Journal of Hyperbolic Differential Equations

We study the free boundary problem for two-dimensional current-vortex sheets in ideal incompressible magneto-hydrodynamics near the transition point between the linearized stability and instability. In order to study the dynamics of the discontinuity near the onset of the instability, Hunter and Thoo have introduced an asymptotic quadratically nonlinear integro-differential equation for the amplitude of small perturbations of the planar discontinuity. For this amplitude equation, a local-in-time existence theory for smooth solutions to the Cauchy problem was established earlier by the authors under a suitable stability condition. However, the solution found therein enjoyed a loss of regularity (of order two) in comparison to the regularity of the initial data. In this work, we are able to prove an existence result with optimal regularity, in the sense that the regularity of the initial data is preserved in the evolution for positive times.

We prove a weak form of the soliton resolution conjecture for — $H^2$ uniformly bounded in time — solutions of semilinear fourth-order Schrödinger equations, in dimensions $n\ge 5$, and with a mass supercritical-energy subcritical power type nonlinearity, by using a strategy developed by T. Tao in 2007. More precisely, we prove that the solutions are decomposed into a sum of two terms: a free solution and a nonradiative term that approaches asymptotically an object that has similar properties to those of a finite sum of solitons. The asymptotic behavior of the nonradiative term is derived from its asymptotic frequency localization and its asymptotic spatial localization. There are two main differences between the present contribution and T. Tao’s earlier work. The first one appears when we prove the asymptotic frequency localization: we fill a gap of regularity by using the better dispersive properties of the high frequency pieces of the free solution. The second one appears when we prove the asymptotic spatial localization. A key estimate depends on the fundamental solution that does not have an explicit form. We overcome the difficulty by introducing a modified fundamental solution and exploiting the symmetries of the characters of the phases.

We investigate local properties for microlocally symmetrizable hyperbolic systems with just time dependent coefficients. Thanks to Paley–Wiener theorem, we establish finite propagation speed by showing precise estimates on the evolution of the support of the solution in terms of suitable norms of the coefficients of the operator and of the symmetrizer. From this result, local existence and uniqueness follow by quite standard methods. Our argument relies on the use of Fourier transform, and it cannot be extended to operators whose coefficients depend also on the space variables. On the other hand, it works under very mild regularity assumptions on the coefficients of the operator and of the symmetrizer.

In this paper, We establish new homogenization results for stochastic nonlinear hyperbolic equations with periodically oscillating coefficients. We use a delicate blending of Tartar’s method of oscillating test functions and deep probabilistic compactness results due to Prokhorov and Skorokhod. We prove that the sequence of solutions of the original problem converges in suitable topologies to the solution of a homogenized stochastic hyperbolic problem with constant coefficients. We also prove the convergence of the associated energies.

The conformal identity for the solution of the Chern–Simons gauged wave equations is derived. As applications of it, we show some decay estimates and the nonexistence of nontrivial solutions.

We prove that the support of a probability measure on the plane that satisfies the Tartar commutation relation for pairs of functions likewise the solutions of the Euler–Poisson–Darboux equation is contained in 16 points.

We study a general system of hyperbolic-parabolic balance laws in $m$ space dimensions ($m\ge 1$). The system has rank deficient viscosity matrices and a lower order term whose Jacobian matrix is rank deficient as well. We consider the Cauchy problem when initial data are small perturbations of a constant equilibrium state. Under a set of reasonable assumptions including Kawashima–Shizuta condition, we establish the existence of solution global in time via energy method. The proposed assumptions are sufficiently general for applications to physical models such as electro-magneto flows and physical gas flows. In particular, we study the gas flow with an internal non-equilibrium mode besides the translational non-equilibrium. The general result in this paper recovers the existing results in literature on hyperbolic-parabolic conservation laws and hyperbolic balance laws, respectively, as two special cases.