arXiv subjects

Jared Speck

Publications and source records attributed to Jared Speck.

34 records · Page 2Linked to original sources

arXiv2016

Shock formation in solutions to the $2D$ compressible Euler equations in the presence of non-zero vorticity

We study the Cauchy problem for the compressible Euler equations in two spatial dimensions under any physical barotropic equation of state except that of a Chaplygin gas. We prove that the well-known phenomenon of shock formation in simple plane wave solutions, starting from smooth initial conditions, is stable under perturbations that break the plane symmetry. Moreover, we provide a sharp asymptotic description of the singularity formation. The new feature of our work is that the perturbed solutions are allowed to have small but non-zero vorticity, even at the location of the shock. Thus, our results provide the first constructive description of the vorticity near a singularity formed from compression: relative to a system of geometric coordinates adapted to the acoustic characteristics, the vorticity remains many times differentiable, all the way up to the shock. In addition, relative to the Cartesian coordinates, the vorticity remains bounded up to the shock. To control the vorticity, we rely on a coalition of new geometric and analytic insights that complement the ones used by Christodoulou in his groundbreaking, sharp proof of shock formation in vorticity-free regions. In particular, we rely on a new formulation of the compressible Euler equations exhibiting remarkable structures. To derive estimates, we construct an eikonal function adapted to the acoustic characteristics (which correspond to sound wave propagation) and a related set of geometric coordinates and differential operators. Thanks to the remarkable structure of the equations, the same set of coordinates and differential operators can be used to analyze the vorticity, whose characteristics are transversal to the acoustic characteristics. In particular, our work provides the first constructive description of shock formation without symmetry assumptions in a system with multiple speeds.

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arXiv2016

The hidden null structure of the compressible Euler equations and a prelude to applications

We derive a new formulation of the compressible Euler equations exhibiting remarkable structures, including surprisingly good null structures. The new formulation comprises covariant wave equations for the Cartesian components of the velocity and the logarithmic density coupled to a transport equation for the specific vorticity, defined to be vorticity divided by density. The equations allow one to use the full power of the geometric vectorfield method in treating the "wave part" of the system. A crucial feature of the new formulation is that all derivative-quadratic inhomogeneous terms verify the strong null condition. The latter is a nonlinear condition signifying the complete absence of nonlinear interactions involving more than one differentiation in a direction transversal to the acoustic characteristics. Moreover, the same good structures are found in the equations verified by the Euclidean divergence and curl of the specific vorticity. This is important because one needs to combine estimates for the divergence and curl with elliptic estimates to obtain sufficient regularity for the specific vorticity, whose derivatives appears as inhomogeneous terms in the wave equations. The above structures collectively open the door for our forthcoming results: exhibiting a stable regime in which initially smooth solutions develop a shock singularity (in particular the first Cartesian coordinate partial derivatives of the velocity and density blow up) while, relative to a system of geometric coordinates adapted to the acoustic characteristics, the solution (including the vorticity) remains many times differentiable, all the way up to the shock. The good null structures, which are often associated with global solutions, are in fact key to proving that the shock singularity forms. Our secondary goal in this article is to overview the central role that the structures play in the proof.

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arXiv2016

Stable shock formation for nearly simple outgoing plane symmetric waves

In an influential 1964 article, P. Lax studied $2 \times 2$ genuinely nonlinear strictly hyperbolic PDE systems (in one spatial dimension). Using the method of Riemann invariants, he showed that a large set of smooth initial data lead to bounded solutions whose first spatial derivatives blow up in finite time, a phenomenon known as wave breaking. In the present article, we study the Cauchy problem for two classes of quasilinear wave equations in two spatial dimensions that are closely related to the systems studied by Lax. When the data have one-dimensional symmetry, Lax's methods can be applied to the wave equations to show that a large set of smooth initial data lead to wave breaking. Here we study solutions with initial data that are close, as measured by an appropriate Sobolev norm, to data belonging to a distinguished subset of Lax's data: the data corresponding to simple plane waves. Our main result is that under suitable relative smallness assumptions, the Lax-type wave breaking for simple plane waves is stable. The key point is that we allow the data perturbations to break the symmetry. Moreover, we give a detailed, constructive description of the asymptotic behavior of the solution all the way up to the first singularity, which is a shock driven by the intersection of null (characteristic) hyperplanes. We also outline how to extend our results to the compressible irrotational Euler equations. To derive our results, we use Christodoulou's framework for studying shock formation to treat a new solution regime in which wave dispersion is not present.

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arXiv2015

A priori estimates for solutions to the relativistic Euler equations with a moving vacuum boundary

We study the relativistic Euler equations on the Minkowski spacetime background. We make assumptions on the equation of state and the initial data that are relativistic analogs of the well-known physical vacuum boundary condition, which has played an important role in prior work on the non-relativistic compressible Euler equations. Our main result is the derivation, relative to Lagrangian (also known as co-moving) coordinates, of local-in-time a priori estimates for the solution. The solution features a fluid-vacuum boundary, transported by the fluid four-velocity, along which the hyperbolicity of the equations degenerates. In this context, the relativistic Euler equations are equivalent to a degenerate quasilinear hyperbolic wave-map-like system that cannot be treated using standard energy methods.

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arXiv2014

Stable Big Bang Formation in Near-FLRW Solutions to the Einstein-Scalar Field and Einstein-Stiff Fluid Systems

We prove a stable singularity formation result for solutions to the Einstein-scalar field and Einstein-stiff fluid systems. Our results apply to small perturbations of the spatially flat FLRW solution with topology $(0,\infty) \times \mathbb{T}^3.$ The FLRW solution models a spatially uniform scalar-field/stiff fluid evolving in a spacetime that expands towards the future and that has a "Big Bang" singularity at $\lbrace 0 \rbrace \times \mathbb{T}^3,$ where its curvature blows up. We place data on a Cauchy hypersurface $Σ_1'$ that are close to the FLRW data induced on $\lbrace 1 \rbrace \times \mathbb{T}^3.$ We study the perturbed solution in the collapsing direction and prove that its basic features closely resemble those of the FLRW solution. In particular, we construct constant mean curvature-transported spatial coordinates for the perturbed solution covering $(t,x) \in (0,1] \times \mathbb{T}^3$ and show that it also has a Big Bang at $\lbrace 0 \rbrace \times \mathbb{T}^3,$ where its curvature blows up. The blow-up confirms Penrose's Strong Cosmic Censorship hypothesis for the "past-half" of near-FLRW solutions. The most difficult aspect of the proof is showing that the solution exists for $(t,x) \in (0,1] \times \mathbb{T}^3,$ and to this end, we derive energy estimates that are allowed to mildly blow-up as $t \downarrow 0.$ To close these estimates, we use the most important ingredient in our analysis: an $L^2-$type energy approximate monotonicity inequality that holds for near-FLRW solutions. In the companion article "A regime of linear stability for the Einstein-scalar field system with applications to nonlinear Big Bang formation," we used the approximate monotonicity to prove a stability result for solutions to linearized versions of the equations. The present article shows that the linear stability result can be upgraded to control the nonlinear terms.

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arXiv2014

Shock Formation in Small-Data Solutions to $3D$ Quasilinear Wave Equations

In his 2007 monograph, D. Christodoulou proved a breakthrough result giving a detailed description of the formation of shocks in solutions to the relativistic Euler equations in three spatial dimensions. He assumed that the data have small $H^N$ norm, where $N$ is a sufficiently large integer. To deduce the shock formation, he also assumed that the data verify a signed integral inequality. In the present monograph, we extend Christodoulou's framework and use it to prove that shock singularities often develop in initially small, regular solutions to two important classes of quasilinear wave equations in three spatial dimensions. Our work also generalizes and unifies earlier work on singularity formation initiated by F. John in the 1970's and continued by L. Hörmander, S. Alinhac, and many others. Specifically, we study $\mbox{i)}$ covariant scalar wave equations of the form $\square_{g(Ψ)} Ψ= 0$ and $\mbox{ii)}$ non-covariant scalar wave equations of the form $(h^{-1})^{αβ}(\partial Φ) \partial_α \partial_β Φ= 0.$ Our main result shows that whenever the nonlinear terms fail Klainerman's classic null condition, shocks develop in solutions arising from an open set of small data. Hence, within the classes $\mbox{i)}$ and $\mbox{ii)}$, our work can be viewed as a sharp converse to the well-known result of Christodoulou and Klainerman, which showed that when the classic null condition is verified, small-data global existence holds.

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arXiv2014

Shock Formation in Small-Data Solutions to $3D$ Quasilinear Wave Equations: An Overview

In his 2007 monograph, D. Christodoulou proved a remarkable result giving a detailed description of shock formation, for small $H^s$-initial conditions ($s$ sufficiently large), in solutions to the relativistic Euler equations in three space dimensions. His work provided a significant advancement over a large body of prior work concerning the long-time behavior of solutions to higher-dimensional quasilinear wave equations, initiated by F. John in the mid 1970's and continued by S. Klainerman, T. Sideris, L. Hörmander, H. Lindblad, S. Alinhac, and others. Our goal in this paper is to give an overview of his result, outline its main new ideas, and place it in the context of the above mentioned earlier work. We also introduce the recent work of J. Speck, which extends Christodoulou's result to show that for two important classes of quasilinear wave equations in three space dimensions, small-data shock formation occurs precisely when the quadratic nonlinear terms fail the classic null condition.

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arXiv2013

The Global Future Stability of the FLRW Solutions to the Dust-Einstein System with a Positive Cosmological Constant

We study small perturbations of the well-known family of Friedman-Lema\^ıtre-Robertson-Walker (FLRW) solutions to the dust-Einstein system with a positive cosmological constant in the case that the spacelike Cauchy hypersurfaces are diffeomorphic to T^3. These solutions model a quiet pressureless fluid in a dynamic spacetime undergoing accelerated expansion. We show that the FLRW solutions are nonlinearly globally future-stable under small perturbations of their initial data. Our analysis takes place relative to a harmonic-type coordinate system, in which the cosmological constant results in the presence of dissipative terms in the evolution equations. Our result extends the results of [38,44,42], where analogous results were proved for the Euler-Einstein system under the equations of state p = c_s^2 ρ, 0<c_s^2 <= 1/3. The dust-Einstein system is the Euler-Einstein system with c_s=0. The main difficulty that we overcome is that the energy density of the dust loses one degree of differentiability compared to the cases 0 < c_s^2 <= 1/3. Because the dust-Einstein equations are coupled, this loss of differentiability introduces new obstacles for deriving estimates for the top-order derivatives of all solution variables. To resolve this difficulty, we commute the equations with a well-chosen differential operator and derive a collection of elliptic estimates that complement the energy estimates of [38,44]. An important feature of our analysis is that we are able to close our estimates even though the top-order derivatives of all solution variables can grow much more rapidly than in the cases 0<c_s^2 <= 1/3. Our results apply in particular to small compact perturbations of the vanishing dust state.

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arXiv2012

The Nonlinear Future-Stability of the FLRW Family of Solutions to the Euler-Einstein System with a Positive Cosmological Constant

In this article, we study small perturbations of the family of Friedmann-Lemaître-Robertson-Walker cosmological background solutions to the 1 + 3 dimensional Euler-Einstein system with a positive cosmological constant. These background solutions describe an initially uniform quiet fluid of positive energy density evolving in a spacetime undergoing accelerated expansion. Our nonlinear analysis shows that under the equation of state pressure = c_s^2 * energy density, with 0 < c_s^2 < 1/3, the background solutions are globally future-stable. In particular, we prove that the perturbed spacetime solutions, which have the topological structure [0,infty) x T^3, are future causally geodesically complete. These results are extensions of previous results derived by the author in a collaboration with I. Rodnianski, in which the fluid was assumed to be irrotational. Our novel analysis of a fluid with non-zero vorticity is based on the use of suitably-defined energy currents.

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arXiv2012

The Stabilizing Effect of Spacetime Expansion on Relativistic Fluids With Sharp Results for the Radiation Equation of State

In this article, we study the 1 + 3 dimensional relativistic Euler equations on a pre-specified conformally flat expanding spacetime background with spatial slices that are diffeomorphic to $\mathbb{R}^3.$ We assume that the fluid verifies the equation of state $p = c_s^2 ρ,$ where $0 \leq c_s \leq \sqrt{1/3}$ is the speed of sound. We also assume that the inverse of the scale factor associated to the expanding spacetime metric verifies a $c_s-$dependent time-integrability condition. Under these assumptions, we use the vectorfield energy method to prove that an explicit family of physically motivated, spatially homogeneous, and spatially isotropic fluid solutions is globally future-stable under small perturbations of their initial conditions. The explicit solutions corresponding to each scale factor are analogs of the well-known spatially flat Friedmann-Lema\^ıtre-Robertson-Walker family. Our nonlinear analysis, which exploits dissipative terms generated by the expansion, shows that the perturbed solutions exist for all future times and remain close to the explicit solutions. This work is an extension of previous results, which showed that an analogous stability result holds when the spacetime is exponentially expanding. In the case of the radiation equation of state $p = (1/3)ρ,$ we also show that if the time-integrability condition for the inverse of the scale factor fails to hold, then the explicit fluid solutions are unstable. More precisely, we show that arbitrarily small smooth perturbations of the explicit solutions' initial conditions can launch perturbed solutions that form shocks in finite time. The shock formation proof is based on the conformal invariance of the relativistic Euler equations when $c_s^2 = 1/3,$ which allows for a reduction to a well-known result of Christodoulou.

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arXiv2010

The Global Stability of the Minkowski Spacetime Solution to the Einstein-Nonlinear Electromagnetic System in Wave Coordinates

In this article, we study the coupling of the Einstein field equations of general relativity to a family of models of nonlinear electromagnetic fields. The family comprises all covariant electromagnetic models that satisfy the following criteria: they are derivable from a sufficiently regular Lagrangian, they reduce to the linear Maxwell model in the weak-field limit, and their corresponding energy-momentum tensors satisfy the dominant energy condition. Our main result is a proof of the global nonlinear stability of the 1 + 3-dimensional Minkowski spacetime solution to the coupled system for any member of the family, which includes the linear Maxwell model. This stability result is a consequence of a small-data global existence result for a reduced system of equations that is equivalent to the original system in our wave coordinate gauge. Our analysis of the spacetime metric components is based on a framework recently developed by Lindblad and Rodnianski, which allows us to derive suitable estimates for tensorial systems of quasilinear wave equations with nonlinearities that satisfy the weak null condition. Our analysis of the electromagnetic fields, which satisfy quasilinear first-order equations, is based on an extension of a geometric energy-method framework developed by Christodoulou, together with a collection of pointwise decay estimates for the Faraday tensor developed in the article. We work directly with the electromagnetic fields, and thus avoid the use of electromagnetic potentials.

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arXiv2010

The Nonlinear Stability of the Trivial Solution to the Maxwell-Born-Infeld System

In this article, we use an electromagnetic gauge-free framework to establish the existence of small-data global solutions to the Maxwell-Born-Infeld (MBI) system on the Minkowski space background in 1 + 3 dimensions. Because the nonlinearities in the system satisfy a version of the null condition, we are also able to show that these solutions decay at exactly the same rates as solutions to the linear Maxwell-Maxwell system. In addition, we show that on any Lorentzian manifold, the MBI system is hyperbolic in the interior of the field-strength regime in which its Lagrangian is real-valued.

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arXiv2010

The Non-relativistic Limit of the Euler Nordström System with Cosmological Constant

In this paper the author studies the singular limit c to infinity of the family of Euler-Nordström systems indexed by the parameters kappa and c (EN_kappa,c), where kappa^2 > 0 is the cosmological constant and c is the speed of light. Using Christodoulou's techniques to generate energy currents, the author develops Sobolev estimates that show that for initial data belonging to an appropriate Sobolev space, as c tends to infinity, the solutions to the EN_kappa,c system converge uniformly on a spacetime slab [0,T]xR^3 to the solution of the Euler-Poisson system with the cosmological constant kappa^2.

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arXiv2010

Hilbert Expansion from the Boltzmann equation to relativistic Fluids

We study the local-in-time hydrodynamic limit of the relativistic Boltzmann equation using a Hilbert expansion. More specifically, we prove the existence of local solutions to the relativistic Boltzmann equation that are nearby the local relativistic Maxwellian constructed from a class of solutions to the relativistic Euler equations that includes a large subclass of near-constant, non-vacuum fluid states. In particular, for small Knudsen number, these solutions to the relativistic Boltzmann equation have dynamics that are effectively captured by corresponding solutions to the relativistic Euler equations.

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arXiv2010

The Stability of the Irrotational Euler-Einstein System with a Positive Cosmological Constant

In this article, we study small perturbations of the family of Friedmann-Lemaître-Robertson-Walker cosmological background solutions to the Euler-Einstein system with a positive cosmological constant in 1 + 3 dimensions. The background solutions describe an initially uniform quiet fluid of positive energy density evolving in a spacetime undergoing accelerated expansion. We show that under the equation of state p = c_s^2*(energy density), 0 < c_s^2 < 1/3, the background solutions are globally future asymptotically stable under small irrotational perturbations. In particular, we prove that the perturbed spacetimes, which have the topological structure [0,infinity) x T^3, are future causally geodesically complete.

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arXiv2008

Well-Posedness for the Euler-Nordstrom System with Cosmological Constant

In this paper the author considers the motion of a relativistic perfect fluid with self-interaction mediated by Nordstrom's scalar theory of gravity. The evolution of the fluid is determined by a quasilinear hyperbolic system of PDEs, and a cosmological constant is introduced in order to ensure the existence of non-zero constant solutions. Accordingly, the initial value problem for a compact perturbation of an infinitely extended quiet fluid is studied. Although the system is neither symmetric hyperbolic nor strictly hyperbolic, Christodoulou's constructive results on the existence of energy currents for equations derivable from a Lagrangian can be adapted to provide energy currents that can be used in place of the standard energy principle available for first-order symmetric hyperbolic systems. After providing such energy currents, the author uses them to prove that the Euler-Nordstrom system with a cosmological constant is well-posed in a suitable Sobolev space.

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