A quasilinear wave with a supersonic shock in a weak solution interrupting the classical development

TL;DR

The paper proves that a 1+1 dimensional quasilinear wave model with carefully chosen smooth initial data forms a shock, and it constructs a unique maximal globally hyperbolic development (MGHD) whose boundary comprises an initial singularity, a singular boundary, and a Cauchy horizon. It also establishes the existence and uniqueness of a global weak entropy solution that develops a shock, demonstrating that the classical and weak solutions agree before the first singularity and diverge afterwards, with a shock development problem governing the transition. The analysis leverages a Burgers’ equation reduction along the outgoing characteristic direction and a Lorentzian acoustical geometry to control caustics, boundary regularity, and causality, including the novel phenomenon of causal bubbles at the singular boundary. The results provide a concrete, fully constructed instance of shock formation and SDP in 1D, clarifying the interplay between Lorentzian geometry, characteristic dynamics, and weak solution theory with connections to Christodoulou’s shock development program in higher dimensions. The work offers a tractable, rigorous framework for understanding shock formation, MGHD uniqueness, and the global behavior of entropy solutions in a setting that mimics key features of multidimensional hyperbolic systems such as the Euler equations.

Abstract

We study the Cauchy problem for classical and weak shock-forming solutions to a model quasilinear wave equation in $1+1$ dimensions arising from a convenient choice of $C^{\infty}$ initial data, which allows us to solve the equation using elementary arguments. The simplicity of our model allows us to succinctly illustrate various phenomena of geometric and analytic significance tied to shocks, which we view as a prototype for phenomena that can occur in more general quasilinear hyperbolic PDE solutions. Our Cauchy problem admits a classical solution that blows up in finite time. The classical solution is defined in a largest possible globally hyperbolic region called a maximal globally hyperbolic development (MGHD), and its properties are tied to the intrinsic Lorentzian geometry of the equation and solution. The boundary of the MGHD contains an initial singularity, a singular boundary along which the solution's second derivatives blow up (the solution and its first derivatives remain bounded), and a Cauchy horizon. Our main results provide the first example of a provably unique MGHD for a shock-forming quasilinear wave equation solution; it is provably unique because its boundary has a favorable global structure that we precisely describe. We also prove that for the same $C^{\infty}$ initial data, the Cauchy problem admits a second kind of solution: a unique global weak entropy solution that has a shock curve separating two smooth regions. Of particular interest is our proof that the classical and weak solutions agree before the shock but differ in a region to the future of the first singularity where both solutions are defined.

Figures (7)

• Figure 1: A figure of the bicharacteristics -- denoted by the orange and blue lines -- of equation \ref{['E:QNLW:INTRO']}. The weak and classical solutions agree in the large region where the bicharacteristics are solid. The weak and classical solution disagree in the wedged region where there is a family of both solid blue and dashed orange characteristics. This is due to the formation of a shock -- the purple line -- across which there are jumps only for the weak solution. The classical solution is inextendible (in a globally hyperbolic manner) to the region with only dashed bicharacteristics and only the weak solution is defined there. The rest of this paper is devoted to explaining this figure in full detail.
• Figure 2: An illustration of the domain and causal structure for the solutions constructed in the proof of Theorem \ref{['thm:main:MGHD']}. The integral curves of $L$ and $\underline{L}$ are depicted in solid orange and blue, respectively. The singular boundary $\mathcal{B}$ is illustrated in red and the Cauchy horizon $\underline{\mathcal{C}}$ is illustrated in green. Both emanate from the initial singularity $\mathcal{S}$, illustrated in black. The classical solution is defined to the right of the singular boundary $\mathcal{B}$ and beneath the Cauchy horizon $\underline{\mathcal{C}}$.
• Figure 3: An illustration of the domain for the solution provided by Theorem \ref{['T:UNIQUEWEAKSOLUTIONS']} which is both unique in the Oleinik entropy class and a solution to the shock development problem. The outgoing characteristics are given in solid orange, while the shock curve is given in purple.
• Figure 4: A figure of the ingoing and outgoing characteristics on the relevant spacetime regions denoted by the solid blue and orange lines, respectively.
• Figure 5: The hypersurface $\{t = x\}$ in $\mathbb{R}^{1+1}$

Theorems & Definitions (22)

• Definition 1.1: The $\mathbf{g}$-null frame
• Theorem 1.2: Characterization of the maximal global hyperbolic development
• Corollary 1.3: Causal bubbles
• Definition 1.4: Weak solutions to \ref{['E:QNLWINDIVERGENCEFORM']}
• Definition 1.5: Class of Oleinik entropy solutions
• Remark 1.6: Possible irregularity of Oleinik entropy solutions
• Definition 1.7: The shock development problem for \ref{['E:QNLWINDIVERGENCEFORM']}
• Remark 1.8: "Global" shock development
• Remark 1.9: The shock development problem in $1D$ or under symmetry reductions
• Theorem 1.10: Unique global entropy and shock developed weak solutions