Gradient blowup of smooth vacuum solutions to 1D compressible Euler equations

1. Problem Statement

The paper investigates the isentropic compressible Euler equations in one dimension for a polytropic gas ($p = \rho^\gamma$) in the half-line domain $\Omega = \{x > 0\}$. The fluid is in contact with a stationary vacuum boundary at $x=0$.

• Governing Equations: The system is written in terms of density $\rho$, velocity $u$, and sound speed $c$:
  $$\rho_t + (\rho u)_x = 0, \quad (\rho u)_t + (\rho u^2 + p)_x = 0$$
  In the non-vacuum region, this is equivalent to a system for velocity $u$ and sound speed $c$.
• Boundary Conditions: The vacuum boundary is stationary ($x=0$) with $c(t,0)=0$ and $u(t,0)=0$.
• The Phenomenon: The authors study gradient blowup (loss of regularity) occurring at the vacuum boundary in finite time. Specifically, they construct solutions that are initially smooth (specifically $C^{\lfloor \beta \rfloor, \beta-\lfloor \beta \rfloor}$) and square-integrable, but evolve to become only $C^{1-\mu}$ Hölder continuous near the boundary at the blowup time $t=0$, where the first derivatives of velocity and sound speed diverge.
• Regularity Parameter: The regularity is characterized by $\mu \in [1/2, 1)$ and $\beta = \frac{1}{1-\mu} \geq 2$. The case $\mu = 1/2$ corresponds to the most stable profile (related to the $C^{1/3}$ cusp singularity in Burgers' equation).

2. Methodology

The proof relies on a stability analysis of self-similar solutions constructed in a previous work [27]. The methodology involves several sophisticated steps:

A. Self-Similar Reformulation

The authors utilize the scaling symmetry of the Euler equations to introduce self-similar variables $(\tau, y)$:
$$\tau = -\log(\mu(-t)), \quad y = \frac{x}{(-t)^\delta}$$
where $\delta = 1/\mu$. The system is transformed into a set of equations for Riemann invariants $(\hat{z}, \hat{w})$. The self-similar solutions constructed in [27] appear as steady states $(\bar{z}, 0)$ in these variables.

B. Localization and Profile Construction

Since the self-similar solutions are unbounded in the far field, the authors construct a localized profile $\mathring{z} = \chi \bar{z}$ using a time-dependent cutoff function $\chi$. This cutoff is transported by the characteristic velocity of the background flow to ensure the error terms decay appropriately.

C. Decomposition of Perturbations

The solution is decomposed into three parts:

1. The Profile: The localized self-similar solution $\mathring{z}$.
2. Modulation Terms: A finite sum of polynomials in $y$ with time-dependent coefficients $(z_i(\tau), w_i(\tau))$. These coefficients correspond to the Taylor expansion of the solution at the origin.
3. Remainder: High-frequency components $(Z, W)$ which vanish at the boundary up to a high order ($n_\mu - 1$).

The decomposition is:
$$\hat{z} = \mathring{z} + \sum_{i=2}^{n_\mu-1} \phi(y) \frac{z_i(\tau)}{i!} y^i + Z, \quad \hat{w} = \sum_{i=2}^{n_\mu-1} \phi(y) \frac{w_i(\tau)}{i!} y^i + W$$

D. Modulation Analysis (ODE System)

The coefficients $(z_i, w_i)$ satisfy a system of ODEs derived by imposing vanishing conditions on the higher-order derivatives at the boundary.

• Stability: The authors identify a finite-dimensional stable manifold for these ODEs. To prevent exponential growth (instability) driven by anti-damping terms, the initial data must satisfy specific algebraic constraints (polynomial relations between $z_i$ and $w_i$).
• Symmetries: The constraints $z_1 = w_1 = 0$ remove time and space translation symmetries, fixing the blowup time to $t=0$.

E. Energy Estimates for the Remainder

For the remainder $(Z, W)$, the authors perform a rigorous nonlinear stability analysis using bootstrap arguments and energy estimates:

• Singular Weights: They employ weighted $L^2$ spaces with singular weights $y^{-n_\mu}$ to handle the degeneracy at the vacuum boundary.
• High-Order Estimates: They derive estimates in $\dot{H}^{n_\mu}$ (Sobolev spaces) to control the regularity.
• Dissipation: The linearized operator around the profile exhibits strict dissipativity for high frequencies due to the transport velocity and the specific choice of the regularity index $n_\mu$.
• Handling Non-Integer $\beta$: Special care is taken when $\beta$ is not an integer (i.e., $\mu \notin S$), where the profile has limited smoothness. In cases where $\beta$ is a half-integer, an additional constraint $w_2 = 0$ is imposed to avoid non-integrable singularities in the interaction terms.

3. Key Contributions and Results

Main Theorem (Theorem 1.1 & 2.7)

The authors prove the existence of a large class of smooth, square-integrable, non-self-similar initial data that evolve into solutions exhibiting gradient blowup at the stationary vacuum boundary.

• Regularity Transition: Solutions start as $C^{\lfloor \beta \rfloor, \beta-\lfloor \beta \rfloor}$ and transition to $C^{1-\mu}$ at $t=0$.
• Blowup Rate: Near the blowup point $(0,0)$, the velocity and sound speed behave asymptotically as:
  $$u(t, x) \sim -\tilde{a} x^{1-\mu}, \quad c(t, x) \sim \tilde{b} x^{1-\mu}$$
  leading to unbounded gradients $\partial_x u, \partial_x c \to \infty$.
• Finite Codimensionality: The set of initial data leading to this specific blowup scenario forms a finite-codimensional subset of the Sobolev space $H^{n_\mu}$.
  • The codimension is $\lfloor \beta + 1 \rfloor$ for generic $\mu$.
  • For $\mu = 1/2$, the codimension is 3, corresponding to the three symmetries (time translation, Galilean boost, scaling).
  • For $\mu \in H$ (half-integer $\beta$), an extra constraint $w_2=0$ is required, increasing the codimension.

Technical Innovations

1. Stability of Self-Similar Solutions: This is the first rigorous proof of the stability of self-similar "waiting time" solutions for the compressible Euler equations with a stationary vacuum boundary, extending previous results on Burgers' equation.
2. Handling Stationary Boundaries: Unlike free boundary problems where the vacuum moves, this work handles the case where the boundary is fixed ($u=0$ at $x=0$) but the gradient blows up, requiring careful analysis of the interaction between the boundary condition and the self-similar profile.
3. Modulation with Singular Weights: The development of energy estimates with singular weights $y^{-n_\mu}$ combined with modulation theory to control the low-frequency modes near the singularity.

4. Significance

• Mathematical Physics: The paper resolves a long-standing question regarding the genericity of gradient blowup in vacuum Euler flows. It demonstrates that gradient blowup is not just a feature of special self-similar solutions but is stable under perturbations of a large class of smooth initial data.
• Connection to Burgers' Equation: The results provide a rigorous link between the formation of cusp singularities in the Burgers' equation and the more complex compressible Euler system, validating the use of Burgers' profiles as base models for singularity formation in gas dynamics.
• Vacuum Boundary Dynamics: It clarifies the behavior of the "waiting time" phenomenon, showing that even with a stationary boundary condition, the system can naturally evolve toward a physical vacuum state (where the boundary would start moving) if the regularity is lost, or maintain the stationary state with infinite gradients.
• Future Directions: The work sets the stage for studying the continuation of these solutions past the blowup time, potentially leading to the formation of free boundaries or weak discontinuities, a problem left for future study.

In summary, this paper provides a rigorous mathematical framework for understanding how smooth, physically relevant initial data in gas dynamics can evolve into singular states with infinite gradients at a vacuum boundary, establishing the stability of these singularities against perturbations.