Global existence of the irrotational Euler-Nordström equations with a positive cosmological constant: The gravitational field equation
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| author | Brauer, Uwe Karp, Lavi |
| author_facet | Brauer, Uwe Karp, Lavi |
| contents | Our aim is to establish the global existence of classical solutions to the nonlinear irrotational Euler--Nordström system, which incorporates a linear equation of state and a cosmological constant. In this setting, gravitation is described by a single scalar field satisfying a specific semilinear wave equation. We restrict attention to spatially periodic perturbations of the background metric and therefore study this equation on the three-dimensional torus $\mathbb{T}^3$, working within the Sobolev spaces $H^m(\mathbb{T}^3)$.
We begin by analysing the Nordström equation in isolation, with a source term generated by an irrotational fluid obeying a linear equation of state. This separation is motivated by the fact that such a fluid produces a source term containing a nonlinear contribution of fractional order.
To obtain a global solution for the gravitational field, the fractional-order nonlinearity $(1+u)^μ$, with $μ\in\mathbb{R}$, must remain smooth throughout the evolution. This condition, in turn, requires that $u$ remain small for all time. We ensure this by introducing a suitably chosen energy functional. We also prove that, asymptotically, the solutions tend to a constant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_02023 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global existence of the irrotational Euler-Nordström equations with a positive cosmological constant: The gravitational field equation Brauer, Uwe Karp, Lavi Analysis of PDEs 35Q31 Our aim is to establish the global existence of classical solutions to the nonlinear irrotational Euler--Nordström system, which incorporates a linear equation of state and a cosmological constant. In this setting, gravitation is described by a single scalar field satisfying a specific semilinear wave equation. We restrict attention to spatially periodic perturbations of the background metric and therefore study this equation on the three-dimensional torus $\mathbb{T}^3$, working within the Sobolev spaces $H^m(\mathbb{T}^3)$. We begin by analysing the Nordström equation in isolation, with a source term generated by an irrotational fluid obeying a linear equation of state. This separation is motivated by the fact that such a fluid produces a source term containing a nonlinear contribution of fractional order. To obtain a global solution for the gravitational field, the fractional-order nonlinearity $(1+u)^μ$, with $μ\in\mathbb{R}$, must remain smooth throughout the evolution. This condition, in turn, requires that $u$ remain small for all time. We ensure this by introducing a suitably chosen energy functional. We also prove that, asymptotically, the solutions tend to a constant. |
| title | Global existence of the irrotational Euler-Nordström equations with a positive cosmological constant: The gravitational field equation |
| topic | Analysis of PDEs 35Q31 |
| url | https://arxiv.org/abs/2509.02023 |