Abstract. We show that the degree-2 Rossby–Haurwitz traveling waves on the Euler equation on [Formula: see text] are orbitally stable. Our proof is short, quantitative, and conceptually easy to follow.
Abstract. In this paper, under violation of the Syrovatskij stability condition, we prove the ill-posedness of the well-known Kelvin–Helmholtz problem of incompressible ideal magnetohydrodynamics (MHD) equations. This result underscores the stabilizing role of magnetic fields satisfying the Syrovatskij condition .
Abstract. We study the insulated conductivity problem with inclusions embedded in a bounded domain in [Formula: see text]. It was known that in the setting of strictly convex inclusions, the gradient of solutions may blow up as the distance between inclusions approaches 0. The optimal blowup rate was proved in [ 10 ] and was achieved in the presence of a uniform background gradient field. In this paper, we demonstrate that when the inclusions are partially flat, the gradient of solutions does not blow up under any uniform background fields.
We consider asymptotics of planar orthogonal polynomials [Formula: see text] (where [Formula: see text]) with respect to the weight [Formula: see text] in the whole complex plane. With [Formula: see text] and [Formula: see text] fixed, we obtain the strong asymptotics of the polynomials, asymptotics for the weighted [Formula: see text] norm, and the limiting zero counting measure. These results apply to the pre-critical phase of the underlying two-dimensional Coulomb gas system, when the support of the equilibrium measure is simply connected. Our method relies on specifying the mother body of the two-dimensional potential problem. It relies too on the fact that the planar orthogonality can be rewritten as a non-Hermitian contour orthogonality. This allows us to perform the Deift–Zhou steepest descent analysis of the associated [Formula: see text] Riemann–Hilbert problem.
Abstract. We study blowup solutions of the “bad” Boussinesq equation and prove that a wide range of asymptotic scenarios can happen. For example, for each [Formula: see text], [Formula: see text] and [Formula: see text], we prove that there exist Schwartz class solutions [Formula: see text] on [Formula: see text] such that [Formula: see text] and [Formula: see text] as [Formula: see text]. We also prove that for any [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], there exist Schwartz class solutions [Formula: see text] on [Formula: see text] such that [Formula: see text] for each [Formula: see text] such that [Formula: see text]; [Formula: see text] for each [Formula: see text] such that [Formula: see text]; and [Formula: see text] as [Formula: see text] for each [Formula: see text] such that [Formula: see text]. In particular, when [Formula: see text], this result establishes the existence of wave-breaking solutions, i.e., solutions that remain bounded but whose [Formula: see text]-derivative blows up in finite time.
