Abstract: We present a robust physical-space approach to establish time decay and global asymptotics of solutions to variable-coefficient Schrödinger equations in (3+1)-dimensions. As an immediate nonlinear application, we obtain new small data global existence and asymptotics results for quasilinear Schrödinger equations with cubic, Hamiltonian nonlinearity, variable coefficients in their linear part, and possibly outside obstacles, even in the presence of trapped bicharacteristics (provided that they are suitably unstable).

Our approach relies on three primary ingredients. First is the concept of a good commutator, which extends Klainerman's classical commuting vector field method, and develops upon earlier approaches of Cuccagna-Georgiev-Visciglia and Rodnianski-Tao in the variable-coefficient case, and of Ifrim-Tataru, Ifrim-Koch-Tataru in the nonlinear case. Second, we apply Ifrim and Tataru's testing-by-wave-packets method, which is key for obtaining global asymptotics and handling sharp decay assumptions on the coefficients. Finally, we introduce a systematic technique for analyzing the control provided by the good commutator, termed two-scale elliptic analysis. Together, these techniques significantly broaden the applicability of physical-space methods to a wider class of variable-coefficient nonlinear problems.

Abstract: We prove small-data global stability of the Minkowski solution to Einstein's equations in a centre-normalised outgoing null-geodesic gauge. Our scheme involves first using the $r^{p}$-estimates of Dafermos-Rodnianski to control certain components of the Weyl tensor which satisfy a decoupled tensorial wave equation. Having established this control, all remaining geometric quantities are controlled by transport equations, taking initial conditions at a regular central axis. This method establishes global stability for initial data which decay only weakly to flat space and can establish additional asymptotic control when the data are assumed to have more structure.

Abstract: We consider a class of scalar quasilinear wave equations in three spatial dimensions satisfying the weak null condition. For solutions arising from small, localized, smooth data, we give an asymptotic formula describing the global asymptotics towards the future. We prove that the late-time asymptotics is given by a continuous superposition of decay rates, in stark contrast to equations satisfying a null condition.

The asymptotic formula we obtain is given in terms of a solution to the linear wave equation. Combining this with analysis on the linear wave equation, we strengthen some rigidity results of the third author, showing in particular that any solution with a faster time decay than expected away from the wave zone must vanish identically.

Abstract: We introduce a new versatile method for constructing solution operators (i.e., right-inverses up to a finite rank operator) for a wide class of underdetermined PDEs $P u = f$, which are regularizing of optimal order and, more interestingly, whose integral kernels have certain prescribed support properties. By duality, we simultaneously obtain integral representation formulas (i.e., left-inverses up to a finite rank operator) for overdetermined PDEs $P^{\ast} v = g$ with analogous properties, which lead to Poincar\'e- or Korn-type inequalities. Our method applies to operators such as the divergence, linearized scalar curvature, and linearized Einstein constraint operators (which are underdetermined), as well as the gradient, Hessian, trace-free part of the Hessian, Killing, and conformal Killing operators (which are overdetermined).

The starting point for our construction is a condition - dubbed the recovery on curves condition (RC) - that leads to Green's functions for $P$ supported on prescribed curves. Then the desired integral solution operators (and, by duality, integral representation formulas) are obtained by taking smooth averages over a suitable family of curves. This procedure generalizes the previous constructions of Bogovskii, Oh-Tataru, and Reshetnyak. We furthermore identify a simple algebraic sufficient condition for (RC), namely, that the principal symbol $p(x, \xi)$ of $P$ is full-rank for all non-zero complex vectors $\xi$ (as opposed to real, as in ellipticity). When the principal symbol has constant coefficients, this is equivalent to (RC) and also to the condition that the formal cokernel of $P$ (without any boundary conditions) is finite-dimensional; for this reason, we call it the finite-dimensional cokernel condition (FC). We give a short proof that all operators above satisfy (FC), and thus (RC).

Various applications will be considered in subsequent papers.

Abstract: We construct finite energy blow-up solutions for the radial self-dual Chern-Simons-Schrödinger equation with a continuum of blow-up rates. Our result stands in stark contrast to the rigidity of blow-up of $H^{3}$ solutions proved by the first author for equivariant index m≥1, where the soliton-radiation interaction is too weak to admit the present blow-up scenarios. It is optimal (up to an endpoint) in terms of the range of blow-up rates and the regularity of the asymptotic profiles in view of the authors' previous proof of $H^{1}$ soliton resolution for the self-dual Chern-Simons-Schrödinger equation in any equivariance class.

Our approach is a backward construction combined with modulation analysis, starting from prescribed asymptotic profiles and deriving the corresponding blow-up rates from their strong interaction with the soliton. In particular, our work may be seen as an adaptation of the method of Jendrej-Lawrie-Rodriguez (developed for energy critical equivariant wave maps) to the Schrödinger case.

Abstract: We prove that the 3-dimensional catenoid is asymptotically stable as a solution to the hyperbolic vanishing mean curvature equation in Minkowski space, modulo suitable translation and boost (i.e., modulation) and with respect to a codimension one set of initial data perturbations. The modulation and the codimension one restriction on the initial data are necessary (and optimal) in view of the kernel and the unique simple eigenvalue, respectively, of the stability operator of the catenoid.

The 3-dimensional problem is more challenging than the higher (specifically, 5 and higher) dimensional case addressed in the previous work of the authors with J. Lührmann, due to slower temporal decay of waves and slower spatial decay of the catenoid. To overcome these issues, we introduce several innovations, such as a proof of Morawetz- (or local-energy-decay-) estimates for the linearized operator with slowly decaying kernel elements based on the Darboux transform, a new method to obtain Price's-law-type bounds for waves on a moving catenoid, as well as a refined profile construction designed to capture a crucial cancellation in the wave-catenoid interaction. In conjunction with our previous work on the higher dimensional case, this paper outlines a systematic approach for studying other soliton stability problems for (3+1)-dimensional quasilinear wave equations.

Abstract: We show that regular solutions to electron-MHD with resistivity can be continued as long as the time integral of the supremum of the current gradient remains finite. This dimensionless continuation criterion is analogous to the celebrated result of Beale-Kato-Majda for the incompressible Euler and Navier-Stokes equations. A similar continuation criterion, formulated in terms of the time integral of the supremum of the vorticity, velocity gradient and current gradient, is established for the Hall-MHD with resistivity as well.

Abstract: It has been shown in our previous work that the incompressible and irresistive Hall- and electron-magnetohydrodynamic (MHD) equations are illposed on flat domains $M=\mathbb{R}^{k} \times \mathbb{T}^{3-k}$ for $0 \leq k \leq 2$. The data and solutions therein were assumed to be independent of one coordinate, which not only significantly simplifies the systems but also allows for a large class of steady states. In this work, we remove the assumption of independence and conclude strong illposedness for compactly supported data in $\mathbb{R}^{3}$. This is achieved by constructing degenerating wave packets for linearized systems around time-dependent axisymmetric magnetic fields. A few main additional ingredients are: a more systematic application of the generalized energy estimate, use of the Bogovskiǐ operator, and a priori estimates for axisymmetric solutions to the Hall- and electron-MHD systems.

Abstract: We introduce a general method for understanding the late time tail for solutions to wave equations on asymptotically flat spacetimes with odd space dimensions. In particular, for a large class of equations, we prove that the precise late time tail is determined by the limits of higher radiation field at future null infinity.

In the setting of stationary linear equations, we recover and generalize the Price law decay rates. In particular, in addition to reproving known results on (3+1)-dimensional black holes, this allows one to obtain the sharp decay rate for the wave equation on higher dimensional black hole spacetimes, which exhibits an anomalous rate due to subtle cancellations. More interesting, our method goes beyond the stationary linear case and applies to both equations on dynamical background and nonlinear equations. In this case, our results can be used to show that in general there is a correction to the Price law rates.

Abstract: We prove the local wellposedness of the Cauchy problems for the electron magnetohydrodynamics equations (E-MHD) without resistivity for possibly large perturbations of nonzero uniform magnetic fields. While the local wellposedness problem for (E-MHD) has been extensively studied in the presence of resistivity (which provides dissipative effects), this seems to be the first such result without resistivity. (E-MHD) is a fluid description of plasma in small scales where the motion of electrons relative to ions is significant. Mathematically, it is a quasilinear dispersive equation with nondegenerate but nonelliptic second-order principal term. Our result significantly improves upon the straightforward adaptation of the classical work of Kenig-Ponce-Rolvung-Vega on the quasilinear ultrahyperbolic Schrödinger equations, as the regularity and decay assumptions on the initial data are greatly weakened to the level analogous to the recent work of Marzuola-Metcalfe-Tataru in the case of elliptic principal term.

A key ingredient of our proof is a simple observation about the relationship between the size of a symbol and the operator norm of its quantization as a pseudodifferential operator when restricted to high frequencies. This allows us to localize the (non-classical) pseudodifferential renormalization operator considered by Kenig-Ponce-Rolvung-Vega, and produce instead a classical pseudodifferential renormalization operator. We furthermore incorporate the function space framework of Marzuola-Metcalfe-Tataru to the present case of nonelliptic principal term.

Abstract: We provide a unified viewpoint on two illposedness mechanisms for dispersive equations in one spatial dimension, namely degenerate dispersion and (the failure of) the Takeuchi-Mizohata condition. Our approach is based on a robust energy- and duality-based method introduced in an earlier work of the authors in the setting of Hall-magnetohydynamics. Concretely, the main results in this paper concern strong illposedness of the Cauchy problem (e.g., non-existence and unboundedness of the solution map) in high-regularity Sobolev spaces for various quasilinear degenerate Schrödinger- and KdV-type equations, including the Hunter-Smothers equation, K(m,n) models of Rosenau-Hyman, and the inviscid surface growth model. The mechanism behind these results may be understood in terms of combination of two effects: degenerate dispersion - which is a property of the principal term in the presence of degenerating coefficients - and the evolution of the amplitude governed by the Takeuchi-Mizohata condition - which concerns the subprincipal term. We also demonstrate how the same techniques yield a more quantitative version of the classical $L^{2}$-illposedness result by Mizohata for linear variable-coefficient Schrödinger equations with failed Takeuchi-Mizohata condition.

Abstract: We give new proofs of general relativistic initial data gluing results on unit-scale annuli based on explicit solution operators for the linearized constraint equation around the flat case with prescribed support properties. These results retrieve and optimize - in terms of positivity, regularity, size and/or spatial decay requirements - a number of known theorems concerning asymptotically flat initial data, including Kerr exterior gluing by Corvino-Schoen and Chruściel-Delay, interior gluing (or "fill-in") by Bieri-Chruściel, and obstruction-free gluing by Czimek-Rodnianski. In particular, our proof of the strengthened obstruction-free gluing theorem relies on purely spacelike techniques, rather than null gluing as in the original approach.

Abstract: We prove strong nonlinear illposedness results for the generalized SQG equation, $\partial_{t} \theta + \nabla^{\perp} \Gamma[\theta] \cdot \nabla \theta = 0$, in any sufficiently regular Sobolev spaces, when Γ is a singular in the sense that its symbol satisfies $|\Gamma(\xi)| \to \infty$ as $|\xi| \to \infty$ with some mild regularity assumptions. The key mechanism is degenerate dispersion, i.e., the rapid growth of frequencies of solutions around certain shear states, and the robustness of our method allows one to extend linear and nonlinear illposedness to fractionally dissipative systems, as long as the order of dissipation is lower than that of $\Gamma$. Our illposedness results are completely sharp in view of various existing wellposedness statements as well as those from our companion paper. Key to our proofs is a novel construction of degenerating wave packets for the class of linear equations $\partial_{t} \phi + i p(t, X, D) \phi = 0$, where $p(t,X,D)$ is a pseudo-differential operator which is self-adjoint in $L^{2}$, degenerate, and dispersive. Degenerating wave packets are approximate solutions to the above linear equation with spatial and frequency support localized at $(X(t), \Xi(t))$, which are solutions to the bicharacteristic ODE system associated with $p(t, x, \xi)$. These wave packets explicitly show degeneration as $X(t)$ approaches a point where p vanishes, which in particular allows us to prove illposedness in topologies finer than $L^{2}$. While the equation for the wave packet can be formally obtained from a Taylor expansion of the symbol near $\xi = \Xi(t)$, the difficult part is to rigorously control the error in sufficiently long timescales, which is obtained by sharp estimates for not only degenerating wave packets but also for oscillatory integrals which naturally appear in the error estimate.

Abstract: We consider the Cauchy problem for the logarithmically singular surface quasi-geostrophic (SQG) equation, introduced by Ohkitani, $\partial_{t} \theta - \nabla^{\perp} \log (10 + (-\Delta)^{\frac{1}{2}}) \theta \cdot \nabla \theta = 0$, and establish local existence and uniqueness of smooth solutions in the scale of Sobolev spaces with exponent decreasing with time. Such a decrease of the Sobolev exponent is necessary, as we have shown in the companion paper that the problem is strongly ill-posed in any fixed Sobolev spaces. The time dependence of the Sobolev exponent can be removed when there is a dissipation term strictly stronger than log. These results improve wellposedness statements by Chae, Constantin, Córdoba, Gancedo, and Wu.

This well-posedness result can be applied to describe the long-time dynamics of the $\delta$-SQG equations, defined by $\partial_{t} \theta + \nabla^{\perp} (10 + (-\Delta)^{\frac{1}{2}})^{-\delta} \theta \cdot \nabla \theta = 0$, for all sufficiently small $\delta > 0$ depending on the size of the initial data. For the same range of $\delta$, we establish global well-posedness of smooth solutions to the logarithmically dissipative counterpart: $\partial_{t} \delta + \nabla^{\perp} (10 + (-\Delta)^{\frac{1}{2}})^{-\delta} \theta \cdot \nabla \theta + \log(10+(-\Delta)^{\frac{1}{2}}) \theta = 0$.

Abstract: We study the problem of stability of the catenoid, which is an asymptotically flat rotationally symmetric minimal surface in Euclidean space, viewed as a stationary solution to the hyperbolic vanishing mean curvature equation in Minkowski space. The latter is a quasilinear wave equation that constitutes the hyperbolic counterpart of the minimal surface equation in Euclidean space. Our main result is the nonlinear asymptotic stability, modulo suitable translation and boost (i.e., modulation), of the $n$-dimensional catenoid with respect to a codimension one set of initial data perturbations without any symmetry assumptions, for $n \geq 5$. The modulation and the codimension one restriction on the data are necessary and optimal in view of the kernel and the unique simple eigenvalue, respectively, of the stability operator of the catenoid.

In a broader context, this paper fits in the long tradition of studies of soliton stability problems. From this viewpoint, our aim here is to tackle some new issues that arise due to the quasilinear nature of the underlying hyperbolic equation. Ideas introduced in this paper include a new profile construction and modulation analysis to track the evolution of the translation and boost parameters of the stationary solution, a new scheme for proving integrated local energy decay for the perturbation in the quasilinear and modulation-theoretic context, and an adaptation of the vectorfield method in the presence of dynamic translations and boosts of the stationary solution.

Abstract: We consider the self-dual Chern-Simons-Schrödinger equation (CSS) under equivariant symmetry, which is an $L^2$-critical equation. It is known that (CSS) admits solitons and finite-time blow-up solutions. In this paper, we show soliton resolution for any solutions with equivariant data in the weighted Sobolev space $H^{1,1}$: every maximal solution decomposes into at most one modulated soliton and a radiation. A striking fact is that the nonscattering part must be a single modulated soliton. To our knowledge, this is the first result on soliton resolution in a class of nonlinear Schrödinger equations which are not known to be completely integrable. The key ingredient is the defocusing nature of the equation in the exterior of a soliton profile. This is a consequence of two distinctive features of (CSS): self-duality and non-local nonlinearity.

Abstract: Motivated by the strong cosmic censorship conjecture, we study the linear scalar wave equation in the interior of subextremal strictly charged Reissner-Nordström black holes by analyzing a suitably-defined "scattering map" at 0 frequency. The method can already be demonstrated in the case of spherically symmetric scalar waves on Reissner-Nordström: we show that assuming suitable ($L^2$-averaged) upper and lower bounds on the event horizon, one can prove ($L^2$-averaged) polynomial lower bound for the solution

1. on any radial null hypersurface transversally intersecting the Cauchy horizon, and
2. along the Cauchy horizon towards timelike infinity.

Taken together with known results regarding solutions to the wave equation in the exterior, (1) above in particular provides yet another proof of the linear instability of the Reissner-Nordström Cauchy horizon. As an application of (2) above, we prove a conditional mass inflation result for a nonlinear system, namely, the Einstein-Maxwell-(real)-scalar field system in spherical symmetry. For this model, it is known that for a generic class of Cauchy data $\mathcal{G}$, the maximal globally hyperbolic future developments are $C^2$-future-inextendible. We prove that if a (conjectural) improved decay result holds in the exterior region, then for the maximal globally hyperbolic developments arising from initial data in $\mathcal{G}$, the Hawking mass blows up identically on the Cauchy horizon.

Abstract: We extend the monumental result of Christodoulou-Klainerman on the global nonlinear stability of the Minkowski spacetime to the global nonlinear stability of a class of large dispersive spacetimes. More precisely, we show that any regular future causally geodesically complete, asymptotically flat solution to the Einstein-scalar field system which approaches the Minkowski spacetime sufficiently fast for large times is future globally nonlinearly stable. Combining our main theorem with results of Luk-Oh, Luk-Oh-Yang and Kilgore, we prove that a class of large data spherically symmetric dispersive solutions to the Einstein-scalar field system are globally nonlinearly stable with respect to small non-spherically symmetric perturbations. This in particular gives the first construction of an open set of large asymptotically flat initial data for which the solutions to the Einstein-scalar field system are future causally geodesically complete.

Abstract: We consider a class of dispersive and dissipative perturbations of the inviscid Burgers equation, which includes the fractional KdV equation of order $\alpha$, and the fractal Burgers equation of order $\beta$, where $\alpha, \beta \in [0, 1)$, and the Whitham equation. For all $\alpha, \beta \in [0, 1)$, we construct solutions whose gradient blows up at a point, and whose amplitude stays bounded, which therefore display a "shock-like" singularity. We moreover provide an asymptotic description of the blow-up. To our knowledge, this constitutes the first proof of gradient blow-up for the fKdV equation in the range $\alpha \in [2/3, 1)$, as well as the first description of explicit blow-up dynamics for the fractal Burgers equation in the range $\beta \in [2/3, 1)$.

Our construction is based on modulation theory, where the well-known smooth self-similar solutions to the inviscid Burgers equation are used as profiles. A somewhat amusing point is that the profiles that are less stable under initial data perturbations (in that the number of unstable directions is larger) are more stable under perturbations of the equation (in that higher order dispersive and/or dissipative terms are allowed) due to their slower rates of concentration. Another innovation of this article, which may be of independent interest, is the development of a streamlined weighted $L^2$-based approach (in lieu of the characteristic method) for establishing the sharp spatial behavior of the solution in self-similar variables, which leads to the sharp Hölder regularity of the solution up to the blow-up time.

Abstract: We consider the finite-time blow-up dynamics of solutions to the self-dual Chern-Simons-Schrödinger (CSS) equation (also referred to as the Jackiw-Pi model) near the radial soliton $Q$ with the least $L^2$-norm (ground state). While a formal application of pseudoconformal symmetry to $Q$ gives rise to an $L^2$-continuous curve of initial data sets whose solutions blow up in finite time, they all have infinite energy due to the slow spatial decay of $Q$. In this paper, we exhibit initial data sets that are smooth finite energy radial perturbations of $Q$, whose solutions blow up in finite time. Interestingly, their blow-up rate differs from the pseudoconformal rate by a power of logarithm. Applying pseudoconformal symmetry in reverse, this also yields a first example of an infinite-time blow-up solution, whose blow-up profile contracts at a logarithmic rate.

Our analysis builds upon the ideas of previous works of the first two authors on (CSS) [21,22], as well as the celebrated works on energy-critical geometric equations by Merle, Raphaël, and Rodnianski [33,38]. A notable feature of this paper is a systematic use of nonlinear covariant conjugations by the covariant Cauchy-Riemann operators in all parts of the argument. This not only overcomes the nonlocality of the problem, which is the principal challenge for (CSS), but also simplifies the structure of nonlinearity arising in the proof.

Abstract: We consider the Cauchy problem for the Schrödinger maps evolution when the domain is the hyperbolic plane. An interesting feature of this problem compared to the more widely studied case on the Euclidean plane is the existence of a rich new family of finite energy harmonic maps. These are stationary solutions, and thus play an important role in the dynamics of Schrödinger maps. The main result of this article is the asymptotic stability of (some of) such harmonic maps under the Schrödinger maps evolution. More precisely, we prove the nonlinear asymptotic stability of a finite energy equivariant harmonic map $Q$ under the Schrödinger maps evolution with respect to non-equivariant perturbations, provided $Q$ obeys a suitable linearized stability condition. This condition is known to hold for all equivariant harmonic maps with values in the hyperbolic plane and for a subset of those maps taking values in the sphere. One of the main technical ingredients in the paper is a global-in-time local smoothing and Strichartz estimate for the operator obtained by linearization around a harmonic map, proved in the companion paper.

Abstract: In this article, we prove various illposedness results for the Cauchy problem for the incompressible Hall- and electron-magnetohydrodynamic (MHD) equations without resistivity. These PDEs are fluid descriptions of plasmas, where the effect of collisions is neglected (no resistivity), while the motion of the electrons relative to the ions (Hall current term) is taken into account. The Hall current term endows the magnetic field equation with a quasilinear dispersive character, which is key to our mechanism for illposedness.

Perhaps the most striking conclusion of this article is that the Cauchy problems for the Hall-MHD (either viscous or inviscid) and the electron-MHD equations, under one translational symmetry, are ill-posed near the trivial solution in any sufficiently high regularity Sobolev space $H^{s}$. This result holds despite obvious wellposedness of the linearized equations near the trivial solution, as well as conservation of the nonlinear energy, by which the $L^{2}$ norm (energy) of the solution stays constant in time. The core illposedness (or instability) mechanism is degeneration of certain high frequency wave packet solutions to the linearization around a class of linearly degenerate stationary solutions of these equations, which are essentially dispersive equations with degenerate principal symbols.

The results in this article are complemented by a companion work, where we provide geometric conditions on the initial magnetic field that ensure wellposedness(!) of the Cauchy problems for the incompressible Hall and electron-MHD equations. In particular, in stark contrast to the results here, it is shown in the companion work that the nonlinear Cauchy problems are well-posed near any nonzero constant magnetic field.

Abstract: We establish global-in-time frequency localized local smoothing estimates for Schrödinger equations on hyperbolic space $\mathbb{H}^{d}$. In the presence of symmetric first and zeroth order potentials, which are possibly time-dependent, possibly large, and have sufficiently fast polynomial decay, these estimates are proved up to a localized lower order error. Then in the time-independent case, we show that a spectral condition (namely, absence of threshold resonances) implies the full local smoothing estimates (without any error), after projecting to the continuous spectrum. In the process, as a means to localize in frequency, we develop a general Littlewood-Paley machinery on $\mathbb{H}^{d}$ based on the heat flow. Our results and techniques are motivated by applications to the problem of stability of solitary waves to nonlinear Schrödinger-type equations on $\mathbb{H}^{d}$.

As a testament of the robustness of approach, which is based on the positive commutator method and a heat flow based Littlewood-Paley theory, we also show that the main results are stable under small time-dependent perturbations, including polynomially decaying second order ones, and small lower order nonsymmetric perturbations.

Abstract: This is the first part of the four-paper sequence, which establishes the Threshold Conjecture and the Soliton Bubbling vs.~Scattering Dichotomy for the energy critical hyperbolic Yang-Mills equation in the (4 + 1)-dimensional Minkowski space-time.
The primary subject of this paper, however, is another PDE, namely the energy critical Yang-Mills heat flow on the 4-dimensional Euclidean space. Our first goal is to establish sharp criteria for global existence and asymptotic convergence to a flat connection for this system in $\dot{H}^{1}$, including the Dichotomy Theorem (i.e., either the above properties hold or a harmonic Yang-Mills connection bubbles off) and the Threshold Theorem (i.e., if the initial energy is less than twice that of the ground state, then the above properties hold). Our second goal is to use the Yang-Mills heat flow in order to define the caloric gauge, which will play a major role in the analysis of the hyperbolic Yang-Mills equation in the subsequent papers.

Abstract: This is the second part in a four-paper sequence, which establishes the Threshold Conjecture and the Soliton Bubbling vs. Scattering Dichotomy for the hyperbolic Yang-Mills equation in the (4+1)-dimensional space-time. This paper provides the key gauge-dependent analysis of the hyperbolic Yang-Mills equation.

We consider topologically trivial solutions in the caloric gauge, which was defined in the first paper using the Yang-Mills heat flow. In this gauge, we establish a strong form of local well-posedness, where the time of existence is bounded from below by the energy concentration scale. Moreover, we show that regularity and dispersive behavior of the solution persists as long as energy dispersion is small. We also observe that fixed-time regularity (but not dispersive) properties in the caloric gauge may be transferred to the temporal gauge without any loss, proving as a consequence small data global well-posedness in the temporal gauge.

The results in this paper are used in the subsequent papers to prove the sharp Threshold Theorem in caloric gauge in the trivial topological class, and the dichotomy theorem in arbitrary topological classes.

Abstract: This is the third part of a four-paper sequence, which establishes the Threshold Conjecture and the Soliton-Bubbling vs. Scattering Dichotomy for the energy critical hyperbolic Yang-Mills equation in the (4+1)-dimensional Minkowski space-time. This paper provides basic tools for considering the dynamics of the hyperbolic Yang-Mills equation in an arbitrary topological class at an optimal regularity. We generalize the standard notion of a topological class of connections on $\mathbb{R}^{d}$, defined via a pullback to the one-point compactification $\mathbb{S}^{d} = \mathbb{R}^{d} \cup \{ \infty \}$, to rough connections with curvature in the critical space $L^{d/2}(\mathbb{R}^{d})$. Moreover, we provide excision and extension techniques for the Yang-Mills constraint (or Gauss) equation, which allow us to efficiently localize Yang-Mills initial data sets. Combined with the results in the previous papers in the sequence, we obtain local well-posedness of the hyperbolic Yang-Mills equation on $\mathbb{R}^{1+d}$ (d≥4) in an arbitrary topological class at optimal regularity in the temporal gauge (where finite speed of propagation holds). In addition, in the energy subcritical case d=3, our techniques provide an alternative proof of the classical finite energy global well-posedness theorem of Klainerman-Machedon, while also removing the smallness assumption in the temporal-gauge local well-posedness theorem of Tao.

Although this paper is a part of a larger sequence, the materials presented in this paper may be of independent and general interest. For this reason, we have organized the paper so that it may be read separately from the sequence.

Abstract: This article represents the fourth and final part of a four-paper sequence whose aim is to prove the Threshold Conjecture as well as the more general Dichotomy Theorem for the energy critical 4+1 dimensional hyperbolic Yang-Mills equation. The Threshold Theorem asserts that topologically trivial solutions with energy below twice the ground state energy are global and scatter. The Dichotomy Theorem applies to solutions in arbitrary topological class with large energy, and provides two exclusive alternatives: Either the solution is global and scatters, or it bubbles off a soliton in either finite time or infinite time.

Using the caloric gauge developed in the first paper, the continuation/scattering criteria established in the second paper, and the large data analysis in an arbitrary topological class at optimal regularity in the third paper, here we perform a blow-up analysis which shows that the failure of global well-posedness and scattering implies either the existence of a soliton with at most the same energy bubbling off, or the existence existence of a nontrivial self-similar solution. The proof is completed by showing that the latter solutions do not exist.

Abstract: This is the first and main paper of a two-part series, in which we prove the $C^{2}$-formulation of the strong cosmic censorship conjecture for the Einstein-Maxwell-(real)-scalar-field system in spherical symmetry for two-ended asymptotically flat data. For this model, it is known through the works of Dafermos and Dafermos-Rodnianski that the maximal globally hyperbolic future development of any admissible two-ended asymptotically flat Cauchy initial data set possesses a non-empty Cauchy horizon, across which the spacetime is $C^{0}$-future-extendible (in particular, the $C^{0}$-formulation of the strong cosmic censorship conjecture is false). Nevertheless, the main conclusion of the present series of papers is that for a generic (in the sense of being open and dense relative to appropriate topologies) class of such data, the spacetime is future-inextendible with a Lorentzian metric of higher regularity (specifically, $C^{2}$).

In this paper, we prove that the solution is $C^{2}$-future-inextendible under the condition that the scalar field obeys an $L^{2}$-averaged polynomial lower bound along each of the event horizons. This, in particular, improves upon a previous result of Dafermos, which required instead a pointwise lower bound. Key to the proof are appropriate stability and instability results in the interior of the black hole region, whose proofs are in turn based on ideas from the work of Dafermos-Luk on the stability of Kerr Cauchy horizon (without symmetry) and from our previous paper on linear instability of Reissner-Nordström Cauchy horizon. In the second paper of the series, which concerns analysis in the exterior of the black hole region, we show that the $L^{2}$-averaged polynomial lower bound needed for the instability result indeed holds for a generic class of admissible two-ended asymptotically flat Cauchy initial data.

Abstract: This is the second and last paper of a two-part series in which we prove the $C^{2}$-formulation of the strong cosmic censorship conjecture for the Einstein-Maxwell-(real)-scalar-field system in spherical symmetry for two-ended asymptotically flat data. In the first paper, we showed that the maximal globally hyperbolic future development of an admissible asymptotically flat Cauchy initial data set is $C^{2}$-future-inextendible provided that an $L^{2}$-averaged (inverse) polynomial lower bound for the derivative of the scalar field holds along each horizon. In this paper, we show that this lower bound is indeed satisfied for solutions arising from a generic set of Cauchy initial data. Roughly speaking, the generic set is open with respect to a (weighted) $C^{1}$ topology and is dense with respect to a (weighted) $C^{\infty}$ topology. The proof of the theorem is based on extensions of the ideas in our previous work on the linear instability of Reissner-Nordström Cauchy horizon, as well as a new large data asymptotic stability result which gives good decay estimates for the difference of the radiation fields for small perturbations of an arbitrary solution.

Abstract: We review our recent work on a construction of spherically symmetric global solutions to the Einstein-scalar field system with large bounded variation norms and large Bondi masses. We show that similar ideas, together with Christodoulou's short pulse method, allow us to prove the following result: Given $M_{i} \geq M_{f} > 0$ and $\epsilon > 0$, there exists a spherically symmetric (black hole) solution to the Einstein-scalar field system such that up to an error of size $\epsilon$, the initial Bondi mass is $M_{i}$ and the final Bondi mass is $M_{f}$. Moreover, if one assumes a continuity property of the final Bondi mass (which in principle follows from known techniques in the literature), then for $M_{i}> $M_{f} >0$, the above result holds without an $\epsilon$-error.

Abstract: It is well-known that small, regular, spherically symmetric characteristic initial data to the Einstein-scalar-field system which are decaying towards (future null) infinity give rise to solutions which are foward-in-time global (in the sense of future causal geodesic completeness). We construct a class of spherically symmetric solutions which are global but the initial norms are consistent with initial data not decaying towards infinity. This gives the following consequences: (1) We prove that there exist foward-in-time global solutions with arbitrarily large (and in fact infinite) initial bounded variation (BV) norms and initial Bondi masses. (2) While general solutions with non-decaying data do not approach Minkowski spacetime, we show using the results of Luk-Oh that if a sufficiently strong asymptotic flatness condition is imposed on the initial data, then the solutions we construct (with large BV norms) approach Minkowski spacetime with a sharp inverse polynomial rate. (3) Our construction can be easily extended so that data are posed at past null infinity and we obtain solutions with large BV norms which are causally geodesically complete both to the past and to the future. Finally, we discuss applications of our method to construct global solutions for other nonlinear wave equations with infinite critical norms.

Abstract: In this paper, we prove global well-posedness of the massless Maxwell-Dirac equation in Coulomb gauge on $\mathbb{R}^{1+4}$ ($d \geq 4$) for data with small scale-critical Sobolev norm, as well as modified scattering of the solutions. Main components of our proof are A) uncovering null structure of Maxwell-Dirac in the Coulomb gauge, and B) proving solvability of the underlying covariant Dirac equation. A key step for achieving both is to exploit (and justify) a deep analogy between Maxwell-Dirac and Maxwell-Klein-Gordon (for which an analogous result was proved earlier by Krieger-Sterbenz-Tataru), which says that the most difficult part of Maxwell-Dirac takes essentially the same form as Maxwell-Klein-Gordon.

Abstract: We establish a general small data global existence and decay theorem for Chern-Simons theories with a general gauge group, coupled with a massive relativistic field of spin 0 or 1/2. Our result applies to a wide range of relativistic Chern-Simons theories considered in the literature, including the abelian/non-abelian self-dual Chern-Simons-Higgs equation and the Chern-Simons-Dirac equation. A key idea is to develop and employ a gauge invariant vector field method for relativistic Chern-Simons theories, which allows us to avoid the long range effect of charge.

Abstract: We establish global well-posedness and scattering for wave maps from d-dimensional hyperbolic space into Riemannian manifolds of bounded geometry for initial data that is small in the critical Sobolev space for d≥4. The main theorem is proved using the moving frame approach introduced by Shatah and Struwe. However, rather than imposing the Coulomb gauge we formulate the wave maps problem in Tao's caloric gauge, which is constructed using the harmonic map heat flow. In this setting the caloric gauge has the remarkable property that the main `gauged' dynamic equations reduce to a system of nonlinear scalar wave equations on ℍd that are amenable to Strichartz estimates rather than tensorial wave equations (which arise in other gauges such as the Coulomb gauge) for which useful dispersive estimates are not known. This last point makes the heat flow approach crucial in the context of wave maps on curved domains.

Abstract: In [Ise13], the first author proposed a strengthening of Onsager’s conjecture on the failure of energy conservation for incompressible Euler flows with Hölder regularity not exceeding 1/3. This stronger form of the conjecture implies that anomalous dissipation will fail for a generic Euler flow with regularity below the Onsager critical space $L^{\infty}_{t} B^{1/3}_{3, \infty}$ due to low regularity of the energy profile.

This paper is the first and the main paper in a series of two papers, whose results may be viewed as first steps towards establishing the conjectured failure of energy regularity for generic solutions with Hölder exponent less than 1/5. The main result of the present paper shows that any given smooth Euler flow can be perturbed in $C^{1/5 - \epsilon}_{t,x}$ on any pre-compact subset of $\mathbb{R} \times \mathbb{R}^{1+3}$ to violate energy conservation. Furthermore, the perturbed solution is no smoother than $C^{1/5 - \epsilon}_{t,x}$. As a corollary of this theorem, we show the existence of nonzero $C^{1/5 - \epsilon}_{t,x}$ solutions to Euler with compact space-time support, generalizing previous work of the first author [Ise12] to the nonperiodic setting.

Abstract: In [Ise13], the first author proposed a strengthening of Onsager’s conjecture on the failure of energy conservation for incompressible Euler flows with Hölder regularity not exceeding 1/3. This stronger form of the conjecture implies that anomalous dissipation will fail for a generic Euler flow with regularity below the Onsager critical space $L^{\infty}_{t} B^{1/3}_{3, \infty}$ due to low regularity of the energy profile.

The present paper is the second in a series of two papers whose results may be viewed as first steps towards establishing the conjectured failure of energy regularity for generic solutions with Hölder exponent less than 1/5. The main result of this paper shows that any non-negative function with compact support and Hölder regularity 1/2 can be prescribed as the energy profile of an Euler flow in the class $C^{1/5 - \epsilon}_{t,x}$. The exponent 1/2 is sharp in view of a regularity result of Isett [Ise13]. The proof employs an improved greedy algorithm scheme that builds upon that in Buckmaster–De Lellis–Székelyhidi [BDLS13].

Abstract: In this paper we continue the analysis of equivariant wave maps from 2-dimensional hyperbolic space into surfaces of revolution that was initiated in earlier papers. When the target is the hyperbolic plane we proved in an earlier work the existence and asymptotic stability of a 1-parameter family of finite energy harmonic maps indexed by how far each map wraps around the target. Here we conjecture that each of these harmonic maps is globally asymptotically stable, meaning that the evolution of any arbitrarily large finite energy perturbation of a harmonic map asymptotically resolves into the harmonic map itself plus free radiation. Since such initial data exhaust the energy space, this is the soliton resolution conjecture for this equation. The main result is a verification of this conjecture for a nonperturbative subset of the harmonic maps

Abstract: This paper is the first part of a trilogy dedicated to a proof of global well-posedness and scattering of the (4+1)-dimensional mass-less Maxwell-Klein-Gordon equation (MKG) for any finite energy initial data. The main result of the present paper is a large energy local well-posedness theorem for MKG in the global Coulomb gauge, where the lifespan is bounded from below by the energy concentration scale of the data. Hence the proof of global well-posedness is reduced to establishing non-concentration of energy. To deal with non-local features of MKG we develop initial data excision and gluing techniques at critical regularity, which might be of independent interest.

Abstract: This article is devoted to the mass-less energy critical Maxwell-Klein-Gordon system in 4+1 dimensions. In earlier work of the second author, joint with Krieger and Sterbenz, we have proved that this problem has global well-posedness and scattering in the Coulomb gauge for small initial data. This article is the second of a sequence of three papers of the authors, whose goal is to show that the same result holds for data with arbitrarily large energy. Our aim here is to show that large data solutions persist for as long as one has small energy dispersion; hence failure of global well-posedness must be accompanied with a non-trivial energy dispersion.

Abstract: This article constitutes the final and main part of a three-paper sequence, whose goal is to prove global well-posedness and scattering of the energy critical Maxwell-Klein-Gordon equation (MKG) on $\mathbb{R}^{1+4}$ for arbitrary finite energy initial data. Using the successively stronger continuation/scattering criteria established in the previous two papers, we carry out a blow-up analysis and deduce that the failure of global well-posedness and scattering implies the existence of a nontrivial stationary or self-similar solution to MKG. Then, by establishing that such solutions do not exist, we complete the proof.

Abstract: The recently established threshold theorem for energy critical wave maps states that wave maps with energy less than that of the ground state (i.e., a minimal energy nontrivial harmonic map) are globally regular and scatter on (1+2)-dimensional Minkowski space. In this note we give a refinement of this theorem when the target is a closed orientable surface by taking into account an additional invariant of the problem, namely the topological degree. We show that the sharp energy threshold for global regularity and scattering is in fact twice the energy of the ground state for wave maps with degree zero, whereas wave maps with nonzero degree necessarily have at least the energy of the ground state. We also give a discussion on the formulation of a refined threshold conjecture for the energy critical SU(2) Yang-Mills equation on (1+4)-dimensional Minkowski space.

Abstract: In this paper we study k-equivariant wave maps from the hyperbolic plane into the 2-sphere as well as the energy critical equivariant SU(2) Yang-Mills problem on 4-dimensional hyperbolic space. The latter problem bears many similarities to a 2-equivariant wave map into a surface of revolution. As in the case of 1-equivariant wave maps considered in an earlier paper, both problems admit a family of stationary solutions indexed by a parameter that determines how far the image of the map wraps around the target manifold. Here we show that if the image of a stationary solution is contained in a geodesically convex subset of the target, then it is asymptotically stable in the energy space. However, for a stationary solution that covers a large enough portion of the target, we prove that the Schrödinger operator obtained by linearizing about such a harmonic map admits a simple positive eigenvalue in the spectral gap. As there is no a priori nonlinear obstruction to asymptotic stability, this gives evidence for the existence of metastable states (i.e., solutions with anomalously slow decay rates) in these simple geometric models.

Abstract: It has long been suggested that solutions to linear scalar wave equation $$ \Box_{g} \phi = 0 $$ on a fixed subextremal Reissner-Nordström spacetime with non-vanishing charge are generically singular at the Cauchy horizon. We prove that generic smooth and compactly supported initial data on a Cauchy hypersurface indeed give rise to solutions with infinite nondegenerate energy near the Cauchy horizon in the interior of the black hole. In particular, the solution generically does not belong to $W^{1,2}_{loc}$. This instability is related to the celebrated blue shift effect in the interior of the black hole. The problem is motivated by the strong cosmic censorship conjecture and it is expected that for the full nonlinear Einstein-Maxwell system, this instability leads to a singular Cauchy horizon for generic small perturbations of Reissner-Nordström spacetime. Moreover, in addition to the instability result, we also show as a consequence of the proof that Price's law decay is generically sharp along the event horizon.

Abstract: The goal for this paper is twofold. Our first main objective is to develop Bahouri-Gerard type profile decompositions for waves on hyperbolic space. Recently, such profile decompositions have proved to be a versatile tool in the study of the asymptotic dynamics of solutions to nonlinear wave equations with large energy. With an eye towards further applications, we develop this theory in a fairly general framework, which includes the case of waves on hyperbolic space perturbed by a time-independent potential. Our second objective is to use the profile decomposition to address a specific nonlinear problem, namely the question of global well-posedness and scattering for the defocusing, energy critical, semi-linear wave equation on three-dimensional hyperbolic space, possibly perturbed by a repulsive time-independent potential. Using the concentration compactness/rigidity method introduced by Kenig and Merle, we prove that all finite energy initial data lead to a global evolution that scatters to linear waves in infinite time. This proof will serve as a blueprint for the arguments in a forthcoming work, where we study the asymptotic behavior of large energy equivariant wave maps on the hyperbolic plane.

Abstract: In this paper we initiate the study of equivariant wave maps from 2d hyperbolic space into rotationally symmetric surfaces. This problem exhibits markedly different phenomena than its Euclidean counterpart due to the exponential volume growth of concentric geodesic spheres on the domain.

In particular, when the target is the 2-sphere, we find a family of equivariant harmonic maps indexed by a parameter that measures how far the image of each harmonic map wraps around the sphere. These maps have energies taking all values between zero and the energy of the unique co-rotational Euclidean harmonic map, Q, from the Euclidean plane to the 2-sphere, given by stereographic projection. We prove that the harmonic maps are asymptotically stable for values of the parameter smaller than a threshold that is large enough to allow for maps that wrap more than halfway around the sphere. Indeed, we prove Strichartz estimates for the operator obtained by linearizing around such a harmonic map. However, for harmonic maps with energies approaching the Euclidean energy of Q, asymptotic stability via a perturbative argument based on Strichartz estimates is precluded by the existence of gap eigenvalues in the spectrum of the linearized operator.

When the target is 2d hyperbolic space, we find a continuous family of asymptotically stable equivariant harmonic maps with arbitrarily small and arbitrarily large energies. This stands in sharp contrast to the corresponding problem on Euclidean space, where all finite energy solutions scatter to zero as time tends to infinity.

Abstract: In this paper, we study the future causally geodesically complete solutions of the spherically symmetric Einstein-scalar field system. Under the a priori assumption that the scalar field $\phi$ scatters locally in the scale-invariant bounded-variation (BV) norm, we prove that $\phi$ and its derivatives decay polynomially. Moreover, we show that the decay rates are sharp. In particular, we obtain sharp quantitative decay for the class of global solutions with small BV norms constructed by Christodoulou. As a consequence of our results, for every future causally geodesically complete solution with sufficiently regular initial data, we show the dichotomy that either the sharp power law tail holds or that the spacetime blows up at infinity in the sense that some scale invariant spacetime norms blow up.

Abstract: We consider the Chern-Simons-Schrödinger model in 1+2 dimensions, and prove scattering for small solutions of the Cauchy problem in the Coulomb gauge. This model is a gauge covariant Schrödinger equation, with a potential decaying like $r^{−1}$ at infinity. To overcome the difficulties due to this long range decay we start by performing $L^{2}$-based estimates covariantly. This gives favorable commutation identities so that only curvature terms, which decay faster than $r^{−1}$, appear in our weighted energy estimates. We then select the Coulomb gauge to reveal a genuinely cubic null structure, which allows us to show sharp decay by Fourier methods.

Abstract: We give a simple proof of Onsager's conjecture concerning energy conservation for weak solutions to the Euler equations on any compact Riemannian manifold, extending the results of Constantin-E-Titi and Cheskidov-Constantin-Friedlander-Shvydkoy in the flat case. When restricted to $\mathbb{T}^{d}$ or $\mathbb{R}^{d}$, our approach yields an alternative proof of the sharp result of the latter authors. Our method builds on a systematic use of a smoothing operator defined via a geometric heat flow, which was considered by Milgram-Rosenbloom as a means to establish the Hodge theorem. In particular, we present a simple and geometric way to prove the key nonlinear commutator estimate, whose proof previously relied on a delicate use of convolutions.

Abstract: In a recent paper, Selberg-Tesfahun proved that the abelian Chern-Simons-Higgs system (CSH) is globally well-posed for finite energy initial data under the Lorenz gauge condition. It has been suspected by Huh, however, that such a result should hold in the Coulomb gauge as well. In this note, we give an affirmative answer to this question by first establishing low regularity local well-posededness of (CSH) in the Coulomb gauge for initial data set $(f, g) \in H^{\gamma}_{x} \times H^{\gamma-1}_{x}$ for any $\gamma > 3/4$. Then by conservation of energy, global well-posedness for (CSH) in the energy space $(f,g) \in H^{1}_{x} \times L^{2}_{x}$ follows rather immediately.

Abstract: n this work, we introduce a novel approach to the problem of gauge choice for the Yang-Mills equation on the Minkowski space $\mathbb{R}^{1+3}$, which uses the Yang-Mills heat flow in a crucial way. As this approach does not possess the drawbacks of the previous approaches, it is expected to be more robust and easily adaptable to other settings.

As the first demonstration of the `structure' offered by this new approach, we will give an alternative proof of the local well-posedness of the Yang-Mills equations for initial data in $(\dot{H}^{1} \cap L^{3}_{x}) \times L^{2}_{x}$, which is a classical result of S. Klainerman and M. Machedon that had been proved using a different method (local Coulomb gauges). The new proof does not involve localization in space-time, which had been the key drawback of the previous method. Based on the results proved in this paper, a new proof of finite energy global well-posedness of the Yang-Mills equations, also using the Yang-Mills heat flow, is established in a companion article.

Abstract: In this work, along with the companion work Oh (2012), we propose a novel approach to the problem of gauge choice for the Yang-Mills equations on the Minkowski space $\mathbb{R}^{1+3}$. A crucial ingredient is the associated Yang-Mills heat flow. As this approach does not possess the drawbacks of the previous approaches (as in Klainerman-Machedon (1995) and Tao (2003)), it is expected to be more robust and easily adaptable to other settings.

Building on the results proved in the companion article Oh (2012), we prove, as one of the first applications of our approach, finite energy global well-posedness of the Yang-Mills equations on $\mathbb{R}^{1+3}$. This is a classical result first proved by S. Klainerman and M. Machedon (1995) using local Coulomb gauges. As opposed to their method, the present approach avoids the use of Uhlenbeck's lemma (1982), and hence does not involve localization in space-time.

Abstract: In this paper, we address the problem of local well-posedness of the Chern-Simons-Dirac (CSD) and the Chern-Simons-Higgs (CSH) equations in the Lorenz gauge for low regularity initial data. One of our main contributions is the uncovering of a null structure of (CSD). Combined with the standard machinery of $X^{s,b}$ spaces, we obtain local well-posedness of (CSD) for initial data $a_{\mu},\psi \in H^{1/4 + \epsilon}_{x}$. Moreover, it is observed that the same techniques applied to (CSH) lead to a quick proof of local well-posedness for initial data $a_{\mu} \in H^{1/4 + \epsilon}_{x}$, $(\phi, \partial_{t} \phi) \in H^{3/4 + \epsilon}_{x} \times H^{-1/4+\epsilon}_{x}$, which improves the previous result of Selberg-Tesfahun (2012).

Abstract: This article surveys recent progress on the long-term behavior of solutions to linear and nonlinear hyperbolic equations, with a focus on robust physical space techniques. We review classical and modern methods for establishing sharp uniform decay rates and precise global asymptotics for the wave equation. We then describe a general method for determining the leading-order late-time tails for waves in odd spatial dimensions, which is applicable to nonlinear problems on dynamical backgrounds. Finally, we discuss a new physical space approach for establishing the global asymptotics of the variable-coefficient Klein–Gordon equation. Throughout, we emphasize methods that are applicable to problems with variable coefficients and nonlinearities, such as those arising in the study of waves on black hole spacetimes or interacting with solitons.

Abstract: This article is devoted to the energy critical hyperbolic Yang-Mills system in the (4+1) dimensional Minkowski space, which is considered by the authors in a sequence of four papers. The final outcome of these papers is twofold: (i) the Threshold Theorem, which asserts that global well-posedness and scattering hold for all topologically trivial initial data with energy below twice the ground state energy, and (ii) the Dichotomy Theorem, which for larger data in arbitrary topological classes provides a choice of two outcomes, either a global, scattering solution or a soliton bubbling off. In the last case, the bubbling off phenomena can happen either (a) in finite time, triggering a finite time blow-up, or (b) in infinite time. Our goal here is to describe these results, and to provide an overview of the flow of ideas within their proofs.

Abstract: In this thesis, we propose a novel choice of gauge for the Yang-Mills equations on the Minkowski space $\mathbb{R}^{1+d}$. A crucial ingredient is the associated Yang-Mills heat flow. Unlike the previous approaches, the new gauge is applicable for large data, while the special analytic structure of the Yang-Mills equations is still manifest. As the first application of the new approach, we shall give new proofs of H1 local well-posedness and finite energy global well-posedness of the Yang-Mills equations on $\mathbb{R}^{1+3}$. These are classical results first proved by S. Klainerman and M. Machedon using the method of local Coulomb gauges, which had been difficult to extend to other settings. As our approach does not possess its drawbacks (in particular the use of Uhlenbeck’s lemma is avoided), it is expected to be more robust and easily applicable to other problems.