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Analytical Blowup Solutions to the 2-dimensional Isothermal Euler-Poisson Equations of Gaseous Stars

We study the Euler-Poisson equations of describing the evolution of the gaseous star in astrophysics. Firstly, we construct a family of analytical blowup solutions for the isothermal case in R^2. Furthermore the blowup rate of the above solutions is also studied and some remarks about the applicability of such solutions to the Navier-Stokes-Poisson equations and the drift-diffusion model in semiconductors are included. Finally, for the isothermal case, the result of Makino and Perthame for the tame solutions is extended to show that the life span of such solutions must be finite if the initial data is with compact support.


Introduction
The evolution of a self-gravitating fluid such as gaseous stars can be formulated by the Euler-Poisson equation of the following form: ρ t +∇ · (ρu) =0, (ρu) t +∇ · (ρu ⊗ u) + ∇P =−ρ∇Φ, S t + u · ∇S = 0, ∆Φ(t, x) = α(N)gρ, where α(N) is a constant related to the unit ball in R N : α(1) = 2; α(2) = 2π; For simplicity, we take the constant term g = 1. For N ≥ 3, α(N) = N(N − 2)V (N) = N(N − 2) π N/2 Γ(N/2 + 1) , where V (N) is the volume of the unit ball in R N and Γ is a Gamma function. As usual, ρ = ρ(t, x), u = u(t, x) ∈ R N and S(t, x) are the density, the velocity and the entropy respectively. P = P (ρ) is the pressure. In the above system, the selfgravitational potential field Φ = Φ(t, x) is determined by the density ρ through the Poisson equation. The equations (1) 1 and (1) 2 are the compressible Euler equation with forcing term. The equation (1) 3 is the Poisson equation through which the gravitational potential is determined by the density distribution of the gas itself. Thus, we called the system (1) the Euler-Poisson equations. Here, the viscosity term does not appear, that is, the viscous effect is neglected. In this case, the equations can be viewed as a prefect gas model. For N = 3, (1) is a classical (nonrelativistic) description of a galaxy, in astrophysics. See [2], [9] for a detail about the system. If we take S(t, x) = ln K, for some fixed K > 0, we have a γ-law on the pressure P (ρ), i.e.
which is a commonly the hypothesis. The constant γ = c P /c v ≥ 1, where c P , c v are the specific heats per unit mass under constant pressure and constant volume respectively, is the ratio of the specific heats, that is, the adiabatic exponent in (2). In particular, the fluid is called isothermal if γ = 1. It can be used for constructing models with non-degenerate isothermal cores, which have a role in connection with the so-called Schonberg-Chandrasekhar limit [8]. For the physical dimension N = 3, we are interested in the hydrostatic equilibrium specified by u = 0, S = ln K. According to [2], the ratio between the core density ρ(0) and the mean density ρ for 6/5 < γ < 2 is given by where y is the solution of the Lane-Emden equation with n = 1/(γ − 1), and z 0 is the first zero of y(z 0 ) = 0. We can solve the Lane-Emden equation analytically for and for the other values, only numerical values can be obtained. It can be shown that for n < 5, the radius of polytropic models is finite; for n ≥ 5, the radius is infinite.
Gambin [6] and Bezard [1] obtained the existence results about the explicitly stationary solution (u = 0) for γ = 6/5 : The Poisson equation (1) 3 can be solved as where G is the Green's function for the Poisson equation in the N-dimensional spaces defined by In the following, we always seek solutions in spherical symmetry. Thus, the Poisson equation (1) 3 is transformed to In this paper, we concern about blowup solutions for the N-dimensional isothermal Euler-Poisson equations, which may describe the phenomenon called the core collapsing in physics. And our aim is to construct a family of such blowup solutions to it.
Historically in astrophysics, Goldreich and Weber [7] constructed the analytical blowup solution (collapsing) solution of the 3-dimensional Euler-Poisson equation for γ = 4/3 for the non-rotating gas spheres. After that, Makino [9] obtained the rigorously mathematical proof of the existence of such kind of blowup solutions. And in [5], we find the extension of the above blowup solutions to the case N ≥ 3 and γ = (2N − 2)/N. In [11], the solutions with a from is written as , for r < a(t)Z µ ; 0, for a(t)Z µ ≤ r. , u(t, r) = In section 2, we obtain the blowup solutions for the Euler-Poisson equation in spherical symmetry in the 2-dimensional case, in the form of the following theorem.
Theorem 2 For the 2-dimensional Euler-Poisson equations in radial symmetry, (5), there exists a family of solutions, where K > 0, µ = 2λ/K with a sufficiently small λ and α are constants.
(1)When λ > 0, the solutions blow up in a finite time T ; The blowup rate of the solution (6) is discussed in section 3. Actually, we have

Theorem 3 The blowup rate of the solution (6) is,
In the last section, we can rewrite (1) in scalar form, The tame solution is introduced in here.
holds on the exterior of the support of ρ.
If only the condition (1) is assumed, the solution is called "classical". For γ > 1, under the transformation Makino and Perthame [10] showed that the life span of tame solution of the Euler-Poisson equation in spherical symmetry is finite. As we are interested in non-global existence for the isothermal case (γ = 1) in 2 dimension, i.e.
the result of Makino and Perthame [10] for the tame solutions is extended as the following theorem.

Separable Blowup Solutions
In this section, before presenting the proof of Theorem 2, we prepare some lemmas. First, we obtain a general class of solutions for the continuity equation of mass in radial symmetry (5) 1 .

Lemma 6 For the equation of conservation of mass in radial symmetry
there exist solutions, with the form with f ≥ 0 ∈ C 1 and a(t) > 0 ∈ C 1 .

If the statement
But sinceä (t) = − λ a(t) ≤ −λ e θ/λ , we integrate this twice to deduce By taking t large enough, we get a(t) < 0.
As a contradiction is met, the statement (1) is true. It is trivial to verify (2).
Lemma 9 There exists a sufficiently small x 0 > 0, such that the equation where σ > 0, µ, and α are constants, has a solution y = y(x, µ) ∈ C 2 [0, Proof. The lemma can be proved by the fixed point theorem. Multiply (14) by x, By using y(0) = α, (14) is reduced to then for any x 0 > 0, we get f ∈ C 1 [0, x 0 ]. and for any y 1, As e y is a C 1 function of y, we can show that the function e y , is Lipschitz-continuous. And we get, By choosing x 0 > 0 to be a sufficiently small number, such that O(1)x 0 < 1, this shows that the mapping T : C[0, X 0 ] → C[0, x 0 ], is a contraction with the supnorm. By the fixed point theorem, there exists a unique y(x) ∈ C[0, x 0 ], such that T y(x) = y(x). The proof is completed.
As we obtained he local existence in Lemma 9, there are two possibilities: Thus, · y(x) is bounded in [0, x 0 ]. Therefore, we can use the fixed point theorem again to obtain a large domain of existence, such that [0, x 0 + δ] for some positive number δ. There is a contradiction. Therefore, (1b) is rejected. Next, we do not accept (1a) because of the following reason: It is impossible that lim Thus, (18) becomes, Since y(x) is bounded below in [0, x 0 ], it contracts the statement (1a), such that lim x→x 0− y(x) = −∞. So, we can exclude the possibility (1).
We claim that the possibility (2b) doesn't exist. It is because Then, we have, By letting x → ∞, (19) turns out to be, Since a contradiction is established, we exclude the possibility (2b). Thus, the equation (15) (14) to the first-order system, we consider the system along the solution curve y = y(x, 0), · y = · y(x, 0), 0 < x 0 ≤ x < +∞. Since the right-hand side of this system is continuously differentiable in x, y, · y and µ. We apply the comparison theorem (Theorem 7.4) in [3], to obtain the following lemma.
Differentiate Q(x) with respect to x, The above result is due to the fact that we choose the following ordinary differential equation: With Q(0) = 0, this implies that Q(x) = 0. By using Lemma 7 about a(t) and Lemma 10 about y(x), we have shown that the family of the solutions blows up in finite time T under the prescribed conditions of Theorem 2. This completes the proof.
Remark 12 Besides, the above solution requiring y(0) > 0 is not necessary for (6) in contrast to (4) in [5], [9]. And the mass of the above solution is where η(N) is the measure of a unit ball. Thus the mass of the solution (6) depends on the initial data y(0). And it is not easy to determine if the mass is finite or infinite with different datum. It is different for the finite mass of the family of solutions (4) for γ = 4/3 and N = 3 which is independent of y(0) in [4].

From the Poisson equation
As the solution is in radially symmetric, (21) becomes, Under the transformation, Furthermore by setting λ = 0 and · a(0) = 0 in Theorem 2, we easily have the following corollary for the non-trivial stationary solutions.
where α is an arbitrary positive constant.
Remark 15 Our blowup solutions only work for the 2-dimensional case. But we do not know what will happen after the critical time that the solutions blow up. If we consider the system of conservation laws with viscosity, i.e. with ν > 0, where β > 0 and ν ≥ 0. The corresponding family of blowup solution is where µ = 2λ/K and α are constants.

Blowup Rate
In this section, we present the confirmation of Theorem 3. The blowup rate of the constructed solutions (6), for the Euler-Poisson equation is studied. It is interesting to investigate how fast the blowup solution tends to infinity as the time tends to the critical value T . Proof of Theorem 3. We choose a finite time t to make a(t) sufficiently small enough, such that −λ ln a + θ > 0. From the lemma 7, it is clear for that there exists a finite time t 0 > 0 such that · a(t) < 0 for t > t 0 . Next, sincë a(t) = − λ a(t) < 0, when t > t 0 + ǫ = t 1 , this means there exists a constant ǫ > 0, such that a(t) ≤ e θ/λ − δ, where δ is a sufficiently small positive constant, we have a smaller upper bound of a(t). On the other hand, as · a = − √ −2λ ln a + θ.
It is easy to see that, We denote δ as a sufficiently small positive number, such that x ∈ (0, δ], where the constant S > 0. Then, (26) becomes, As y(0) = α, we can estimate the blowup rate at the origin by Thus, we obtain, lim with η < 2, and complete the proof.

Non-global Existence for Tame Solutions
In this section, we present the proof for Theorem 5. The technique to Makino and Perthame's [10] is similar.
Proof of Theorem 5 . We denote the radius of the compact support of the tame solution by R(t). Under the assumption that the initial data has a compact support, i.e. ρ(0, r) = u(0, r) = 0 for r ≥ R(0), from (9) 2 , we must have Kρ r = 0.