arXiv subjects

William Golding

Publications and source records attributed to William Golding.

13 recordsLinked to original sources

arXiv2026

Global smooth solutions to the inhomogeneous Landau-Fermi-Dirac equation

We consider the spatially inhomogeneous Landau-Fermi-Dirac equation with Coulomb potential, a quantum modification of the classical Landau equation for fermions. Mathematically, the Pauli exclusion principle manifests as an additional a priori $L^\infty$-bound for solutions. Using this bound, we propagate polynomial decay in velocity, yielding unconditional upper bounds on the local mass and energy densities, thereby ruling out the possibility of implosions in the hydrodynamic quantities. Combining this estimate with a modified mass spreading method that yields desaturation, we deduce the existence of global-in-time classical solutions for rough initial data with polynomial decay in velocity. This result stands in stark contrast to the theory for the classical Landau and Boltzmann equations, for which no comparable nonperturbative global existence result is known despite sustained effort. Our treatment is almost entirely self-contained, using only robust, generic estimates for linear kinetic equations. In particular, our proof of local existence, in contrast to prior works, more closely mirrors the theory for parabolic equations using weak solutions and simpler function spaces. It may provide a concise roadmap to organizing, adapting, and applying the various linear and nonlinear estimates to obtain well-posedness for kinetic equations.

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arXiv2026

A sharp rigidity/flexibility threshold for the isotropic Landau equation

We establish a sharp rigidity/flexibility threshold for stationary solutions of the Krieger--Strain equation, an isotropic model of the Landau--Coulomb equation. For every $1< p < \frac65$, we use Nash iteration to construct nontrivial, nonnegative solutions in $L^1(\mathbb{R}^3) \cap L^p(\mathbb{R}^3)$ with arbitrarily strong exponential localization. Conversely, every stationary weak solution in $L^{\frac65}(\mathbb{R}^3)$ is trivial, identifying $L^{\frac65}(\mathbb{R}^3)$ as a new sharp integrability threshold. To our knowledge, this is the first use of Nash iteration for a nonlinear equation from collisional kinetic theory. The construction is based on a high--high--low cancellation within the Krieger--Strain operator and suggests that such mechanisms may occur more broadly in kinetic theory. The construction must accommodate kinetic features unusual for the method including a strongly nonlocal collision operator; an equation fundamentally posed on the whole space---not the periodic box; and a positive scalar unknown. At this low level of regularity, the usual formulations of the collision operator are not a priori well-defined, so a central part of the problem is specifying in what sense the constructed objects solve the equation. We isolate the notion of mollifier confluence, a simple and canonical way to interpret a nonlinearity below naive thresholds related to multiplying distributions. We complement this definition with a systematic treatment of weak solution notions and several explicit formal computations and clarifying examples that may be of independent interest.

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arXiv2026

Pointwise bounds and obstructions to blowup for the Landau and Boltzmann equations

We establish a new a priori estimate on solutions to the space-inhomogeneous Landau and Boltzmann equations. As a consequence, we prove a new continuation criterion, based on a weighted $L^\infty$-norm, without requiring bounds on the hydrodynamic quantities. This complements existing conditional regularity results from a rather different perspective. Consequently, we show that the singularities present in the fluid equations are largely incompatible with the Boltzmann and Landau equations. More precisely, we largely rule out ``lifting a singularity'' from the 3D Euler equations to the physical range of kinetic equations, a widely expected mechanism for singularity formation. Under general considerations, this mechanism is essentially excluded for soft potentials, whereas for hard potentials the situation is more nuanced: one cannot produce blowup through the standard hydrodynamic ansatz using known imploding solutions to the Euler equations.

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arXiv2025

On Hydrodynamic Implosions and the Landau-Coulomb Equation

We study the inhomogeneous Landau equation with Coulomb potential and derive a new continuation criterion: a smooth solution can be uniquely continued for as long as it remains bounded. This provides, to our knowledge, the first continuation criterion based on a quantity not controlling the mass density. Consequently, we are able to rule out a potential singularity formation scenario known as tail fattening, in which an implosion occurs due to the loss of decay at large $v$. More generally, we are able to rule out all Type II approximately self-similar blow-up rates that are slower than the Type I blow-up rate, without any assumption of decay on the inner profile, complementing existing Type I blow-up analysis in the literature. Heuristically, this suggests that it should be impossible to directly use the hydrodynamic limit connection with the 3D compressible Euler equations to construct a singular solution to the Landau equation with Coulomb potential. Such a potential implosion scenario -- based on either an isentropic or nonisentropic implosion for the 3D Euler equations -- would naturally result in a slow Type II approximately self-similar blow-up scenario, falling well within the range our theorem. This preprint has been subsumed by a more recent work by the authors and Luis Silvestre titled ``Pointwise bounds and obstructions to blowup for the Landau and Boltzmann equations,'' arXiv:2605.20426. This manuscript will remain a permanent preprint; all references should be directed to the more recent work.

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arXiv2024

Production of the Fisher information for the Landau-Coulomb equation with L1 initial data

We consider the Landau-Coulomb equation for initial data with bounded mass, finite numbers of moments, and entropy. We show the existence of a global weak solution that has bounded Fisher information for positive times. This solution is therefore a global strong solution away from the initial time. We propose an alternative approach, based on already existing estimates, to the study of the appearance of Fisher information recently performed by Ji in [12].

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arXiv2024

Global smooth solutions to the Landau-Coulomb equation in $L^{3/2}$

We consider the homogeneous Landau equation in $\mathbb{R}^3$ with Coulomb potential and initial data in polynomially weighted $L^{3/2}$. We show that there exists a smooth solution that is bounded for all positive times. The proof is based on short-time regularization estimates for the Fisher information, which, combined with the recent result of Guillen and Silvestre, yields the existence of a global-in-time smooth solution. Additionally, if the initial data belongs to $L^p$ with $p>3/2$, there is a unique solution. At the crux of the result is a new $\varepsilon$-regularity criterion in the spirit of the Caffarelli-Kohn-Nirenberg theorem: a solution which is small in weighted $L^{3/2}$ is regular. Although the $L^{3/2}$ norm is a critical quantity for the Landau-Coulomb equation, using this norm to measure the regularity of solutions presents significant complications. For instance, the $L^{3/2}$ norm alone is not enough to control the $L^\infty$ norm of the competing reaction and diffusion coefficients. These analytical challenges caused prior methods relying on the parabolic structure of the Landau-Coulomb to break down. Our new framework is general enough to handle slowly decaying and singular initial data, and provides the first proof of global well-posedness for the Landau-Coulomb equation with rough initial data.

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arXiv2023

Local-in-time strong solutions of the homogeneous Landau-Coulomb equation with $L^p$ initial datum

We consider the homogeneous Landau equation with Coulomb potential and general initial data $f_{in} \in L^p$, where $p$ is arbitrarily close to $3/2$. We show the local-in-time existence and uniqueness of smooth solutions for such initial data. The constraint $p > 3/2$ has appeared in several related works and appears to be the minimal integrability assumption achievable with current techniques. We adapt recent ODE methods and conditional regularity results appearing in [arXiv:2303.02281] to deduce new short time $L^p \to L^\infty$ smoothing estimates. These estimates enable us to construct local-in-time smooth solutions for large $L^p$ initial data, and allow us to show directly conditional regularity results for solutions verifying \emph{unweighted} Prodi-Serrin type conditions. As a consequence, we obtain additional stability and uniqueness results for the solutions we construct.

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arXiv2023

Nonlinear asymptotic stability in $L^\infty$ for Lipschitz solutions to scalar conservation laws

In this note, we show nonlinear stability in $L^\infty$ for Lipschitz solutions to genuinely nonlinear, multi-dimensional scalar conservation laws. As an application, we are able to compute explicit algebraic decay rates of the $L^\infty$ norm of perturbations of global-in-time Lipschitz solutions, including perturbations of planar rarefaction waves. Our analysis uses the De Giorgi method applied to the kinetic formulation and is an extension of the method introduced recently by Silvestre in [Comm. Pure Appl. Math., 72(6):1321-1348, 2019].

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arXiv2023

Nonlinear regularization estimates and global well-posedness for the Landau-Coulomb equation near equilibrium

We consider the Landau equation with Coulomb potential in the spatially homogeneous case. We show short time propagation of smallness in $L^p$ norms for $p>3/2$ and instantaneous regularization in Sobolev spaces. This yields new short time quantitative a priori estimates that are unconditional near equilibrium. We combine these estimates with existing literature on global well-posedness for regular data to extend the well-posedness theory to small $L^p$ data with $p$ arbitrarily close to $3/2$. The threshold $p = 3/2$ agrees with previous work on conditional regularity for the Landau equation in the far from equilibrium regime. In light of the monotonicity of the Fisher information shown in the recent preprint [arXiv:2311.09420], our primary nonlinear regularization estimate holds even in the far-from-equilibrium regime. As a consequence, we obtain exponential convergence to equilibrium for suitably localized solutions in every Sobolev norm.

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arXiv2022

Unconditional regularity and trace results for the isentropic Euler equations with $γ= 3$

In this paper, we study the regularity properties of bounded entropy solutions to the isentropic Euler equations with $γ= 3$. First, we use a blow-up technique to obtain a new trace theorem for all such solutions. Second, we use a modified De Giorgi type iteration on the kinetic formulation to show a new partial regularity result on the Riemann invariants. We are able to conclude that in fact for any bounded entropy solution $u$, the density $ρ$ is almost everywhere upper semicontinuous away from vacuum. To our knowledge, this is the first example of a nonlinear hyperbolic system, which fails to be Temple class, but has the property that generic $L^\infty$ initial data give rise to bounded entropy solutions with a form of near classical regularity. This provides one example that $2\times 2$ hyperbolic systems can possess some of the more striking regularizing effects known to hold generically in the genuinely nonlinear, multidimensional scalar setting. While we are not able to use our regularity results to show unconditional uniqueness, the results substantially lower the likelihood that current methods of convex integration can be used in this setting.

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arXiv2022

Existence of smooth solutions to the Landau-Fermi-Dirac equation with Coulomb potential

In this paper, we prove global-in-time existence and uniqueness of smooth solutions to the homogeneous Landau-Fermi-Dirac equation with Coulomb potential. The initial conditions are nonnegative, bounded and integrable. We also show that any weak solution converges towards the steady state given by the Fermi-Dirac statistics. Furthermore, the convergence is algebraic, provided that the initial datum is close to the steady state in a suitable weighted Lebesgue norm.

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arXiv2020

Sharp a-contraction estimates for small extremal shocks

In this paper, we study the $a$-contraction property of small extremal shocks for 1-d systems of hyperbolic conservation laws endowed with a single convex entropy, when subjected to large perturbations. We show that the weight coefficient $a$ can be chosen with amplitude proportional to the size of the shock. The main result of this paper is a key building block in the companion paper, [{arXiv:2010.04761}, 2020], in which uniqueness and BV-weak stability results for $2\times 2$ systems of hyperbolic conservation laws are proved.

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arXiv2020

Uniqueness Criteria for the Oseen Vortex in the 3d Navier-Stokes Equations

In this paper, we consider the uniqueness of solutions to the 3d Navier-Stokes equations with initial vorticity given by $ω_0 = αe_z δ_{x = y = 0}$, where $δ_{x=y= 0}$ is the one dimensional Hausdorff measure of an infinite, vertical line and $α\in \mathbb R$ is an arbitrary circulation. This initial data corresponds to an idealized, infinite vortex filament. One smooth, mild solution is given by the self-similar Oseen vortex column, which coincides with the heat evolution. Previous work by Germain, Harrop-Griffiths, and the first author implies that this solution is unique within a class of mild solutions that converge to the Oseen vortex in suitable self-similar weighted spaces. In this paper, the uniqueness class of the Oseen vortex is expanded to include any solution that converges to the initial data in a sufficiently strong sense. This gives further evidence in support of the expectation that the Oseen vortex is the only possible mild solution that is identifiable as a vortex filament. The proof is a 3d variation of a 2d compactness/rigidity argument in $t \searrow 0$ originally due to Gallagher and Gallay.

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