arXiv subjects

Steve Shkoller

Publications and source records attributed to Steve Shkoller.

At least 19 recordsLinked to original sources

arXiv2026

A new class of Euler explosions

We study the global-in-time continuation, past the singularity, of the smooth, non-isentropic, radially symmetric imploding solutions of the compressible Euler equations recently constructed by Chen, Shkoller, and Vicol. In three space dimensions, for all physically relevant adiabatic exponents $\gamma>1$, we consider the Euler solution that evolves smoothly until an implosion singularity forms at the origin at time $t=0$. We then prove that this solution can be uniquely continued for $t>0$ as a reflected outward-propagating shock, sometimes called a reflected blast wave. For $t>0$, the continuation is a globally forward self-similar weak solution of the Euler equations, selected by the Rankine--Hugoniot conditions and the Lax entropy inequality; it is smooth away from the expanding shock sphere and the spatial origin. The structure at the center of symmetry distinguishes these explosions from the classical Guderley reflected shock. In Guderley's continuation, the reflected blast wave leaves a point vacuum at the origin, where the density vanishes. The solutions constructed here exhibit the opposite behavior: for every fixed $t>0$ the density is unbounded at $r=0$ (though it remains locally integrable), while the pressure stays bounded and the temperature vanishes there.

math.AP↗

arXiv2026

Smooth and stable Euler implosions

We construct a new class of self-similar implosion profiles for the multi-dimensional compressible Euler equations. These profiles are smooth, genuinely non-isentropic, radially/spherically symmetric, and have explicit (closed-form) similarity exponents. We prove that the exact Euler solution corresponding to the ground state implosion profile is stable to radially symmetric perturbations, as a solution to the full nonlinear compressible Euler equations, modulo a one-dimensional compatibility condition on the initial data. For perturbations of the Euler solution corresponding to the ground state implosion profile of a monatomic or diatomic gas, that do not obey any symmetry assumptions, we provide a complete characterization of the set of initial data that yield nonlinear stability.

math.AP↗

arXiv2026

Intermittent Sub-grid Wave Correction from Differentiated Riemann Variables

We introduce a low-cost every-$K$-step correction for one-dimensional Euler computations. The correction uses differentiated Riemann variables (DRVs) -- characteristic derivatives that isolate the left acoustic wave, the contact, and the right acoustic wave -- to locate the current wave packet, sample the surrounding constant states, perform a short Newton update for the intermediate pressure and contact speed, and conservatively remap a sharpened profile back onto the grid. The ingredients are elementary -- filtered centered differences, local state sampling, a single Newton step, and conservative cell averaging -- yet the effect on accuracy is disproportionate. On a long-time severe-expansion benchmark ($N=900$, $t=0.4$), intermittent correction drives the intermediate-state errors from $O(10^{-2})$ to $O(10^{-13})$, i.e. to machine precision. On a long-time LeBlanc benchmark ($N=800$, $t=1$), the method crosses a qualitative threshold: one-shot final-time reconstruction fails entirely (shock location error $2.7\times 10^{-1}$), whereas correction every three steps recovers an almost exact sharp solution with contact and shock positions accurate to machine precision. The same detector-and-Newton mechanism handles two-shock and two-rarefaction packets without case-specific logic, with plateau improvements of four to sixteen orders of magnitude. In an unoptimized Python prototype the wall-clock overhead is below a factor of two even on the most aggressively corrected benchmark. To our knowledge, no comparably lightweight fixed-grid add-on has been shown to recover this level of coarse-grid accuracy on the long-time LeBlanc and related near-vacuum problems.

physics.comp-ph↗

arXiv2026

Sub-cell Wave Reconstruction from Differentiated Riemann Variables

We introduce a postprocessing procedure that recovers sub-cell wave geometry from a standard one-dimensional Euler shock-capturing computation using differentiated Riemann variables (DRVs) -- characteristic derivatives that separate the three wave families into distinct localized spikes. Filtered DRV surrogates detect the waves, plateau sampling extracts the local states, and a pressure-wave-function Newton closure completes the geometry. The entire pipeline adds less than $0.25\%$ to the cost of a baseline WENO--5/HLLC solve. For Sod, a severe-expansion problem, and the LeBlanc shock tube, wave locations are recovered to within roundoff or $O(10^{-4})$ and the contact is sharpened to one cell width; a pattern-agnostic extension handles all four Riemann configurations with errors at the $10^{-6}$--$10^{-8}$ level. Direct comparison with MUSCL--THINC--BVD and WENO-Z--THINC--BVD shows that neither reproduces the combination of sharp contacts, small contact-window internal-energy error, and elimination of the LeBlanc positive overshoot achieved by the DRV reconstruction.

physics.comp-ph↗

arXiv2026

Incompressible Euler Blowup at the $C^{1,\frac{1}{3}}$ Threshold

We prove finite-time Type--I blowup for the three-dimensional incompressible Euler equations in the axisymmetric no-swirl class, with initial velocity in $C^{1,\alpha}(\mathbb{R}^3)\cap L^2(\mathbb{R}^3)$, odd symmetry in $z$, and $0<\alpha<\tfrac13$, for an explicit class of finite-energy initial data. The singularity forms at a stagnation point on the symmetry axis. The axial strain and the global vorticity norm blow up at the Type--I rates $-\partial_z u_z(0,0,t)\simeq (T^*-t)^{-1}$ and $\|\omega(\cdot,t)\|_{L^\infty}\simeq (T^*-t)^{-1}$, while the meridional Jacobian collapses according to $J(t)\simeq (T^*-t)^{1/(1-3\alpha)}$. The proof is organized around a Lagrangian clock-and-driver framework. The clock is the meridional Jacobian $J(t)$, and the driver is the compressive axial strain $-\partial_z u_z(0,0,t)$. These variables satisfy, to leading order, a closed Riccati-clock system: the axial strain drives the collapse of $J(t)$, while the collapse of $J(t)$ amplifies the axial strain. We prove that the Euler flow tracks this clock-and-driver model up to the singular time. The main nonlocal obstruction is the pressure Hessian; it is controlled by a non-perturbative strain--pressure Hessian comparison showing that pressure cannot cancel the quadratic compressive strain responsible for collapse. This gives a dynamical explanation of the threshold $\alpha=\tfrac13$. The blowup mechanism is structurally stable and persists for an open set of admissible angular functions in a weighted H\"older topology.

math.AP↗

arXiv2025

Classical Euler flows generate the strong Guderley imploding shock wave

We prove that Guderley's self-similar imploding shock solution for the compressible Euler equations with ideal--gas law ($\gamma>1$) arises from classical, radially symmetric, shock--free data. For such data prescribed at initial time $\mathrm{T_{in}} < 0$, we prove that the flow remains smooth up to a first singular time $t=\mathrm{T}_* \in (\mathrm{T_{in}}, 0)$, where a preshock forms with a $C^{1/3}$ cusp in the fast acoustic variable. From this preshock a unique, initially weak, regular shock is born, whose strength can be made arbitrarily large on a controlled time interval; the front then deforms onto the Guderley shock and implodes at the origin at the collapse time $t=0$. There exists a matching time $t=\mathrm{T_{fin}} \in (\mathrm{T}_*,0)$ such that on $[\mathrm{T_{fin}},0)$ the solution coincides exactly with the classical Guderley self--similar profile, and at $t=\mathrm{T_{fin}}$ the shock trajectory matches the self--similar front to all orders. As $t \to 0^-$, the Euler solution implodes at the center, and continues for $t>0$ as a reflected blast wave, providing a global-in-time unique Euler solution which evolves from regular initial conditions.

math.AP↗

arXiv2024

Gradient catastrophes and an infinite hierarchy of H\"older cusp-singularities for 1D Euler

We establish an infinite hierarchy of finite-time gradient catastrophes for smooth solutions of the 1D Euler equations of gas dynamics with non-constant entropy. Specifically, for all integers $n\geq 1$, we prove that there exist classical solutions, emanating from smooth, compressive, and non-vacuous initial data, which form a cusp-type gradient singularity in finite time, in which the gradient of the solution has precisely $C^{0,\frac{1}{2n+1}}$ H\"older-regularity. We show that such Euler solutions are codimension-$(2n-2)$ stable in the Sobolev space $W^{2n+2,\infty}$.

math.AP↗

arXiv2024

Vorticity blowup in 2D compressible Euler equations

We prove finite-time vorticity blowup for smooth solutions of the 2D compressible Euler equations with smooth, localized, and non-vacuous initial data. The vorticity blowup occurs at the time of the first singularity, and is accompanied by an axisymmetric implosion in which the swirl velocity enjoys full stability, as opposed to finite co-dimension stability.

math.AP↗

arXiv2024

The geometry of maximal development and shock formation for the Euler equations in multiple space dimensions

We construct a fundamental piece of the boundary of the maximal globally hyperbolic development (MGHD) of Cauchy data for the multi-dimensional compressible Euler equations, which is necessary for the local shock development problem. For an open set of compressive and generic $H^7$ initial data, we construct unique $H^7$ solutions to the Euler equations in the maximal spacetime region below a given time-slice, beyond the time of the first singularity; at any point in this spacetime, the solution can be smoothly and uniquely computed by tracing both the fast and slow acoustic characteristic surfaces backward-in-time, until reaching the Cauchy data prescribed along the initial time-slice. The future temporal boundary of this spacetime region is a singular hypersurface, containing the union of three sets: first, a co-dimension-$2$ surface of ``first singularities'' called the pre-shock; second, a downstream hypersurface called the singular set emanating from the pre-shock, on which the Euler solution experiences a continuum of gradient catastrophes; third, an upstream hypersurface consisting of a Cauchy horizon emanating from the pre-shock, which the Euler solution cannot reach. We develop a new geometric framework for the description of the acoustic characteristic surfaces which is based on the Arbitrary Lagrangian Eulerian (ALE) framework, and combine this with a new type of differentiated Riemann variables which are linear combinations of gradients of velocity, sound speed, and the curvature of the fast acoustic characteristic surfaces. With these new variables, we establish uniform $H^7$ Sobolev bounds for solutions to the Euler equations without derivative loss and with optimal regularity.

math.AP↗

arXiv2023

A fast dynamic smooth adaptive meshing scheme with applications to compressible flow

We develop a fast-running smooth adaptive meshing (SAM) algorithm for dynamic curvilinear mesh generation, which is based on a fast solution strategy of the time-dependent Monge-Ampère (MA) equation, $\det \nabla ψ(x,t) = \mathsf{G} \circψ(x,t)$. The novelty of our approach is a new so-called perturbation formulation of MA, which constructs the solution map $ψ$ via composition of a sequence of near-identity deformations of a reference mesh. Then, we formulate a new version of the deformation method that results in a simple, fast, and high-order accurate numerical scheme and a dynamic SAM algorithm that is of optimal complexity when applied to time-dependent mesh generation for solutions to hyperbolic systems such as the Euler equations of gas dynamics. We perform a series of challenging 2$D$ and 3$D$ mesh generation experiments for grids with large deformations, and demonstrate that SAM is able to produce smooth meshes comparable to state-of-the-art solvers, while running approximately 200 times faster. The SAM algorithm is then coupled to a simple Arbitrary Lagrangian Eulerian (ALE) scheme for 2$D$ gas dynamics. Specifically, we implement the $C$-method and develop a new ALE interface tracking algorithm for contact discontinuities. We perform numerical experiments for both the Noh implosion problem as well as a classical Rayleigh-Taylor instability problem. Results confirm that low-resolution simulations using our SAM-ALE algorithm compare favorably with high-resolution uniform mesh runs.

physics.comp-ph↗

arXiv2023

A new type of stable shock formation in gas dynamics

From an open set of initial data, we construct a family of classical solutions to the 1D nonisentropic compressible Euler equations which form $C^{0,ν}$ cusps as a first singularity, for any $ν\in [1/2,1)$. For this range of $ν$, this is the first result demonstrating the stable formation of such $C^{0,ν}$ cusp-type singularities, also known as pre-shocks. The proof uses a new formulation of the differentiated Euler equations along the fast acoustic characteristic, and relies on a novel set of $L^p$ energy estimates for all $1 < p < \infty$, which may be of independent interest.

math.AP↗

arXiv2023

3D Interface Models for Rayleigh-Taylor Problems

We derive interface models for 3D Rayleigh-Taylor instability (RTI), making use of a novel asymptotic expansion in the non-locality of the fluid flow. These interface models are derived for the purpose of studying universal features associated to RTI such as the Froude number in single-mode RTI, the predicted quadratic growth of the interface amplitude under multi-mode random perturbations, the optimal (viscous) mixing rates induced by the RTI and the self-similarity of horizontally averaged density profiles, and the remarkable stabilization of the mixing layer growth rate which arises for the three-fluid two-interface heavy-light-heavy configuration, in which the addition of a third fluid bulk slows the growth of the mixing layer to a linear rate. Our interface models can capture the formation of small-scale structures induced by severe interface roll-up, reproduce experimental data in a number of different regimes, and study the effects of multiple interface interactions even as the interface separation distance becomes exceedingly small. Compared to traditional numerical schemes used to study such phenomena, our models provide a computational speed-up of at least two orders of magnitude.

physics.flu-dyn↗

arXiv2023

A characteristics approach to shock formation in 2D Euler with azimuthal symmetry and entropy

We provide a detailed analysis of the shock formation process for the non-isentropic 2d Euler equations in azimuthal symmetry. We prove that from an open set of smooth and generic initial data, solutions of Euler form a first singularity or gradient blow-up or shock. This first singularity is termed a Hölder $C^{\frac{1}{3}}$ pre-shock, and our analysis provides the first detailed description of this cusp solution. The novelty of this work relative to [Buckmaster-Drivas-Shkoller-Vicol, 2022] is that we herein consider a much larger class of initial data, allow for a non-constant initial entropy, allow for a non-trivial sub-dominant Riemann variable, and introduce a host of new identities to avoid apparent derivative loss due to entropy gradients. The method of proof is also new and robust, exploring the transversality of the three different characteristic families to transform space derivatives into time derivatives. Our main result provides a fractional series expansion of the Euler solution about the pre-shock, whose coefficients are computed from the initial data.

math.AP↗

arXiv2021

Simultaneous development of shocks and cusps for 2D Euler with azimuthal symmetry from smooth data

A fundamental question in fluid dynamics concerns the formation of discontinuous shock waves from smooth initial data. We prove that from smooth initial data, smooth solutions to the 2d Euler equations in azimuthal symmetry form a first singularity, the so-called $C^{\frac{1}{3}} $ pre-shock. The solution in the vicinity of this pre-shock is shown to have a fractional series expansion with coefficients computed from the data. Using this precise description of the pre-shock, we prove that a discontinuous shock instantaneously develops after the pre-shock. This regular shock solution is shown to be unique in a class of entropy solutions with azimuthal symmetry and regularity determined by the pre-shock expansion. Simultaneous to the development of the shock front, two other characteristic surfaces of cusp-type singularities emerge from the pre-shock. These surfaces have been termed weak discontinuities by Landau & Lifschitz [Chapter IX, §96], who conjectured some type of singular behavior of derivatives along such surfaces. We prove that along the slowest surface, all fluid variables except the entropy have $C^{1, {\frac{1}{2}} }$ one-sided cusps from the shock side, and that the normal velocity is decreasing in the direction of its motion; we thus term this surface a weak rarefaction wave. Along the surface moving with the fluid velocity, density and entropy form $C^{1, {\frac{1}{2}} }$ one-sided cusps while the pressure and normal velocity remain $C^2$; as such, we term this surface a weak contact discontinuity.

math.AP↗

arXiv2020

Shock formation and vorticity creation for 3d Euler

We analyze the shock formation process for the 3d non-isentropic Euler equations with the ideal gas law, in which sounds waves interact with entropy waves to produce vorticity. Building on our theory for isentropic flows in [3,4], we give a constructive proof of shock formation from smooth initial data. Specifically, we prove that there exist smooth solutions to the non-isentropic Euler equations which form a generic stable shock with explicitly computable blowup time, location, and direction. This is achieved by establishing the asymptotic stability of a generic shock profile in modulated self-similar variables, controlling the interaction of wave families via: (i) pointwise bounds along Lagrangian trajectories, (ii) geometric vorticity structure, and (iii) high-order energy estimates in Sobolev spaces.

math.AP↗

arXiv2020

Formation of point shocks for 3D compressible Euler

We consider the 3D isentropic compressible Euler equations with the ideal gas law. We provide a constructive proof of shock formation from smooth initial datum of finite energy, with no vacuum regions, with nontrivial vorticity present at the shock, and under no symmetry assumptions. We prove that for an open set of Sobolev-class initial data which are a small $L^ \infty $ perturbation of a constant state, there exist smooth solutions to the Euler equations which form a generic stable shock in finite time. The blow up time and location can be explicitly computed, and solutions at the blow up time are smooth except for a single point, where they are of cusp-type with Hölder $C^ {\frac{1}{3}}$ regularity. Our proof is based on the use of modulated self-similar variables that are used to enforce a number of constraints on the blow up profile, necessary to establish the stability in self-similar variables of the generic shock profile.

math.AP↗

arXiv2020

Affine motion of 2d incompressible fluids surrounded by vacuum and flows in ${\rm SL}(2,{\mathbb R})$

The affine motion of two-dimensional (2d) incompressible fluids surrounded by vacuum can be reduced to a completely integrable and globally solvable Hamiltonian system of ordinary differential equations for the deformation gradient in ${\rm SL}(2,{\mathbb R})$. In the case of perfect fluids, the motion is given by geodesic flow in ${\rm SL}(2,{\mathbb R})$ with the Euclidean metric, while for magnetically conducting fluids (MHD), the motion is governed by a harmonic oscillator in ${\rm SL}(2,{\mathbb R})$. A complete classification of the dynamics is given including rigid motions, rotating eddies with stable and unstable manifolds, and solutions with vanishing pressure. For perfect fluids, the displacement generically becomes unbounded, as $t\to\pm\infty$. For MHD, solutions are bounded and generically quasi-periodic and recurrent.

math.AP↗

arXiv2019

A multiscale model for Rayleigh-Taylor and Richtmyer-Meshkov instabilities

We develop a novel multiscale model of interface motion for the Rayleigh-Taylor instability (RTI) and Richtmyer-Meshkov instability (RMI) for two-dimensional, inviscid, compressible flows with vorticity, which yields a fast-running numerical algorithm that produces both qualitatively and quantitatively similar results to a resolved gas dynamics code, while running approximately two orders of magnitude (in time) faster. Our multiscale model is founded upon a new compressible-incompressible decomposition of the velocity field $u=v+w$. The incompressible component $w$ of the velocity is also irrotational and is solved using a new asymptotic model of the Birkhoff-Rott singular integral formulation of the incompressible Euler equations, which reduces the problem to one spatial dimension. This asymptotic model, called the higher-order $z$-model, is derived using small nonlocality as the asymptotic parameter, allows for interface turn-over and roll-up, and yields a significant simplification for the equation describing the evolution of the amplitude of vorticity. This incompressible component $w$ of the velocity controls the small scale structures of the interface and can be solved efficiently on fine grids. Meanwhile, the compressible component of the velocity $v$ remains continuous near contact discontinuities and can be computed on relatively coarse grids, while receiving subgrid scale information from $w$. We first validate the incompressible higher-order $z$-model by comparison with classical RTI experiments as well as full point vortex simulations. We then consider both the RTI and the RMI problems for our multiscale model of compressible flow with vorticity, and show excellent agreement with our high-resolution gas dynamics solutions.

physics.comp-ph↗