Local existence of solutions to the Euler–Poisson system, including densities without compact support

Abstract

Abstract. Local existence and well posedness for a class of solutions for the Euler Poisson system is shown. These solutions have a density ρ which either falls off at infinity or has compact support. The solutions have finite mass, finite energy functional and include the static spherical solutions for γ = 6 5 . The result is achieved by using weighted Sobolev spaces of fractional order and a new non linear estimate which allows to estimate the physical density by the regularised non linear matter variable. Gamblin also has studied this setting but using very different functional spaces. However we believe that the functional setting we use is more appropriate to describe a physical isolated body and more suitable to study the Newtonian limit.

The global existence issue for the compressible Euler system with Poisson or Helmholtz couplings

“…A number of works have been devoted to the local existence issue for the Euler-Poisson system, in various functional settings (see e.g. Makino [33], Gamblin [17], Bézard [5] and Brauer-Karp [7]). However, to the best of our knowledge, none of them treats also the case µ = 0 and Sobolev spaces with fractional regularity (furthermore, our data are not exactly in uniformly local Sobolev spaces).…”

“…This introduces the classical difficulty of vacuum (as first observed by Kato [25]) when symmetrizing the system. Despite this, the corresponding Cauchy problem for Euler-Poisson with vacuum for strong solutions was solved locally in time in the eighties by various authors, among them: Makino [33,34], Makino-Ukai [37], Makino-Pertame [36], Gamblin [17], Bézard [5], Braun and Karp [7] (see also [35] for a clear survey).…”

Local well-posedness of Euler-Poisson dominated molecular clouds in astrophysics

“…However, there is a severe difficulty when vacuum appears. The vanishing of density leads to the degenerated or unbounded coefficients of the symmetric system (detailed explanations of this difficulty and the ideas on how to overcome this difficulty can be found, for example, in [6,32,37]). Let us briefly recall Makino's ideas [37] in this section.…”

“…(3) of regular solutions). There are some improvements of the regularities of solutions with extra exterior velocity constraints in [6,33,44]. However, for simplicity, we currently only use above cited Makino's local existence theorem 2.9 directly for the followings and leave improvements in near future.…”

“…We point out that [38] specified Makino's conjecture that any tame solution, including nonsymmetric tame solution, will become not tame after a finite time, and this current article can be viewed as a partial answer. Oliynyk [44], Brauer and Karp [6] and the references cited therein have improved the local existence and uniqueness of solutions to Euler-Poisson(-Makino) system with vacuum. Brauer [5], by generalizing the method in [4,9], gives a strong continuation principle for tame solutions, especially, a more detailed classification of the type of blowups.…”