We are concerned with the global existence of finite-energy entropy solutions of the one-dimensional compressible Euler equations with (possibly) damping, alignment forces, and the nonlocal interactions of Newtonian repulsion and quadratic confinement. Both the polytropic gas law and the general gas law are analyzed. This is achieved by constructing a sequence of solutions of the one-dimensional compressible Navier–Stokes-type equations with density-dependent viscosity on expanding intervals with the stress-free boundary condition and then taking the vanishing viscosity limit. The main difficulties in this paper arise from the appearance of the nonlocal terms. In particular, some uniform higher moment estimates of the solutions for the compressible Navier–Stokes equations on the expanding intervals with stress-free boundary condition are obtained by careful design of the approximate initial data.

One of the unexplored benefits of studying layer potentials on smooth, closed hypersurfaces of Euclidean space is the factorization of the Neumann-Poincaré operator into a product of two self-adjoint transforms. Resurrecting some pertinent indications of Carleman and M. G. Krein, we exploit this grossly overlooked structure by confining the spectral analysis of the Neumann-Poincaré operator to the amenable L2-space setting, rather than bouncing back and forth the computations between Sobolev spaces of negative or positive fractional order. An enhanced, fresh new look at symmetrizable linear transforms enters into the picture in the company of geometric/microlocal analysis techniques. The outcome is manyfold, complementing recent advances on the theory of layer potentials, in the smooth boundary setting.

We consider the homogeneous Landau equation in R3 with Coulomb potential and initial data in polynomially weighted L3/2. We show that there exists a smooth solution that is bounded for all positive times. The proof is based on short-time regularization estimates for the Fisher information, which, combined with the recent result of Guillen and Silvestre, yields the existence of a global-in-time smooth solution. Additionally, if the initial data belongs to Lp with p>3/23/2$$\\end{document}]]>, there is a unique solution. At the crux of the result is a new ε-regularity criterion in the spirit of the Caffarelli–Kohn–Nirenberg theorem: a solution which is small in weighted L3/2 is regular. Although the L3/2 norm is a critical quantity for the Landau–Coulomb equation, using this norm to measure the regularity of solutions presents significant complications. For instance, the L3/2 norm alone is not enough to control the L∞ norm of the competing reaction and diffusion coefficients. These analytical challenges caused prior methods relying on the parabolic structure of the Landau–Coulomb to break down. Our new framework is general enough to handle slowly decaying and singular initial data, and provides the first proof of global well-posedness for the Landau–Coulomb equation with rough initial data.