In these lecture notes we describe the propagation of singularities of tempered distributional solutions $u\in {𝒮}^{\text{'}}$ of $(H-\lambda )u=0$, where $H$ is a many-body hamiltonian $H=\Delta +V$, $\Delta \ge 0$, $V={\sum }_a{V}_a$, and $\lambda$ is not a threshold of $H$, under the assumption that the inter-particle (e.g. two-body) interactions $V_a$ are real-valued polyhomogeneous symbols of order $-1$ (e.g. Coulomb-type with the singularity at the origin removed). Here the term “singularity” provides a microlocal description of the lack of decay at infinity. Our result is then that the...

We investigate the propagation of the uniform spatial Gevrey $G^{\sigma }$, σ ≥ 1, regularity for t → +∞ of solutions to evolution equations like generalizations of the Euler equation and the semilinear Schrödinger equation with polynomial nonlinearities. The proofs are based on direct iterative arguments and nonlinear Gevrey estimates.

The propagation of weak discontinuities for quasilinear systems with coefficients functionally dependent on the solution is studied. We demonstrate that, similarly to the case of usual quasilinear systems, the transport equation for the intensity of weak discontinuity is quadratic in this intensity. However, the contribution from the (nonlocal) functional dependence appears to be in principle linear in the jump intensity (with some exceptions). For illustration, several examples, including two hyperbolic...

Motivated by the study of resolvent estimates in the presence of trapping, we prove a semiclassical propagation theorem in a neighborhood of a compact invariant subset of the bicharacteristic flow which is isolated in a suitable sense. Examples include a global trapped set and a single isolated periodic trajectory. This is applied to obtain microlocal resolvent estimates with no loss compared to the nontrapping setting.

Proper orthogonal decomposition (POD) is a powerful technique for model reduction of non-linear systems. It is based on a Galerkin type discretization with basis elements created from the dynamical system itself. In the context of optimal control this approach may suffer from the fact that the basis elements are computed from a reference trajectory containing features which are quite different from those of the optimally controlled trajectory. A method is proposed which avoids this problem of unmodelled...

In this paper we propose a new concept of quasi-uniform monotonicity weaker than the uniform monotonicity which has been developed in the study of nonlinear operator equation $Au=b$. We prove that if $A$ is a quasi-uniformly monotone and hemi-continuous operator, then $A^{-1}$ is strictly monotone, bounded and continuous, and thus the Galerkin approximations converge. Also we show an application of a quasi-uniformly monotone and hemi-continuous operator to the proof of the well-posedness and convergence...

We construct fundamental solutions of some partial differential equations of order higher than two and examine properties of these solutions and of some related integrals. The results will be used in our next paper concerning boundary-value problems for these equations.

This work is devoted to the concept of statistical solution of the Navier-Stokes equations, proposed as a rigorous mathematical object to address the fundamental concept of ensemble average used in the study of the conventional theory of fully developed turbulence. Two types of statistical solutions have been proposed in the 1970’s, one by Foias and Prodi and the other one by Vishik and Fursikov. In this article, a new, intermediate type of statistical solution is introduced and studied. This solution...

Let ${\ell }_j:=-d²/dx²+k²q_j\left(x\right)$, k = const > 0, j = 1,2, $0<essinf{q}_j\left(x\right)\le esssup{q}_j\left(x\right)<\infty$. Suppose that (*) ${\int }_0^{1}p\left(x\right)u₁(x,k)u₂(x,k)dx=0$ for all k > 0, where p is an arbitrary fixed bounded piecewise-analytic function on [0,1], which changes sign finitely many times, and $u_j$ solves the problem ${\ell }_j{u}_j=0$, 0 ≤ x ≤ 1, $u_j^{\text{'}}(0,k)=0$, $u_j(0,k)=1$. It is proved that (*) implies p = 0. This result is applied to an inverse problem for a heat equation.