Partial differential equations

We prove global existence and uniqueness for arbitrary smooth admissible initial data in the three-dimensional, one-species relativistic Vlasov–Maxwell system. The particle density is initially compactly supported, and the electromagnetic fields have finite energy and bounded derivatives of all orders. The solution remains smooth on every finite time interval. This resolves the large-data global classical regularity problem for this model, without size or symmetry restrictions on the data.

We prove nonuniqueness for the three-dimensional periodic hard-sphere Boltzmann equation among global entropy solutions satisfying exact local conservation of mass, momentum, and kinetic energy. We construct one nonnegative initial density with bounded velocity support and finite mass, energy, and absolute entropy that gives rise to two distinct such solutions. Both are strongly continuous in L¹, and their collision gain and loss terms are integrable with every polynomial velocity weight on every bounded time interval.

We prove nonuniqueness for the periodic hard-sphere Boltzmann equation by constructing two distinct global renormalized solutions with the same nonnegative initial density on \(\mathbb T^3\times\mathbb R^3\). This density has bounded velocity support and finite mass, energy, and absolute entropy. Both solutions conserve local mass and total momentum and satisfy the global energy and entropy-dissipation inequalities. On a common initial interval, they are strongly continuous in L¹, and their collision gains and losses are integrable.

We derive the nonlinear Boltzmann equation from a grand-canonical Newtonian gas with initial pair exclusion. We treat stable, finite-range radial potentials that are C² except for an allowed repulsive singularity at the origin, and C¹ initial probability densities with spatially summable Gaussian bounds on the density and its spatial gradient. All fixed-order rescaled factorial marginals converge in L¹, uniformly throughout every finite interval on which the classical kinetic solution has a uniform Gaussian bound. The result allows attractive wells and dynamically formed clusters, without restrictions on the differential scattering cross-section.

We prove a finite-dimensional central limit theorem away from equilibrium for a deterministic grand-canonical hard-sphere gas in three dimensions. The initial probability density is smooth, with spatially summable Gaussian velocity bounds on the density and its spatial gradient. The limit holds on every finite interval on which the classical Boltzmann solution has uniform Gaussian velocity decay. The empirical measure is centered by its exact microscopic expectation. Its fluctuations converge to the Gaussian solution of the linear fluctuating Boltzmann equation.

We prove that zero-frequency boundary measurements on any nonempty open boundary patch determine both a smooth Riemannian metric and a smooth unitary connection on the trivial Hermitian rank-two bundle over a compact connected smooth manifold of dimension at least three with smooth boundary. Both inputs and observations are restricted to the same patch. The metric and connection are determined up to a diffeomorphism and a unitary gauge that restrict to the identity on the measured patch.

We resolve the smooth anisotropic Calderón uniqueness problem with arbitrary same-patch measurements. A smooth Riemannian metric on a compact connected manifold of dimension at least three is determined, up to a diffeomorphism fixing the measured patch, by its zero-frequency Dirichlet-to-Neumann energy form with both input and observation on any nonempty open boundary patch.

We construct two distinct uniformly positive bounded measurable scalar conductivities on a ball in ℝ³ with the same full Dirichlet-to-Neumann operator. Both conductivities equal one near the boundary. This gives nonuniqueness in the scalar Calderón problem at bounded measurable regularity.

We prove the interior regularity assertion of the planar Mumford–Shah conjecture for reduced absolute minimizers with bounded fidelity data. Every interior point of the closed discontinuity set has a neighborhood consisting of a \(C^{1,\alpha}\) arc, an arc ending at that point, or three such arcs meeting at 120 degrees. Only finitely many global connected components meet any relatively compact open set.

We prove that seven is the critical dimension for the one-phase Bernoulli problem: every nonzero one-homogeneous global minimizer in dimensions at most six is flat, while a nonflat one-homogeneous global minimizer exists in dimension seven. It follows that the interior free boundary of a local minimizer is smooth in dimensions at most six. In dimension n ≥ 7, its singular set has Hausdorff dimension at most \(n-7\), and this bound is sharp. In dimension seven, the singular set is locally finite.

We resolve the three-dimensional Ball–Evans approximation problem for every finite p > 2. Every \(W^{1,p}\) homeomorphism between arbitrary bounded domains in ℝ³ can be approximated strongly in \(W^{1,p}\) by smooth diffeomorphisms onto the same target.

We resolve the three-dimensional Ball–Evans approximation problem for \(1\le p\le2\). Every \(W^{1,p}\) homeomorphism between arbitrary bounded domains in ℝ³ is a strong \(W^{1,p}\) limit of smooth diffeomorphisms onto the same target.

We prove the strict hot spots conjecture for smooth bounded simply connected planar domains. More precisely, every nonzero eigenfunction for the first positive Neumann eigenvalue has nonvanishing gradient in the interior, so all its global maxima and minima lie on the boundary. This holds even when the eigenvalue is multiple.

We prove the subcritical Hénon–Lane–Emden conjecture: for every dimension n ≥ 2, positive powers p, q, and real weights A, B, the system has no strictly positive entire solution when \((n+A)/(p+1)+(n+B)/(q+1)\gt n-2\). Solutions need only be continuous at the origin and classical elsewhere, without a condition at infinity. For n ≥ 3 and \(A,B\gt -2\), combining this result with the radial existence theorem of Bidaut-Véron and Giacomini proves Phan's Conjecture C in full: a positive radial entire solution exists whenever the strict inequality fails.

We prove stable self-similar finite-time blowup for a supercritical defocusing nonlinear Schrödinger equation on the twelve-dimensional torus. For a sufficiently large odd power, the blowup initial data contain a nonempty open set in a high Sobolev space. As a consequence, Gaussian Fourier initial data of arbitrarily high Sobolev regularity can have positive probability of finite-time blowup.

We prove that the full static displacement-to-traction map uniquely determines both real smooth Lamé moduli on every bounded connected smooth domain in ℝ³, provided μ > 0 and \(3\lambda+2\mu\gt 0\) on the closure. This resolves the smooth three-dimensional isotropic elastic Calderón uniqueness problem under these positivity assumptions.

We construct a smooth compactly supported probability density on ℝ³, with a smooth compactly supported square root, for which the three-marginal Coulomb transport minimum is not attained by any pair of measure-preserving Borel maps. Nevertheless, the Monge and Kantorovich infima agree: we construct preserving maps whose costs approach the Kantorovich minimum. For every d ≥ 2 and s > 0, we also construct a smooth identical marginal with the same nonattainment and equality-of-infima properties for the inverse-power interaction \(\sum_{i\lt j}|x_i-x_j|^{-s}\) on ℝ^d.

For the uniform probability measure ρ on a compact convex body in ℝ^d, d ≥ 2, quadratic optimal transport maps are one-third Hölder continuous in \(L^2(\rho)\) with respect to the target's 2-Wasserstein distance. The constant is uniform over all targets supported in a fixed compact set, and the exponent is optimal. A three-atom family on a fixed cube disproves Letrouit's conjectured uniform square-root estimate.

We resolve De Giorgi's conjecture positively in dimension eight: every entire C² solution \(u:\mathbb R^8\to(-1,1)\) of \(\Delta u=u^3-u\) with an everywhere positive directional derivative is a planar heteroclinic. We prove that every stable solution \(v:\mathbb R^7\to[-1,1]\) of this equation is a constant well or a planar heteroclinic, without an energy-growth assumption.

We construct smooth forces for three-dimensional incompressible Navier–Stokes flow, from rest at any fixed positive computable viscosity, whose velocity field detects whether a prescribed machine halts. On the unit flat torus, a pointwise test of the third velocity component uses successively shorter stirring intervals in a fixed spatial region. On \(\mathbb R^2\times\mathbb T\), an integral test uses an expanding array of translation regions and a force with globally bounded mixed derivatives. The vertical velocity solves an advection–diffusion equation: short bursts control its pointwise error in the first construction, and a moving cutoff controls the total escaped mass in the second.

At every fixed positive computable viscosity, we construct smooth mean-zero forces on the flat three-torus that become stationary after time one and make a fixed particle, initially in a fluid at rest, enter a fixed open set exactly when a prescribed Turing machine halts. The force has bounded derivatives of every order and a finite effective description. A reversible recording table is realized on whole planar rectangles by Hamiltonian motions, then driven by a mean-zero spatial clock. Separate choices give periodic forcing from time zero or a force whose derivatives are square-integrable in time.

We construct smooth forces of fixed compact spatial support for three-dimensional incompressible Navier–Stokes flow from rest whose particle at the origin detects halting by entering a fixed half-space. Every mixed derivative of the force and velocity decays at rate \(O((1+t)^{-1-j})\) for time order j. Three scalar potentials lift a coded rectangle, perform its instruction, and lower its image; an unbounded logarithmic clock supplies the decay. We also realize area-changing prefix instructions on an invariant torus plane, and construct a planar Hamiltonian processor admitting periodic forcing, stationary forcing after startup, and a velocity-field detector through a companion diffusion theorem.

Finite prefix instructions may change area and erase information. We give explicit history processors and smooth incompressible motions that retain that information and realize every instruction on its full domain. A first application assigns each machine and finite input a smooth mean-zero force on the flat unit three-torus, at any fixed positive computable viscosity. The solution starts from rest; a fixed particle enters a fixed open strip exactly when the machine halts. The force repeats with period one after an initial loading interval. We then prove alternative realizations using full boxes, normal compensation, invariant planes, and a spatial clock. The constructions specify their initialization, comparison class, derivative bounds, and continuous-time observation. Exact formulas and effective cutoffs provide finite descriptions of the fields and all their derivatives.

We realize finite positive diagonal affine maps of determinant one by effective smooth incompressible flows on neighborhoods of entire closed rational solid boxes. The source and target families are each disjoint, but may overlap each other. The construction uses localized curls, evacuation to storage and obstacle detours. A balanced three-stack recorder then assigns every machine and finite input a smooth Navier–Stokes force with one compact spatial support, periodic after a loading interval, at any fixed positive computable viscosity. The fluid starts at rest, and one fixed particle enters one fixed open cube exactly when the machine halts. Further constructions give fixed torus charts, periodicity from time zero, alternative history guards and bounded or slab observers. Onto slow clocks yield separate decaying forces. Each construction includes an all-time observation proof, effective derivative bounds and a stated pressure comparison class.

We construct smooth solenoidal mean-zero forces, periodic from time zero, for which a fixed particle on a flat three-torus reaches a fixed open strip exactly when a given machine halts. The initial fluid velocity and the pressure are zero. We also realize positive diagonal maps on coding sheets by incompressible shears, with explicit normal compensation for changes of planar area, and reciprocal maps on whole boxes of positive thickness. Complete local inverses, initialization rules, and intermediate trajectory bounds connect these geometric constructions to finite computations.

We realize finite reciprocal affine instruction maps by smooth incompressible shear flows on a flat three-torus. Applied to a reversible one-head recorder with a finite transition table, this gives a complete machine-to-fluid construction: a fixed particle enters a fixed open strip exactly when a given machine halts, with zero initial velocity, zero pressure, and a solenoidal mean-zero force that is periodic after initialization. The construction acts on full closed rectangles and controls every intermediate trajectory. Separate heights resolve overlap between sources and targets, and an endpoint-size estimate controls all excursions independently of the reciprocal scaling factor.

At any fixed positive computable viscosity, smooth forces can make a fixed fluid particle detect the halting of an arbitrary machine, starting from rest, while every mixed derivative of the force and velocity decreases faster than every inverse power of time. We give three complete memory constructions: compact moving curls, alternating fractional coordinates, and a periodic lattice on the flat three-torus. Each machine step takes one unit of physical time. The constructions retain earlier records at separated spatial scales and use an exact open detector with a shrinking signal.

We construct smooth external forces for incompressible Navier–Stokes flow on a fixed flat three-torus at any fixed positive computable viscosity, from zero initial velocity, such that a fixed particle enters a fixed open set exactly when a prescribed Turing machine halts. Every mixed derivative of the force and velocity is bounded and square-integrable in time in spatial supremum norm. The construction uses the machine's ordinary instructions, records their history in a third coordinate, and compensates for finer spatial gates by longer time steps.

We prove a uniform interior \(C^{1,\alpha_d}\) estimate for bounded infinity-harmonic functions in every dimension d ≥ 3. For some \(\alpha_d\in(0,1/3]\) depending only on the dimension, the gradient's supremum norm and α_d-Hölder seminorm on B_1/2 are bounded by a dimension-dependent constant times the oscillation on B₁. The exponent is not explicit, and no endpoint or boundary regularity claim is made. In particular, infinity-harmonic functions are locally C¹ on every open subset of ℝ^d, for d ≥ 3.