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We investigate interesting connections between Mizohata type vector fields and microlocal regularity of nonlinear first-order PDEs, establishing results in Denjoy–Carleman classes and real analyticity results in the linear case.
In the following note, we focus on the problem of existence of continuous solutions vanishing at infinity to the equation div v = f for f ∈ Ln(ℝn) and satisfying an estimate of the type ||v||∞ ⩽ C||f||n for any f ∈ Ln(ℝn), where C > 0 is related to the constant appearing in the Sobolev–Gagliardo–Nirenberg inequality for functions with bounded variation (BV functions).
This review paper is concerned with the stability analysis of the continuity equation in the DiPerna–Lions setting in which the advecting velocity field is Sobolev regular. Quantitative estimates for the equation were derived only recently, but optimality was not discussed. We revisit the results from our 2017 paper, compare the new estimates with previously known estimates for Lagrangian flows and demonstrate how these can be applied to produce optimal bounds in applications from physics, engineering and numerical analysis.
The scope of the paper is twofold. We show that for a large class of measurable vector fields in the sense of Weaver (i.e. derivations over the algebra of Lipschitz functions), called in the paper laminated, the notion of integral curves may be naturally defined and characterized (when appropriate) by an ordinary differential equation. We further show that for such vector fields the notion of a flow of the given positive Borel measure similar to the classical one generated by a smooth vector field (in a space with smooth structure) may be defined in a reasonable way, so that the measure ‘flows along’ the appropriately understood integral curves of the given vector field and the classical continuity equation is satisfied in the weak sense.
We consider an initial–boundary value problem that involves a partial differential equation with a functional term. The problem is motivated by a cell division model for size structured cell cohorts in which growth and division occur. Although much is known about the large time asymptotic behaviour of solutions to these problems for constant growth rates, general solution techniques are rare. We analyse the case where the growth rate is linear and the division rate is a monomial, and we develop a method to determine the general solution for a general class of initial data. The large time dynamics of solutions for this case are significantly different from the constant growth rate case. We show that solutions approach a time-dependent attracting solution that is periodic in the time variable.
We consider the ergodic (or additive eigenvalue) problem for the Neumann-type boundary-value problem for Hamilton–Jacobi equations and the corresponding discounted problems. Denoting by uλ the solution of the discounted problem with discount factor λ > 0, we establish the convergence of the whole family to a solution of the ergodic problem as λ → 0, and give a representation formula for the limit function via the Mather measures and Peierls function. As an interesting by-product, we introduce Mather measures associated with Hamilton–Jacobi equations with the Neumann-type boundary conditions. These results are variants of the main results in a recent paper by Davini et al., who study the same convergence problem on smooth compact manifolds without boundary.