arXiv subjects

Leonardo Abbrescia

Publications and source records attributed to Leonardo Abbrescia.

7 recordsLinked to original sources

arXiv2026

Two global stability theorems for small-data $3D$ compressible Euler solutions: Unique maximal globally hyperbolic developments with gradient-blowup & smooth global existence in the entire spacetime

We prove a pair of complementary small-data stability theorems describing the global future and past structure of classical solutions to the isentropic, spherically symmetric $3D$ compressible Euler equations. We allow any equation of state with positive sound speed, except that our shock-formation results do not apply to the Chaplygin gas. Our first theorem concerns open sets of smooth Cauchy data that are perturbations of trivial data with vanishing velocity and constant positive density. The perturbed data have finite kinetic energy, and the perturbed density is equal to a positive constant plus a signed ``asymptotically flat'' tail. Our main theorem yields a complete description of the existence and uniqueness of the maximal globally hyperbolic development (MGHD) of the data. The boundary of the MGHD contains hypersurfaces that extend to spatial infinity on which our solutions develop gradient-singularities. This is the first MGHD existence-uniqueness-stability result for shock-forming solutions for any multi-$D$ quasilinear wave system. While prior works have yielded local portions of ``candidate'' MGHDs, neither the existence nor the uniqueness of an MGHD can be inferred from local considerations. Our second theorem considers analogous data with the opposite sign of the $1/r$ density tail. We prove global existence throughout spacetime, to both the future and past. The solution's asymptotic behavior towards null infinity is not linear, but rather is distorted by a logarithmic bending of the sound cones away from the flat Minkowski ones, leading to logarithmically enhanced dispersion. This is the first small-data global existence result in the entire spacetime for any $3D$ quasilinear wave system that fails to satisfy the null condition and the weak null condition. The main mechanism of stabilization is a global-in-spacetime rarefaction effect, tied to the $1/r$ tail.

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arXiv2026

The emergence of the Cauchy horizon from the crease for $3D$ compressible Euler flow

We study open sets of nearly plane symmetric data for the $3D$ compressible Euler equations with dynamic entropy and non-trivial vorticity. In our prior work, we proved that the solutions develop a gradient singularity along a singular boundary, which is a hypersurface that emanates from a co-dimension 2, spacelike submanifold called the crease. In the present paper, we prove that a null hypersurface, called a Cauchy horizon, emanates from the crease, and propagates in a direction transverse to the singular boundary. The solution remains smooth up to the Cauchy horizon, even though it ``feels'' the influence of the gradient singularity at the crease. The union of the Cauchy horizon, the crease, and the singular boundary, make up a connected portion of the boundary of a maximal globally hyperbolic development (MGHD) of the data. Roughly, an MGHD is the ``largest'' way that smooth data can evolve into a classical solution. Known examples show that the only way one can guarantee uniqueness of an MGHD is by proving that certain structural properties hold along its entire boundary. We prove that a local version of the needed properties are satisfied along the Cauchy horizon and singular boundary. Our work provides the first description of the formation, structure, and stability of an $O(1)$-size portion of the Cauchy horizon in $3D$ without symmetry assumptions. The presence of dynamic entropy and vorticity stretching, both of which are absent in simpler settings such as $2D$ isentropic Euler, introduces significant difficulties that we resolve with novel techniques. Our approach relies on new foliations of spacetime dynamically adapted to the shape of the crease, and a PDE framework based on double-null foliations and the energy identities we derived in earlier work. Collectively, our techniques allow us to handle a challenging new solution regime where the crease lacks strict convexity.

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arXiv2025

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

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.

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arXiv2023

The relativistic Euler equations: ESI notes on their geo-analytic structures and implications for shocks in $1D$ 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.

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arXiv2020

Remarkable localized integral identities for $3D$ compressible Euler flow and the double-null framework

We derive new, localized geometric integral identities for solutions to the $3D$ compressible Euler equations under an arbitrary equation of state when the sound speed is positive. The identities are coercive in the first derivatives of the specific vorticity and the second derivatives of the entropy, and the error terms exhibit remarkable regularity and null structures. Our framework allows one to simultaneously unleash the full power of the geometric vectorfield method for both the wave- and transport- parts of the flow on compact regions. In particular, the integral identities yield localized control over one additional derivative of the vorticity and entropy compared to standard results, assuming that the initial data enjoy the same gain. Similar results hold for the solution's higher derivatives. We derive the identities in detail for two classes of spacetime regions that frequently arise in PDE applications: i) compact spacetime regions that are globally hyperbolic with respect to the acoustical metric and ii) compact regions covered by double-acoustically null foliations. Our results have implications for the geometry and regularity of solutions, the formation of shocks, the structure of the maximal classical development of the data, and for controlling solutions whose state along a pair of intersecting characteristic hypersurfaces is known. Our analysis relies on a recent new formulation of the compressible Euler equations that splits the flow into a geometric wave-part coupled to a div-curl-transport part. Our main new contribution is our analysis of the positive co-dimension, spacelike boundary integrals that arise in the div-curl identities. By exploiting interplay between the elliptic and hyperbolic parts of the new formulation, we observe several crucial cancellations, which in total show that the boundary terms have a good sign.

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arXiv2020

Global nearly-plane-symmetric solutions to the membrane equation

We prove that any simple planar travelling wave solution to the membrane equation in spatial dimension $d \geq 3$ with bounded spatial extent is globally nonlinearly stable under sufficiently small compactly-supported perturbations, where the smallness depends on the size of the support of the perturbation as well as on the initial travelling wave profile. The main novelty of the argument is the lack of higher-order peeling in our vector-field based method. In particular, the higher order energies (in fact, all energies at order $2$ or higher) are allowed to grow polynomially (but in a controlled way) in time. This is in contrast with classical global stability arguments where only the "top" order energies used in the bootstrap argument exhibit growth, and reflects the fact that the background travelling wave solution has "infinite energy" and the coefficients of the perturbation equation are not asymptotically Lorentz invariant. Nonetheless, we can prove that the perturbation converges to zero in $C^2$ by carefully analyzing the nonlinear interactions and exposing a certain "vestigial" null structure in the equations.

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arXiv2019

Global versions of Gagliardo-Nirenberg-Sobolev inequality and applications to wave and Klein-Gordon equations

We prove global, or space-time weighted, versions of the Gagliardo-Nirenberg interpolation inequality, with $L^p$ ($p < \infty$) endpoint, adapted to a hyperboloidal foliation. The corresponding versions with $L^\infty$ endpoint was first introduced by Klainerman and is the basis of the classical vector field method, which is now one of the standard techniques for studying long-time behavior of nonlinear evolution equations. We were motivated in our pursuit by settings where the vector field method is applied to an energy hierarchy with growing higher order energies. In these settings the use of the $L^p$ endpoint versions of Sobolev inequalities can allow one to gain essentially one derivative in the estimates, which would then give a corresponding gain of decay rate. The paper closes with the analysis of one such model problem, where our new estimates provide an improvement.

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