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

Abstract: We consider the isentropic compressible Euler equations in the half-line which govern the motion of gaseous fluids in contact with stationary vacuum boundary. We construct a large class of solutions that are initially smooth and square-integrable, and which, in finite time, transition to $C<sup>{1-μ}$ regularity for $μ\in [1/2,1)$ near the boundary, leading to the gradient blowup at the boundary. It is based on stability analysis of self-similar waiting time solutions \cite{JLN2025} recently constructed by the authors.

• The paper rigorously constructs a finite-codimensional class of smooth initial data that develop finite-time gradient blowup at the vacuum boundary.
• It employs self-similar reduction, profile localization, and modulation analysis to derive precise power-law asymptotics near the singularity.
• Energy estimates and stability analysis ensure that the interior remains uniformly regular while the singularity is strictly localized at the boundary.

Rigorous Analysis of Gradient Blowup in 1D Compressible Euler Equations with Vacuum Boundary

Introduction and Context

This paper addresses the regularity and singularity formation in the one-dimensional isentropic compressible Euler equations with stationary vacuum boundary conditions. Specifically, the authors analyze the transition of smooth, square-integrable vacuum solutions—initially more regular than $C^2$—to solutions exhibiting finite-time gradient blowup with Hölder regularity $C^{1-\mu}$ for $\mu \in [1/2, 1)$ at the vacuum boundary. The setting involves polytropic gases, with the pressure following a $\gamma$-law ($p=\rho^\gamma$, $\gamma\in(1,3)$), in the half-space $\Omega = (0,\infty)$, subject to initial and boundary conditions enforcing vanishing sound speed and velocity at the contact with vacuum.

The compressible Euler equations' dynamics near vacuum have generated significant interest, as the interplay between nonlinearity, degeneracy near vacuum, and boundary geometry leads to complex singular behaviors. Building upon prior work—including the construction of self-similar "waiting time" solutions in [Jang, Liu, Masmoudi 2025]—this manuscript settles a major open problem: the rigorous construction and stability analysis of a broad class of nonself-similar, smooth vacuum solutions that undergo a loss of regularity due to self-similar blowup, even for classes of initial data that are compactly supported and integrable.

Problem Formulation and Main Theorem

The authors consider solutions $(\rho, u, p)$ to the 1D isentropic Euler equations:

$\begin{split} \rho_t + (\rho u)_x &= 0 \ (\rho u)_t + (\rho u^2 + p)_x &= 0 \end{split}$

with pressure $p = \rho^\gamma$, $C^{1-\mu}$0, and domain $C^{1-\mu}$1. The boundary at $C^{1-\mu}$2 corresponds to contact with vacuum, with non-moving boundary conditions:

$C^{1-\mu}$3

where $C^{1-\mu}$4 is the sound speed. Initial data are assumed to satisfy $C^{1-\mu}$5 for $C^{1-\mu}$6, $C^{1-\mu}$7.

The central result is the construction of a large class of smooth initial data resulting in solutions that remain strong and regular up to a finite time $C^{1-\mu}$8, at which the solution's regularity at $C^{1-\mu}$9 drops from $\mu \in [1/2, 1)$0 to $\mu \in [1/2, 1)$1. Here $\mu \in [1/2, 1)$2 parametrizes the regularity exponent, and $\mu \in [1/2, 1)$3. Additionally, the $\mu \in [1/2, 1)$4 norm (kinetic and acoustic energy) remains bounded up to blowup, the density stays strictly positive in the interior, and the gradient singularity is localized at $\mu \in [1/2, 1)$5.

Self-Similar Structure and Analytical Techniques

The analysis exploits the Euler system's scaling symmetry, introducing appropriately renormalized self-similar variables and Riemann invariants $\mu \in [1/2, 1)$6 and $\mu \in [1/2, 1)$7. The study focuses on the behavior near the vacuum boundary $\mu \in [1/2, 1)$8, exploiting self-similar steady states derived from previous work. The main approach can be decomposed as follows:

1. Self-Similar Reduction: Utilize variable transformations leading to a steady-state ODE for the self-similar profile $\mu \in [1/2, 1)$9 (see Eq. (2.8)), revealing its precise power-law asymptotics near $\gamma$0 and at infinity.
2. Profile Localization: Construct a localized profile $\gamma$1, using a dynamical cutoff transported along the self-similar characteristics, ensuring square-integrability and compatibility with physical vacuum boundary behavior.
3. Decomposition and Modulation: Decompose the solution into a localized self-similar part, a finite-dimensional sum of polynomials modulating boundary behavior, and a rapidly decaying remainder with vanishing derivatives at $\gamma$2. This precision allows for quantifying the finite codimensional set of initial data that yield the prescribed blowup.
4. Stability and Energy Estimates: Develop detailed weighted $\gamma$3 and high-order Sobolev energy estimates for the remainder, leveraging the strictly outgoing character of the self-similar transport operator. The analysis capitalizes on the coercivity properties and damping effects present in the linearized dynamics.
5. Control of Nonlinear Effects: Address nonlinear contributions through bootstrap arguments and careful modulation analysis. Constraints on the modulation variables yield the finite codimension of admissible initial data, and additional degeneracies at critical values of $\gamma$4 are handled via further restrictions (e.g., $\gamma$5 at half-integer $\gamma$6).

The principal technical innovation lies in tracking the local evolution of regularity and singularity formation from smooth, integrable initial data, rigorously linking it to the self-similar blowup phenomena characterized by Burgers-type equations. The study leverages modern approaches to singularity formation via modulation and stability theory, detailed spectral analysis, and localization by characteristically transported cutoffs.

Quantitative and Qualitative Results

The main quantitative outcome is the explicit description of a finite-codimensional class of $\gamma$7 initial data for which the solution experiences its first regularity breakdown at $\gamma$8, with

$\gamma$9

near $p=\rho^\gamma$0 as $p=\rho^\gamma$1. The exponent $p=\rho^\gamma$2 parametrizes a continuum of singular behaviors, and the set of admissible data is shown to have codimension $p=\rho^\gamma$3 or $p=\rho^\gamma$4 depending on additional regularity constraints.

These solutions are globally square-integrable up to blowup; the density remains strictly positive in the interior, and the singularity is entirely localized at the boundary. Moreover, the methods extend stability results for self-similar blowup profiles in Burgers and Euler models to the setting with a vacuum boundary, confining both the singularity's structure and its persistence under robust classes of perturbations.

Implications and Future Directions

The rigorous construction of smooth solutions whose gradients blow up at a vacuum boundary elucidates the transition from smooth to singular dynamics in compressible flows, especially relevant for vacuum-bounded physical systems arising in gas dynamics, astrophysics, and shallow water theory. The results extend the classically understood 'physical vacuum' interface motion, providing evidence that nontrivial gradient singularities develop generically from smooth initial data in the presence of a stationary boundary.

Future research may address:

• The continuation of such $p=\rho^\gamma$5 boundary data as weak solutions in the post-blowup regime, potentially using free boundary or weak discontinuity frameworks.
• Extension to higher dimensions, alternative equations of state, or coupling to gravitating fields.
• Application of these techniques to singularity formation in more complex hyperbolic or dispersive systems, where self-similar or modulation-driven blowup is suspected.

Conclusion

This paper delivers a rigorous, technical analysis of finite-time gradient blowup for smooth, integrable vacuum flows in the 1D compressible Euler equations, providing an explicit and stable description of singularity formation driven by self-similar local dynamics. The approach bridges Burgers and Euler singularity theory and refines the understanding of vacuum boundary problems, offering new tools and perspectives for singularity formation in nonlinear hyperbolic PDEs (2605.00407).