Search for: All records

Abstract In this article, we provide notes that complement the lectures on the relativistic Euler equations and shocks that were given by the second author at the programMathematical Perspectives of Gravitation Beyond the Vacuum Regime, which was hosted by the Erwin Schrödinger International Institute for Mathematics and Physics in Vienna in February 2022. We set the stage by introducing a standard first-order formulation of the relativistic Euler equations and providing a brief overview of local well-posedness in Sobolev spaces. Then, using Riemann invariants, we provide the first detailed construction of a localized subset of the maximal globally hyperbolic developments of an open set of initially smooth, shock-forming isentropic solutions in 1D, with a focus on describing the singular boundary and the Cauchy horizon that emerges from the singularity. Next, we provide an overview of the new second-order formulation of the 3Drelativistic Euler equations derived in Disconzi and Speck (2019Ann. Henri Poincare202173–270), its rich geometric and analytic structures, their implications for the mathematical theory of shock waves, and their connection to the setup we use in our 1Danalysis of shocks. We then highlight some key prior results on the study of shock formation and related problems. Furthermore, we provide an overview of how the formulation of the flow derived in Disconzi and Speck (2019Ann. Henri Poincare202173–270) can be used to study shock formation in multiple spatial dimensions. Finally, we discuss various open problems tied to shocks. 

We prove that the time of classical existence of smooth solutions to the relativistic Euler equations can be bounded from below in terms of norms that measure the “(sound) wave-part” of the data in Sobolev space and “transport-part” in higher regularity Sobolev space and Hölder spaces. The solutions are allowed to have nontrivial vorticity and entropy. We use the geometric framework from [M. M. Disconzi and J. Speck, The relativistic Euler equations: Remarkable null structures and regularity properties, Ann. Henri Poincaré 20(7) (2019) 2173–2270], where the relativistic Euler flow is decomposed into a “wave-part”, that is, geometric wave equations for the velocity components, density and enthalpy, and a “transport-part”, that is, transport-div-curl systems for the vorticity and entropy gradient. Our main result is that the Sobolev norm [Formula: see text] of the variables in the “wave-part” and the Hölder norm [Formula: see text] of the variables in the “transport-part” can be controlled in terms of initial data for short times. We note that the Sobolev norm assumption [Formula: see text] is the optimal result for the variables in the “wave-part”. Compared to low-regularity results for quasilinear wave equations and the three-dimensional (3D) non-relativistic compressible Euler equations, the main new challenge of the paper is that when controlling the acoustic geometry and bounding the wave equation energies, we must deal with the difficulty that the vorticity and entropy gradient are four-dimensional space-time vectors satisfying a space-time div-curl-transport system, where the space-time div-curl part is not elliptic. Due to lack of ellipticity, one cannot immediately rely on the approach taken in [M. M. Disconzi and J. Speck, The relativistic Euler equations: Remarkable null structures and regularity properties, Ann. Henri Poincaré 20(7) (2019) 2173–2270] to control these terms. To overcome this difficulty, we show that the space-time div-curl systems imply elliptic div-curl-transport systems on constant-time hypersurfaces plus error terms that involve favorable differentiations and contractions with respect to the four-velocity. By using these structures, we are able to adequately control the vorticity and entropy gradient with the help of energy estimates for transport equations, elliptic estimates, Schauder estimates and Littlewood–Paley theory. 

Consider a one-dimensional simple small-amplitude solution (ϱ(bkg), v1(bkg)) to the isentropic compressible Euler equations which has smooth initial data, coincides with a constant state outside a compact set, and forms a shock in finite time. Viewing (ϱ(bkg), v1(bkg)) as a plane-symmetric solution to the full compressible Euler equations in three dimensions, we prove that the shock-formation mechanism for the solution (ϱ(bkg), v1(bkg)) is stable against all sufficiently small and compactly supported perturbations. In particular, these perturbations are allowed to break the symmetry and have nontrivial vorticity and variable entropy. Our approach reveals the full structure of the set of blowup-points at the first singular time: within the constant-time hypersurface of first blowup, the solution’s first-order Cartesian coordinate partial derivatives blow up precisely on the zero level set of a function that measures the inverse foliation density of a family of characteristic hypersurfaces. Moreover, relative to a set of geometric coordinates constructed out of an acoustic eikonal function, the fluid solution and the inverse foliation density function remain smooth up to the shock; the blowup of the solution’s Cartesian coordinate partial derivatives is caused by a degeneracy between the geometric and Cartesian coordinates, signified by the vanishing of the inverse foliation density (i.e., the intersection of the characteristics). 

For $(t,x)\in <\#comment/>(0,\mathrm{\infty <\#comment/>})\times <\#comment/>{\mathbb{T}}^{\mathfrak{D}}$, the generalized Kasner solutions (which we refer to as Kasner solutions for short) are a family of explicit solutions to various Einstein-matter systems that, exceptional cases aside, start out smooth but then develop a Big Bang singularity as $t\downarrow <\#comment/>0$, i.e., a singularity along an entire spacelike hypersurface, where various curvature scalars blow up monotonically. The family is parameterized by the Kasner exponents ${\stackrel{~<\#comment/>}{q}}_1,\cdots <\#comment/>,{\stackrel{~<\#comment/>}{q}}_{\mathfrak{D}}\in <\#comment/>\mathbb{R}$, which satisfy two algebraic constraints. There are heuristics in the mathematical physics literature, going back more than 50 years, suggesting that the Big Bang formation should be dynamically stable, that is, stable under perturbations of the Kasner initial data, given say at $\{t=1\}$, as long as the exponents are “sub-critical” in the following sense: $\underset{\begin{array}{c}I,J,B=1,\cdots <\#comment/>,\mathfrak{D}\\ I>J\end{array}}{max}\{{\stackrel{~<\#comment/>}{q}}_I+{\stackrel{~<\#comment/>}{q}}_J-<\#comment/>{\stackrel{~<\#comment/>}{q}}_B\}>1$. Previous works have rigorously shown the dynamic stability of the Kasner Big Bang singularity under stronger assumptions: (1) the Einstein-scalar field system with $\mathfrak{D}=3$and ${\stackrel{~<\#comment/>}{q}}_1\approx <\#comment/>{\stackrel{~<\#comment/>}{q}}_2\approx <\#comment/>{\stackrel{~<\#comment/>}{q}}_3\approx <\#comment/>1/3$, which corresponds to the stability of the Friedmann–Lemaître–Robertson–Walker solution’s Big Bang or (2) the Einstein-vacuum equations for $\mathfrak{D}\ge <\#comment/>38$with $\underset{I=1,\cdots <\#comment/>,\mathfrak{D}}{max}|{\stackrel{~<\#comment/>}{q}}_I|>1/6$. In this paper, we prove that the Kasner singularity is dynamically stable forallsub-critical Kasner exponents, thereby justifying the heuristics in the literature in the full regime where stable monotonic-type curvature-blowup is expected. We treat in detail the $1+\mathfrak{D}$-dimensional Einstein-scalar field system for all $\mathfrak{D}\ge <\#comment/>3$and the $1+\mathfrak{D}$-dimensional Einstein-vacuum equations for $\mathfrak{D}\ge <\#comment/>10$; both of these systems feature non-empty sets of sub-critical Kasner solutions. Moreover, for the Einstein-vacuum equations in $1+3$dimensions, where instabilities are in general expected, we prove that all singular Kasner solutions have dynamically stable Big Bangs under polarized $U\left(1\right)$-symmetric perturbations of their initial data. Our results hold for open sets of initial data in Sobolev spaces without symmetry, apart from our work on polarized $U\left(1\right)$-symmetric solutions. Our proof relies on a new formulation of Einstein’s equations: we use a constant-mean-curvature foliation, and the unknowns are the scalar field, the lapse, the components of the spatial connection and second fundamental form relative to a Fermi–Walker transported spatial orthonormal frame, and the components of the orthonormal frame vectors with respect to a transported spatial coordinate system. In this formulation, the PDE evolution system for the structure coefficients of the orthonormal frame approximately diagonalizes in a way that sharply reveals the significance of the Kasner exponent sub-criticality condition for the dynamic stability of the flow: the condition leads to the time-integrability of many terms in the equations, at least at the low derivative levels. At the high derivative levels, the solutions that we study can be much more singular with respect to $t$, and to handle this difficulty, we use $t$-weighted high order energies, and we control non-linear error terms by exploiting monotonicity induced by the $t$-weights and interpolating between the singularity-strength of the solution’s low order and high order derivatives. Finally, we note that our formulation of Einstein’s equations highlights the quantities that might generate instabilities outside of the sub-critical regime.