Structured coagulation-fragmentation equation in the space of radon measures: Unifying discrete and continuous models

We present a structured coagulation-fragmentation model which describes the population dynamics of oceanic phytoplankton. This model is formulated on the space of Radon measures equipped with the bounded Lipschitz norm and unifies the study of the discrete and continuous coagulation-fragmentation models. We prove that the model is well-posed and show it can reduce down to the classic discrete and continuous coagulation-fragmentation models. To understand the interplay between the physical processes of coagulation and fragmentation and the biological processes of growth, reproduction, and death, we establish a regularity result for the solutions and use it to show that stationary solutions are absolutely continuous under some conditions on model parameters. We develop a semi-discrete approximation scheme which conserves mass and prove its convergence to the unique weak solution. We then use the scheme to perform numerical simulations for the model.

Global well-posedness and long-time behavior of the fractional NLS

In this paper, our discussion mainly focuses on equations with energy supercritical nonlinearities. We establish probabilistic global well-posedness (GWP) results for the cubic Schrödinger equation with any fractional power of the Laplacian in all dimensions. We consider both low and high regularities in the radial setting, in dimension $$\ge 2$$ ≥ 2 . In the high regularity result, an Inviscid - Infinite dimensional (IID) limit is employed while in the low regularity global well-posedness result, we make use of the Skorokhod representation theorem. The IID limit is presented in details as an independent approach that applies to a wide range of Hamiltonian PDEs. Moreover we discuss the adaptation to the periodic settings, in any dimension, for smooth regularities.

Variational Approach to Regularity of Optimal Transport Maps: General Cost Functions

We extend the variational approach to regularity for optimal transport maps initiated by Goldman and the first author to the case of general cost functions. Our main result is an $$\epsilon $$ ϵ -regularity result for optimal transport maps between Hölder continuous densities slightly more quantitative than the result by De Philippis–Figalli. One of the new contributions is the use of almost-minimality: if the cost is quantitatively close to the Euclidean cost function, a minimizer for the optimal transport problem with general cost is an almost-minimizer for the one with quadratic cost. This further highlights the connection between our variational approach and De Giorgi’s strategy for $$\epsilon $$ ϵ -regularity of minimal surfaces.

Fractional diffusion-wave equations: Hidden regularity for weak solutions

We prove a “hidden” regularity result for weak solutions of time fractional diffusion-wave equations where the Caputo fractional derivative is of order α ∈ (1, 2). To establish such result we analyse the regularity properties of the weak solutions in suitable interpolation spaces.

Local regularity for concave homogeneous complex degenerate elliptic equations dominating the Monge–Ampère equation

In this paper, we establish a local regularity result for $$W^{2,p}_{{\mathrm {loc}}}$$ W loc 2 , p solutions to complex degenerate nonlinear elliptic equations $$F(D^2_{\mathbb {C}}u)=f$$ F ( D C 2 u ) = f when they dominate the Monge–Ampère equation. Notably, we apply our result to the so-called k-Monge–Ampère equation.

Maximal Regularity for Non-autonomous Evolutionary Equations

We discuss the issue of maximal regularity for evolutionary equations with non-autonomous coefficients. Here evolutionary equations are abstract partial-differential algebraic equations considered in Hilbert spaces. The catch is to consider time-dependent partial differential equations in an exponentially weighted Hilbert space. In passing, one establishes the time derivative as a continuously invertible, normal operator admitting a functional calculus with the Fourier–Laplace transformation providing the spectral representation. Here, the main result is then a regularity result for well-posed evolutionary equations solely based on an assumed parabolic-type structure of the equation and estimates of the commutator of the coefficients with the square root of the time derivative. We thus simultaneously generalise available results in the literature for non-smooth domains. Examples for equations in divergence form, integro-differential equations, perturbations with non-autonomous and rough coefficients as well as non-autonomous equations of eddy current type are considered.

A regularity result for minimal configurations of a free interface problem

We prove a regularity result for minimal configurations of variational problems involving both bulk and surface energies in some bounded open region $$\varOmega \subseteq {\mathbb {R}}^n$$ Ω ⊆ R n . We will deal with the energy functional $${\mathscr {F}}(v,E):=\int _\varOmega [F(\nabla v)+1_E G(\nabla v)+f_E(x,v)]\,dx+P(E,\varOmega )$$ F ( v , E ) : = ∫ Ω [ F ( ∇ v ) + 1 E G ( ∇ v ) + f E ( x , v ) ] d x + P ( E , Ω ) . The bulk energy depends on a function v and its gradient $$\nabla v$$ ∇ v . It consists in two strongly quasi-convex functions F and G, which have polinomial p-growth and are linked with their p-recession functions by a proximity condition, and a function $$f_E$$ f E , whose absolute valuesatisfies a q-growth condition from above. The surface penalization term is proportional to the perimeter of a subset E in $$\varOmega $$ Ω . The term $$f_E$$ f E is allowed to be negative, but an additional condition on the growth from below is needed to prove the existence of a minimal configuration of the problem associated with $${\mathscr {F}}$$ F . The same condition turns out to be crucial in the proof of the regularity result as well. If (u, A) is a minimal configuration, we prove that u is locally Hölder continuous and A is equivalent to an open set $${\tilde{A}}$$ A ~ . We finally get $$P(A,\varOmega )={\mathscr {H}}^{n-1}(\partial {\tilde{A}}\cap \varOmega $$ P ( A , Ω ) = H n - 1 ( ∂ A ~ ∩ Ω ).