Local Existence of Solutions to the Euler–Poisson System, Including Densities Without Compact Support

Abstract

Local existence and uniqueness for a class of solutions for the Euler Poisson system is shown, whose properties can be described as follows. Their density ρ either falls off at infinity or has compact support. Their mass and the energy functional is finite and they also include the static spherical solutions for $\gamma =\frac {6}{5}$. The result is achieved by using weighted Sobolev spaces of fractional order and a new non-linear estimate that allows to estimate the physical density by the regularised non linear matter variable.

U. B. gratefully acknowledges support from Grant MTM2016-75465 by MINECO, Spain and UCM-GR17-920894.

Appendix A Useful Propositions

The following Proposition was proved by Kateb in the H ^s spaces.

Proposition A.1

Let u ∈ H _s,δ ∩ L ^∞, 1 < β, $0<s<\beta +\frac {1}{2}$ , and $\delta \in \mathbb {R}$ , then

$$\displaystyle \begin{aligned} \||u|{}^\beta\|{}_{H_{s,\delta}}\leq C(\|u\|{}_{L^\infty}) \|u\|{}_{H_{s,\delta}}. \end{aligned} $$

(A.1)

Proposition A.2 (Sobolev Embedding)

• (i) If $\frac {3}{2}<s$ and $\beta \leq \delta +\frac {3}{2}$ , then $\|u\|{ }_{L^\infty _\beta }\leq C \|u\|{ }_{H_{s,\delta }}.$

• (ii) Let m be a nonnegative integer, $m+\frac {3}{2}<s$ , and $\beta \leq \delta +\frac {3}{2}$ , then

  $$\displaystyle \begin{aligned} \|u\|{}_{C^m_\beta}\leq C \|u\|{}_{H_{s,\delta}}. \end{aligned}$$

Proposition A.3 (Nonlinear Estimate of Power of Functions)

Suppose that w ∈ H _s,δ, 0 ≤ w and β is a real number greater or equal 2. Then

• 1. If β is an integer, $\frac {3}{2}<s$ and $\frac {2}{\beta -1}-\frac {3}{2}\leq \delta$ , then

  $$\displaystyle \begin{aligned} \|w^\beta\|{}_{H_{s-1,\delta+2}}\leq C_n \left(\|w\|{}_{H_{s,\delta}}\right)^\beta. \end{aligned} $$

  (A.2)

• 2. If $\beta \not \in {{\mathbb N}}$, $\frac {5}{2}<s<\beta -[\beta ]+\frac {5}{2}$ and $\frac {2}{[\beta ]-1}-\frac {3}{2}\leq \delta$ , then

  $$\displaystyle \begin{aligned} \|w^\beta\|{}_{H_{s-1,\delta+2}}\leq C_n \left(\|w\|{}_{H_{s,\delta}}\right)^{[\beta]}. \end{aligned} $$

  (A.3)

Theorem A.4 (Well Posedness of First Order Hyperbolic Symmetric Systems in H _s,δ)

Let $\frac {5}{2}<s$, $-\frac {3}{2}\leq \delta$ , U ₀ ∈ H _s,δ , and F(t, ⋅) ∈ C([0, T ⁰], H _s,δ) for some positive T ⁰ . Then there exists a positive T ≤ T ⁰ and a unique solution U to the system

$$\displaystyle \begin{aligned} \begin{cases} & \displaystyle{\partial_t U + A^a(U)\partial_a U + B(U)U = F(t,x)}\\ & U(0,x)=U_0(x) \end{cases}, \end{aligned} $$

(A.4)

such that

$$\displaystyle \begin{aligned} U\in C([0,T],H_{s,\delta})\cap C^1([0,T],H_{s-1,\delta+1}). \end{aligned}$$