<p>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 positive cosmological constant. In this setting, gravitation is described by a single scalar field satisfying a specific semi-linear wave equation. We restrict attention to spatially periodic perturbations of the background metric and therefore study this equation on the three-dimensional torus <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {T}^3$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation>, working within the Sobolev spaces <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$H^m(\mathbb {T}^3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mi>m</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mn>3</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. 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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(1+u)^\mu $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mi>μ</mi> </msup> </math></EquationSource> </InlineEquation>, with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mu \in \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>, must remain smooth throughout the evolution. This condition, in turn, requires that <i>u</i> 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.</p>

Global existence of solutions to the irrotational Euler–Nordström equations with a positive cosmological constant: the gravitational field equation

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发表日期：

2026年7月1日

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Uwe Brauer,
Lavi Karp

摘要

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 positive cosmological constant. In this setting, gravitation is described by a single scalar field satisfying a specific semi-linear wave equation. We restrict attention to spatially periodic perturbations of the background metric and therefore study this equation on the three-dimensional torus $\mathbb {T}^3$ $T^{3}$ , working within the Sobolev spaces $H^m(\mathbb {T}^3)$ $H^{m} (T^{3})$ . 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)^\mu $ ${(1 + u)}^{μ}$ , with $\mu \in \mathbb {R}$ $μ \in R$ , 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.