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In this paper we study the Cauchy problem for the Landau Hamiltonian wave equation, with time dependent irregular (distributional) electromagnetic field and similarly irregular velocity. For such equations, we describe the notion of a `very weak solution adapted to the type of solutions that exist for regular coefficients. The construction is based on considering Friedrichs-type mollifier of the coefficients and corresponding classical solutions, and their quantitative behaviour in the regularising parameter. We show that even for distributional coefficients, the Cauchy problem does have a very weak solution, and that this notion leads to classical or distributional type solutions under conditions when such solutions also exist.
We prove existence and uniqueness of distributional, bounded, nonnegative solutions to a fractional filtration equation in ${mathbb R}^d$. With regards to uniqueness, it was shown even for more general equations in [19] that if two bounded solutions
This paper discusses some regularity of almost periodic solutions of the Poissons equation $-Delta u = f$ in $mathbb{R}^n$, where $f$ is an almost periodic function. It has been proved by Sibuya [Almost periodic solutions of Poissons equation. Proc.
We investigate the stationary diffusion equation with a coefficient given by a (transformed) Levy random field. Levy random fields are constructed by smoothing Levy noise fields with kernels from the Matern class. We show that Levy noise naturally ex
We study the long time behaviour of the kinetic Fokker-Planck equation with mean field interaction, whose limit is often called Vlasov-Fkker-Planck equation. We prove a uniform (in the number of particles) exponential convergence to equilibrium for t
We consider the energy-critical non-linear focusing wave equation in dimension N=3,4,5. An explicit stationnary solution, $W$, of this equation is known. The energy E(W,0) has been shown by C. Kenig and F. Merle to be a threshold for the dynamical be