Search

We prove a uniqueness theorem for solutions to the wave map equation in the natural class, namely (u, ∂tu) ∈ C([0, T); Ḣd/2) × C1([0, T); Ḣd/2-1) in dimension d ≥ 4. This is achieved by estimating the difference of two solutions at a lower regularity level. In order to reduce to the Coulomb gauge, one has to localize the gauge change in suitable cones, as well as to estimate the difference between the frames and connections associated with each solution and to take advantage of the assumption that the target manifold has bounded curvature. [ABSTRACT FROM AUTHOR]

For hyperbolic differential operators P with double characteristics we study the relations between the maximal Gevrey index for the strong Gevrey well-posedness and the Hamilton map and flow of the associated principal symbol p. If the Hamilton map admits a Jordan block of size 4 on the double characteristic manifold denoted by Σ and by assuming that the Hamilton flow does not approach Σ tangentially, we proved earlier that the Cauchy problem for P is well-posed in the Gevrey class 1 ≤ s < 4 for any lower order term. In the present paper, we remove this restriction on the Hamilton flow and establish that the Cauchy problem for P is well-posed in the Gevrey class 1 ≤ s < 3 for any lower order term and we check that the Gevrey index 3 is optimal. Combining this with results already proved for the other cases, we conclude that the Hamilton map and flow completely characterizes the threshold for the strong Gevrey well-posedness and vice versa. [ABSTRACT FROM AUTHOR]

We derive a quadratically nonlinear equation that describes the motion of a tangential discontinuity in incompressible magnetohydrodynamics near the onset of a Kelvin-Helmholtz instability. We show that nonlinear effects enhance the instability and give a criterion for it to occur in terms of a longitudinal strain of the fluid motion along the discontinuity. [ABSTRACT FROM AUTHOR]

The method of generalized characteristics yields an elementary proof of uniqueness of dissipative solutions to the Cauchy problem for the Hunter-Saxton equation. [ABSTRACT FROM AUTHOR]

A notion of entropy quasisolution is introduced for the Euler equations of isothermal gas flows. Such a solution is obtained by means of nonlinear parabolic approximation with a small parameter ε. Compensated compactness argument is applied to justify the passage to limit as ε → 0 for the case when the mass density is strictly positive. It is verified that smooth entropy quasisolution is necessarily a classic solution. An example of entropy solution with a shock front is constructed to reveal that it is not an entropy quasisolution. The study is motivated by the explosion physics experiments in which the mass conservation law may be violated at a shock front passing through the gas. [ABSTRACT FROM AUTHOR]

We study the evolution of a many-particle system whose wave function obeys the N-body Schrödinger equation under Bose symmetry. The system Hamiltonian describes pairwise particle interactions in the absence of an external potential. We derive a priori dispersive estimates that express the overall repulsive nature of the particle interactions. These estimates hold for a wide class of two-body interaction potentials which are independent of the particle number, N. We discuss applications of these estimates to the BBGKY hierarchy for reduced density matrices analyzed by Elgart, Erdős, Schlein and Yau. [ABSTRACT FROM AUTHOR]

We develop a version of Haar and Holmgren methods which applies to discontinuous solutions of nonlinear hyperbolic systems and allows us to control the L1 distance between two entropy solutions. The main difficulty is to cope with linear hyperbolic systems with discontinuous coefficients. Our main observation is that, while entropy solutions contain compressive shocks only, the averaged matrix associated with two such solutions has compressive or undercompressive shocks, but no rarefaction-shocks — which are recognized as a source for non-uniqueness and instability. Our Haar–Holmgren-type method rests on the geometry associated with the averaged matrix and takes into account adjoint problems and wave cancellations along generalized characteristics. It generalizes the method proposed earlier by LeFloch et al. for genuinely nonlinear systems. In the present paper, we cover solutions with small total variation and a class of systems with general flux that need not be genuinely nonlinear and includes for instance fluid dynamics equations. We prove that solutions generated by Glimm or front tracking schemes depend continuously in the L1 norm upon their initial data, by exhibiting an L1 functional controlling the distance between two solutions. [ABSTRACT FROM AUTHOR]

Dedicates the September 2006 issue of the "Journal of Hyperbolic Differential Equations" to Tai-Ping Liu by his friends, collaborators and former students on the occasion of his sixtieth birthday. Scope of Tai-Ping's scientific work; Wish for Tai-Ping.