We consider the Bloch–Torrey operator in $L^2(\mathrm{\Omega },{ℝ}^3)$, where $\mathrm{\Omega }\subseteq {ℝ}^k$. After normalization, this operator takes the form $-{𝜖}^2\mathrm{\Delta }+b\wedge$, where and $\mathit{b}$ represents a magnetic vector field. For $\mathrm{\Omega }={ℝ}^k$ we give natural conditions under which this operator can be defined as a maximally accretive operator, characterize its domain and obtain its spectral properties in some special cases where we manage to show that the essential spectrum is $[0,+\infty )$. This result lies in contrast with the $L^2(\mathrm{\Omega },{ℝ}^2)$ case considered in previous works.

In the asymptotic limit $𝜖\to 0$ and for $k=1$, assuming that $b(x)$ is an affine function, we give accurate estimates for the location of the discrete spectrum in the cases $\mathrm{\Omega }=ℝ$ or when $\mathrm{\Omega }$ is a finite interval. Resolvent estimates are established as well.

We consider a sequence of open quantum graphs, with uniformly bounded data, and we are interested in the asymptotic distribution of their scattering resonances. Supposing that the number of leads in our quantum graphs is small compared to the total number of edges, we show that most resonances are close to the real axis. More precisely, the asymptotic distribution of resonances of our open quantum graphs is the same as the asymptotic distribution of the square root of the eigenvalues of the closed quantum graphs obtained by removing all the leads.

Here we develop a method for investigating global strong solutions of partially dissipative hyperbolic systems in the critical regularity setting. Compared to the recent works by Kawashima and Xu, we use hybrid Besov spaces with different regularity exponents in low and high frequencies. This allows us to consider more general data and to track the exact dependency on the dissipation parameter for the solution. Our approach enables us to go beyond the $L^2$ framework in the treatment of the low frequencies of the solution, which is totally new, to the best of our knowledge.

The focus is on the one-dimensional setting (the multidimensional case will be considered in a forth-coming paper) and, for expository purposes, the first part of the paper is devoted to a toy model that may be seen as a simplification of the compressible Euler system with damping. More elaborate systems (including the compressible Euler system with general increasing pressure law) are considered at the end of the paper.

We consider various filtered time discretizations of the periodic Korteweg–de Vries equation: a filtered exponential integrator, a filtered Lie splitting scheme, as well as a filtered resonance-based discretization, and establish error estimates at low regularity. Our analysis is based on discrete Bourgain spaces and allows us to prove convergence in $L^2$ for rough data $u_0\in H^s$, , with an explicit convergence rate.

For geometric systems of real principal type, we define a subprincipal symbol and derive a transport equation for polarizations which, in the scalar case, is a well-known equation of Duistermaat and Hörmander. We apply the transport equation to propagation of polarization in transmission problems of elastodynamics, to interior bulk waves as well as to free (Rayleigh) surface waves. Using spectral factorizations of matrix polynomials having real spectrum, we establish reflection and refraction laws of polarizations at the boundary and at interior interfaces. The results are not limited to isotropic elasticity.