On the Fractional Density Gradient Blow-Up Conjecture of Rendall

Abstract

On exponentially expanding Friedmann–Lemaître–Robertson–Walker (FLRW) spacetimes, there is a distinguished family of spatially homogeneous and isotropic solutions to the relativistic Euler equations with a linear equation of state of the form p=σρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=\\sigma \\rho $$\\end{document}, where σ∈[0,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma \\in [0,1]$$\\end{document} is the square of the sound speed. Restricting these solutions to a constant time hypersurface yields initial data that uniquely generates them. In this article, we show, for sound speeds satisfying 13<σ<k+13k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{1}{3}<\\sigma <\\frac{k+1}{3k}$$\\end{document} with k∈Z>32\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\in {\\mathbb {Z}}{}_{>\\frac{3}{2}}$$\\end{document}, that T2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {T}}{}^2$$\\end{document}-symmetric initial data that is chosen sufficiently close to spatially homogeneous and isotropic data uniquely generates a T2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {T}}{}^2$$\\end{document}-symmetric solution of the relativistic Euler equations that exists globally to the future. Moreover, provided k∈Z>52\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\in {\\mathbb {Z}}{}_{>\\frac{5}{2}}$$\\end{document}, we show that there exist open sets of T2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {T}}{}^2$$\\end{document}-symmetric initial data for which the fractional density gradient becomes unbounded at timelike infinity. This rigorously confirms, in the restricted setting of relativistic fluids on exponentially expanding FLRW spacetimes, the fractional density gradient blow-up scenario conjectured by Rendall (Ann Henri Poincaré 5(6):1041–1064, 2004).