Ingo Witt

On the Factorization of Meromorphic Mellin Symbols

It is proved that meromorphic, parameter-dependent elliptic Mellin symbols can be factorized in a particular way. The proof depends on the availability of logarithms of pseudodifferential operators. As a byproduct, we obtain a characterization of the group generated by pseudodifferential operators admitting a logarithm. The factorization has applications to the theory of pseudodifferential operators on spaces with conical singularities, e.g., to the index theory and the construction of various sub-calculi of the cone calculus.

On the existence and cusp singularity of solutions to semilinear generalized Tricomi equations with discontinuous initial data

In this paper, we are concerned with the local existence and singularity structures of low regularity solution to the semilinear generalized Tricomi equation with typical discontinuous initial data (u(0, x), ∂tu(0, x)) = (0, φ(x)), where m ∈ ℕ, x = (x1,…,xn), n ≥ 2, and f(t, x, u) is C∞ smooth on its arguments. When the initial data φ(x) is homogeneous of degree zero or piecewise smooth along the hyperplane {t = x1 = 0}, it is shown that the local solution u(t, x) ∈ L∞([0, T] × ℝn) exists and is C∞ away from the forward cuspidal conic surface or the cuspidal wedge-shaped surfaces respectively. On the other hand, for n = 2 and piecewise smooth initial data φ(x) along the two straight lines {t = x1 = 0} and {t = x2 = 0}, we establish the local existence of a solution and further show that in general due to the degenerate character of the equation under study, where . This is an essential difference to the well-known result for solution to the two-dimensional semilinear wave equation with (v(0, x), ∂tv(0, x)) = (0, φ(x)), where Σ0 = {t = |x|}, and .

Global existence and blowup of smooth solutions of 3-D potential equations with time-dependent damping

In this paper, we are concerned with the global existence and blowup of smooth solutions of the 3-D compressible Euler equation with time-depending damping

Coordinate Invariance of the Cone Algebra with Asymptotics

\partial_t\rho+\operatorname{div}(\rho u)=0, \quad \partial_t(\rho u)+\operatorname{div}\left(\rho u\otimes u+p\,I_{3}\right)=-\,\frac{\mu}{(1+t)^{\lambda}}\,\rho u, \quad \rho(0,x)=\bar \rho+\varepsilon\rho_0(x),\quad u(0,x)=\varepsilon u_0(x),

Global singularity structure of weak solutions to 3-D semilinear dispersive wave equations with discontinuous initial data☆

where

Propagation of Smoothness for Edge-degenerate Wave Equations

x\in\mathbb R^3

Global Multidimensional Shock Waves of 2-Dimensional and 3-Dimensional Unsteady Potential Flow Equations

,

Small data solutions of 2-D quasilinear wave equations under null conditions

\mu>0