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Register a new userSemiclassical diffraction by conormal potential singularities
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We establish propagation of singularities for the semiclassical Schrodinger equation, where the potential is conormal to a hypersurface. We show that semiclassical wavefront set propagates along generalized broken bicharacteristics, hence reflection of singularities may occur along trajectories reaching the hypersurface transversely. The reflected wavefront set is weaker, however, by a power of $h$ that depends on the regularity of the potential. We also show that for sufficiently regular potentials, wavefront set may not stick to the hypersurface, but rather detaches from it at points of tangency to travel along ordinary bicharacteristics.
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We consider the Schrodinger operator [ P=h^2 Delta_g + V ] on $mathbb{R}^n$ equipped with a metric $g$ that is Euclidean outside a compact set. The real-valued potential $V$ is assumed to be compactly supported and smooth except at conormal singularities of order $-1-alpha$ along a compact hypersurface $Y.$ For $alpha>2$ (or even $alpha>1$ if the classical flow is unique), we show that if $E_0$ is a non-trapping energy for the classical flow, then the operator $P$ has no resonances in a region [ [E_0 - delta, E_0 + delta] - i[0, u_0 h log(1/h)]. ] The constant $ u_0$ is explicit in terms of $alpha$ and dynamical quantities. We also show that the size of this resonance-free region is optimal for the class of piecewise-smooth potentials on the line.
We investigate the singularities of the trace of the half-wave group, $mathrm{Tr} , e^{-itsqrtDelta}$, on Euclidean surfaces with conical singularities $(X,g)$. We compute the leading-order singularity associated to periodic orbits with successive degenerate diffractions. This result extends the previous work of the third author cite{Hil} and the two-dimensional case of the work of the first author and Wunsch cite{ForWun} as well as the seminal result of Duistermaat and Guillemin cite{DuiGui} in the smooth setting. As an intermediate step, we identify the wave propagators on $X$ as singular Fourier integral operators associated to intersecting Lagrangian submanifolds, originally developed by Melrose and Uhlmann cite{MelUhl}.
The paper is a comprehensive study of the $L_p$ and the Schauder estimates for higher-order divergence type parabolic systems with discontinuous coefficients in the half space and cylindrical domains with conormal derivative boundary condition. For the $L_p$ estimates, we assume that the leading coefficients are only bounded measurable in the $t$ variable and $VMO$ with respect to $x$. We also prove the Schauder estimates in two situations: the coefficients are Holder continuous only in the $x$ variable; the coefficients are Holder continuous in both variables.
For ideal fluid flow with zero surface tension and gravity, it remains unknown whether local singularities on the free surface can develop in well-posed initial value problems with smooth initial data. This is so despite great advances over the last 25 years in the mathematical analysis of the Euler equations for water waves. Here we expand our earlier work (Chin. Ann. Math. Ser. B 40 (2019) 925) and review the mathematical literature and some of the history concerning Dirichlets ellipsoids and related hyperboloids associated with jet formation and flip-through, splash singularities, and recent constructions of singular free surfaces that however violate the Taylor sign condition for linear well-posedness. We illustrate some of these phenomena with numerical computations of 2D flow based upon a conformal mapping formulation (whose derivation is detailed and discussed in an appendix). Additional numerical evidence strongly suggests that corner singularities may form in an unstable self-similar way from specially prepared initial data.
A 2D problem of acoustic wave scattering by a segment bearing impedance boundary conditions is considered. In the current paper (the first part of a series of two) some preliminary steps are made, namely, the diffraction problem is reduced to two matrix Riemann-Hilbert problems with exponential growth of unknown functions (for the symmetrical part and for the antisymmetrical part). For this, the Wiener--Hopf problems are formulated, they are reduced to auxiliary functional problems by applying the embedding formula, and finally the Riemann-Hilbert problems are formulated by applying the Hurds method. In the second part the Riemann-Hilbert problems will be solved by a novel method of OE-equation.
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