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  1. Abstract

    This article represents a 1st step toward understanding the well-posedness of the dispersive Hunter–Saxton equation, which arises in the study of nematic liquid crystals. Although the equation has formal similarities with the KdV equation, the lack of $L^2$ control gives it a quasilinear character. Further, the lack of spatial decay obstructs access to dispersive tools, including local smoothing estimates. Here, we give the 1st proof of local and global well-posedness for the Cauchy problem. Secondly, we improve our well-posedness results with respect to the low regularity of the initial data. The key techniques we use include constructing modified energies to realize a normal form analysis in our quasilinear setting, and frequency envelopes to prove continuous dependence with respect to the initial data.

  2. Abstract

    We consider the Kuramoto–Sivashinsky equation (KSE) on the two-dimensional torus in the presence of advection by a given background shear flow. Under the assumption that the shear has a finite number of critical points and there are linearly growing modes only in the direction of the shear, we prove global existence of solutions with data in$$L^2$$L2, using a bootstrap argument. The initial data can be taken arbitrarily large.

  3. Free, publicly-accessible full text available October 1, 2023
  4. This paper proves that the motion of small-slope vorticity fronts in the two-dimensional incompressible Euler equations is approximated on cubically nonlinear timescales by a Burgers–Hilbert equation derived by Biello and Hunter (2010) using formal asymptotic expansions. The proof uses a modified energy method to show that the contour dynamics equations for vorticity fronts in the Euler equations and the Burgers–Hilbert equation are both approximated by the same cubically nonlinear asymptotic equation. The contour dynamics equations for Euler vorticity fronts are also derived.
    Free, publicly-accessible full text available August 1, 2023
  5. Free, publicly-accessible full text available August 1, 2023
  6. Suppose f ∈ K [ x ] f \in K[x] is a polynomial. The absolute Galois group of K K acts on the preimage tree T \mathrm {T} of 0 0 under f f . The resulting homomorphism ϕ f : Gal K → Aut ⁡ T \phi _f\colon \operatorname {Gal}_K \to \operatorname {Aut} \mathrm {T} is called the arboreal Galois representation. Odoni conjectured that for all Hilbertian fields K K there exists a polynomial f f for which ϕ f \phi _f is surjective. We show that this conjecture is false.
    Free, publicly-accessible full text available August 1, 2023
  7. Free, publicly-accessible full text available August 1, 2023
  8. Free, publicly-accessible full text available July 1, 2023
  9. Let $G$ be a graph with vertex set $\{1,2,\ldots,n\}$. Its bond lattice, $BL(G)$, is a sublattice of the set partition lattice. The elements of $BL(G)$ are the set partitions whose blocks induce connected subgraphs of $G$. In this article, we consider graphs $G$ whose bond lattice consists only of noncrossing partitions. We define a family of graphs, called triangulation graphs, with this property and show that any two produce isomorphic bond lattices. We then look at the enumeration of the maximal chains in the bond lattices of triangulation graphs. Stanley's map from maximal chains in the noncrossing partition lattice to parking functions was our motivation. We find the restriction of his map to the bond lattice of certain subgraphs of triangulation graphs. Finally, we show the number of maximal chains in the bond lattice of a triangulation graph is the number of ordered cycle decompositions.
    Free, publicly-accessible full text available July 1, 2023
  10. Let 𝐺 be a complex semisimple Lie group and 𝘏 a complex closed connected subgroup. Let g and h be their Lie algebras. We prove that the regular representation of 𝐺 in 𝐿²(𝐺/𝘏) is tempered if and only if the orthogonal of h in g contains regular elements by showing simultaneously the equivalence to other striking conditions, such as h has a solvable limit algebra.
    Free, publicly-accessible full text available June 7, 2023