Shock formation for 2D2D quasilinear wave systems featuring multiple speeds: Blowup for the fastest wave, with non-trivial interactions up to the singularity

We prove a stable shock formation result for a large class of systems of quasilinear wave equations in two spatial dimensions. We give a precise description of the dynamics all the way up to the singularity. Our main theorem applies to systems of two wave equations featuring two distinct wave speeds and various quasilinear and semilinear nonlinearities, while the solutions under study are (non-symmetric) perturbations of simple outgoing plane symmetric waves. The two waves are allowed to interact all the way up to the singularity. Our approach is robust and could be used to prove shock formation results for other related systems with many unknowns and multiple speeds, in various solution regimes, and in higher spatial dimensions. However, a fundamental aspect of our framework is that it applies only to solutions in which the "fastest wave" forms a shock while the remaining solution variables do not. Our approach is based on an extended version of the geometric vectorfield method developed by D. Christodoulou in his study of shock formation for scalar wave equations as well as the framework developed in our recent joint work with J. Luk, in which we proved a shock formation result for a quasilinear wave-transport system featuring a single wave operator. A key new difficulty that we encounter is that the geometric vectorfields that we use to commute the equations are, by necessity, adapted to the wave operator of the (shock-forming) fast wave and therefore exhibit very poor commutation properties with the slow wave operator, much worse than their commutation properties with a transport operator. To overcome this difficulty, we rely on a first-order reformulation of the slow wave equation, which, though somewhat limiting in the precision it affords, allows us to avoid uncontrollable commutator terms.Comment: 117 pages, 3 figure

A regime of linear stability for the Einstein-scalar field system with applications to nonlinear Big Bang formation

We linearize the Einstein-scalar field equations, expressed relative to constant mean curvature (CMC)-transported spatial coordinates gauge, around members of the well-known family of Kasner solutions on $(0,\infty) \times \mathbb{T}^3$. The Kasner solutions model a spatially uniform scalar field evolving in a (typically) spatially anisotropic spacetime that expands towards the future and that has a "Big Bang" singularity at $\lbrace t = 0 \rbrace$. We place initial data for the linearized system along $\lbrace t = 1 \rbrace \simeq \mathbb{T}^3$ and study the linear solution's behavior in the collapsing direction $t \downarrow 0$. Our first main result is the proof of an approximate $L^2$ monotonicity identity for the linear solutions. Using it, we prove a linear stability result that holds when the background Kasner solution is sufficiently close to the Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) solution. In particular, we show that as $t \downarrow 0$, various time-rescaled components of the linear solution converge to regular functions defined along $\lbrace t = 0 \rbrace$. In addition, we motivate the preferred direction of the approximate monotonicity by showing that the CMC-transported spatial coordinates gauge can be viewed as a limiting version of a family of parabolic gauges for the lapse variable; an approximate monotonicity identity and corresponding linear stability results also hold in the parabolic gauges, but the corresponding parabolic PDEs are locally well-posed only in the direction $t \downarrow 0$. Finally, based on the linear stability results, we outline a proof of the following result, whose complete proof will appear elsewhere: the FLRW solution is globally nonlinearly stable in the collapsing direction $t \downarrow 0$ under small perturbations of its data at $\lbrace t = 1 \rbrace$.Comment: 73 page

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\^{\i}tre-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 \rho, 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.Comment: In the latest version, we added a few references and corrected some typo