Shock Formation in Small-Data Solutions to 3D3D Quasilinear Wave Equations: An Overview

In his 2007 monograph, D. Christodoulou proved a remarkable result giving a detailed description of shock formation, for small $H^s$-initial conditions ($s$ sufficiently large), in solutions to the relativistic Euler equations in three space dimensions. His work provided a significant advancement over a large body of prior work concerning the long-time behavior of solutions to higher-dimensional quasilinear wave equations, initiated by F. John in the mid 1970's and continued by S. Klainerman, T. Sideris, L. H\"ormander, H. Lindblad, S. Alinhac, and others. Our goal in this paper is to give an overview of his result, outline its main new ideas, and place it in the context of the above mentioned earlier work. We also introduce the recent work of J. Speck, which extends Christodoulou's result to show that for two important classes of quasilinear wave equations in three space dimensions, small-data shock formation occurs precisely when the quadratic nonlinear terms fail the classic null condition

Finite-time degeneration of hyperbolicity without blowup for quasilinear wave equations

In three spatial dimensions, we study the Cauchy problem for the wave equation −∂[superscript 2][subscript t]Ψ + (1+Ψ)[superscript P] ΔΨ=0 for P∈{1,2}. We exhibit a form of stable Tricomi-type degeneracy formation that has not previously been studied in more than one spatial dimension. Specifically, using only energy methods and ODE techniques, we exhibit an open set of data such that Ψ is initially near 0, while 1+Ψ vanishes in finite time. In fact, generic data, when appropriately rescaled, lead to this phenomenon. The solution remains regular in the following sense: there is a high-order L[superscript 2]-type energy, featuring degenerate weights only at the top-order, that remains bounded. When P = 1, we show that any C[superscript 1] extension of Ψ to the future of a point where 1 + Ψ = 0 must exit the regime of hyperbolicity. Moreover, the Kretschmann scalar of the Lorentzian metric corresponding to the wave equation blows up at those points. Thus, our results show that curvature blowup does not always coincide with singularity formation in the solution variable. Similar phenomena occur when P = 2, where the vanishing of 1 + Ψ corresponds to the failure of strict hyperbolicity, although the equation is hyperbolic at all values of Ψ. The data are compactly supported and are allowed to be large or small as measured by an unweighted Sobolev norm. However, we assume that initially the spatial derivatives of Ψ are nonlinearly small relative to |∂[subscript t]Ψ|, which allows us to treat the equation as a perturbation of the ODE (d[superscript 2]/dt[superscript 2])Ψ = 0. We show that for appropriate data, ∂tΨ remains quantitatively negative, which simultaneously drives the degeneracy formation and yields a favorable spacetime integral in the energy estimates that is crucial for controlling some top-order error terms. Our result complements those of Alinhac and Lindblad, who showed that if the data are small as measured by a Sobolev norm with radial weights, then the solution is global. Keywords: degenerate hyperbolic; strictly hyperbolic; Tricomi equation; weakly hyperbolicNational Science Foundation (U.S.) (Grant DMS-1162211)National Science Foundation (U.S.) (Grant DMS-1454419

The relativistic Euler equations: ESI notes on their geo-analytic structures and implications for shocks in 1D1D and multi-dimensions

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 program Mathematical Perspectives of Gravitation Beyond the Vacuum Regime, which was hosted by the Erwin Schrodinger 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 3D relativistic Euler equations derived in [41], 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 1D analysis 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 [41] can be used to study shock formation in multiple spatial dimensions. Finally, we discuss various open problems tied to shocks.Comment: 61 page