von Karman-Tsien gas) model was used as an approximation when studying shock curves (see [7]). The model also finds its applications in recent study of the dark energy in cosmology (see [3,6]).

In the theory of conservation laws, system (1.1) can be written as

The symbol of the system is

where ω = ( ω 1 , ω 2 ) and ∣ ω ∣ = ( ω 1 ) 2 + ( ω 2 ) 2 = 1 . The eigenvalues of A ( ω ) are

and the corresponding right eigenvectors are

It can be verified that, for the Chaplygin gas model (1.2), there holds that

That is, all the characteristic families are linearly degenerate. Thus system (1.3) is totally linearly degenerate (see [28]).

Majda posed the following conjecture on Page 89 of [19]:

If the d ( d ≥ 2 ) dimensional nonlinear symmetric system is totally linearly degenerate, then it typically has smooth global solutions when the initial data are in H s ( R d ) with s > d 2 + 1 unless the solution itself blows up in finite time. In particular, the shock formation never happens for any smooth initial data.

This is quite different from the polytropic gas models where p = ρ γ = v − γ with γ > 1 and some characteristic families are “genuinely nonlinear.” In general, shocks will form and derivatives of the solution will blow up no matter how smooth the initial data are. More information can be found in [8,28].

1.2 Presentation of problem

In this article, we shall consider the axisymmetric case for the Chaplygin gas model in two space dimensions. In the polar coordinates ( r , θ ) , we can decompose the velocity u as

with u r = u r ( t , r ) , u θ = u θ ( t , r ) , and e r = ( cos θ , sin θ ) ⊤ , e θ = ( − sin θ , cos θ ) ⊤ . System (1.1) can be written as

Under the axisymmetry, we pose the following compatibility conditions around r = 0 :

As r → ∞ , we assume that the gas is in the equilibrium lim r → ∞ p = 0 , or equivalently

It is direct to check that ( v ¯ , u ¯ ) = ( v ¯ , u ¯ r e r + u ¯ θ e θ ) is a steady solution to (1.4) if

More precisely, given u ¯ θ satisfying ∫ 0 ∞ ∣ u ¯ θ ( r ) ∣ 2 r d r < ∞ , the specific volume v ¯ can be solved by

That is, there exists a large family of steady states without vacuum to systems (1.1) and (1.4), the angular velocity of which can be rather general and large. For example, we can take u ¯ θ to be smooth with compact support, or decaying as 1 ( 1 + r ) ε with ε > 0 .

To study our problem around some steady state, we pose the following conditions on the initial data ( v 0 , u 0 r , u 0 θ ) to system (1.4). Assume that there exist two constants M 0 ≥ 1 , ε 0 ≥ 0 such that

For simplicity, the constant C can depend on M 0 in the rest of the article.

To prove the theorem, we shall use the rescaled Lagrangian coordinates ( T , S ) defined in (2.2) and (2.8). System (1.4) can be reformulated as a single wave-type equation of f = r 2 S 2 :

Since we want to prove the global existence result for the case with large angular velocities, there is a nontrivial steady solution f ¯ to equation (1.20), which corresponds to the steady state ( v ¯ , u ¯ ) satisfying (1.7). The steady state f ¯ satisfies a semilinear elliptic equation. We shall use an iteration procedure to prove the existence of f ¯ .

As for the perturbation f ˜ = f − f ¯ , we have a wave equation with variable coefficients

The Laplacian D S 2 + 3 S D S indicates that the equation can be seen as radial symmetric in four space dimensions. Since we have variable coefficients f ¯ = f ¯ ( S ) which is not necessarily around some constant, only D T commutes well with the wave operator. Here we shall use a Morawetz-type inequality to derive the L T 2 L S 2 space-time estimates of f ˜ . The advantage is that we only need to prove the decay of f ˜ in S rather than decay in T to close the estimates. The term β f ˜ with β ≥ 0 is a defocusing term, although the effect is really weak, that is, 0 ≤ β ( S ) ≲ 1 ( 1 + S ) 2 . The defocusing term β f ˜ actually poses potential damage in the estimates. Together with the standard energy estimates, we can prove the global existence of the perturbation f ˜ . Then Theorem 1.1 can be proved when we switch back from f to the original variables ( v , u ) .

1.3 History

Before the end of the introduction, we review some related results.

For the Euler equations, there exist three basic wave families: pressure waves, shear waves, and entropy waves. The pressure waves travel with the speed of the sound and satisfy wave-type equations. The shear waves and entropy waves travel slower and satisfy some transport equations. This multi-speeds nature is one of the main difficulties in the study of the Euler equations.

The local existence of solutions is well understood now, see e.g. [19]. In general, singularities will form [24,26] and the solutions cannot remain smooth globally. The mechanisms behind are one of the most important fields in the theory of gas dynamics and conservation laws. When the flow is curl-free and isentropic, there are only pressure waves. The Euler equations can be written into a single wave equation of the velocity potential. Under the polytropic gas state assumption, that is, p ( ρ ) = ρ γ with γ > 1 , the wave equation does not satisfy the null condition (see [16]) and the derivatives of solutions will blow up in finite time. The readers are suggested to consult [12] for a more detailed review. The breakthrough was made in [4] by Christodoulou where geometric methods were used to study the shock formation from small initial data (see also [5,29]). The methods were also applied to the case with nonzero vorticities [18] and more general wave equations with different speeds [30]. In [23], Miao and Yu managed to prove a blowup result for a class of large initial data called “short pulses.” Recently, Merle et al. constructed examples where the solution itself blows up in finite time in [20,21,22].

If the gas is assumed to be the Chaplygin gas (1.2), then the wave equation of the pressure satisfies the null condition. Here comes the Majda’s conjecture in the beginning of our discussion. For the isentropic curl-free flows where only pressure waves exist, the global existence of classical solutions can be proved (see [32] for a geometric approach). When the flow is assumed to be radial symmetric and curl-free, Godin in [11] and Ding et al. in [10] proved the global existence of classical solutions for nonisentropic Euler equations in 3D and 2D, respectively. In the two space dimensions, the shear waves and the entropy waves travel along the particle trajectory and remain bounded. It is possible to obtain a global existence result in 2D with nontrivial u θ and prove the Majda’s conjecture. In a series of papers [13,14,15] where u θ was assumed to be small, Hou and Yin introduced delicate weights and proved some global existence results in the axisymmetric setting. The general case without symmetry assumption is still quite open.

In our article, we treat the 2D isentropic Euler equations where the angular velocity u θ is only assumed to be bounded and with a little decay at infinity. This generalizes the result in [13] where u θ was assumed to be small. Our method can also be applied to the nonisentropic Euler equations which will generalize the results in [14]. Here we consider the problem in the Lagrangian coordinates which greatly simplify the proof. The resulting equation is a wave equation with variable coefficients (1.20). However, the wave equations with variable wave speeds are complicated even on the linear level (see [2,25] for some classical studies and [9] for an introduction of waves on black hole backgrounds). We believe our methods here can find more applications in the fields of wave equations in inhomogeneous materials where the wave speeds can vary significantly.

The rest of the article is organized as follows. In Section 2, we reformulate the Euler equations (1.4) in the Lagrangian coordinates and obtain a single wave-type equation. The steady state satisfies a semilinear elliptic equation and will be investigated in Section 3. To tackle the wave equation of the perturbation, we first present some preliminary estimates in Section 4, then we finish the energy estimates for the perturbation in Section 5. Theorem 1.1 is proved in Section 6. In Appendix A, we prove some Sobolev inequalities in the axisymmetric setting. In Appendix B, we construct an auxiliary function φ used in (4.8) in the Morawetz-type estimates. In Appendix C, we give an estimate of the initial energy of the perturbation.

2.1 Lagrangian coordinates

To study system (1.4), we shall use the Lagrangian coordinates ( T , R ) with t = T and r = r ( T , R ) satisfying

Then we can define

Since D T r D R r v = 0 from the first equation in (1.4), we have

Denote the angular momentum of the flow by Γ = r u θ . The third equation in (1.4) implies that D T Γ = 0 . Hence,

That is, we have the conservation of the angular momentum.

Noting that D T r = u r in (2.1), by taking the another D T derivative to both sides, we have from the second equation in (1.4) that

where we have used (1.2) and (2.3)–(2.4). The initial-boundary data of r are

To simplify equation (2.5), we shall define a scaled spatial operator

that is,

This is possible since M 0 − 1 ≤ v 0 ≤ M 0 from the assumptions (1.8). Furthermore, it is direct to verify that

Then equation (2.5) can be written in the ( T , S ) coordinates as

To study equation (2.10) with (2.6) in a suitable functional space, we first multiply both sides of (2.10) by r to obtain that

Setting f = r 2 S 2 , the resulting system for f is

This can be seen as an radially symmetric defocusing wave equation in R + × R 4 . In the following, we shall just use ( T , S ) to indicate the coordinates in R + × R 4 .

The original variables ( v , u r , u θ ) of (1.4) can be recovered by (2.8) and

2.2 Decomposition of f

Since the angular velocity is large, the background solution is nontrivial. We first define a radially symmetric function f ¯ = f ¯ ( S ) in R 4 satisfying

which is the steady version of (2.11). The construction and properties of f ¯ are stated in Proposition 3.1 and proved in Section 3.

To simplify notations, we define three functions as follows:

The function f can be decomposed as f = f ¯ + f ˜ with the perturbation f ˜ = f ˜ ( T , S ) satisfying the following wave equation in R + × R 4 :

where the initial data of f ˜ are

The energy estimates of f ˜ are given in Proposition 5.1 and proved in Section 5.

Notations: ⟨ S ⟩ = ( 1 + S 2 ) 1 / 2 . We define two weight functions by

The inequality f ≲ g means that f ≤ C g for some constant C > 0 independent of all the parameters or f , g ; f ≳ g means g ≲ f ; f ∼ g means f ≲ g and g ≲ f . Denote the Sobolev spaces of radially symmetric functions in R 4 by

4.1 Standard energy estimates

Multiplying both sides of (4.1) by 2 α D T ψ S 3 , one has