This paper considers two types of boundary control problems for linear transport equations. The first one shows that transport solutions on a subdomain of a domain X can be controlled exactly from incoming boundary conditions for X under appropriate convexity assumptions. This is in contrast with the only approximate control one typically obtains for elliptic equations by an application of a unique continuation property, a property which we prove does not hold for transport equations. We also consider the control of an outgoing solution from incoming conditions, a transport notion similar to the Dirichlet-to-Neumann map for elliptic equations. We show that for well-chosen coefficients in the transport equation, this control may not be possible. In such situations and by (Fredholm) duality, we obtain the existence of non-trivial incoming conditions that are compatible with vanishing outgoing conditions.
In this paper we develop a boundary $\varepsilon$-regularity theory for optimal transport maps between bounded open sets with $C^{1,α}$-boundary. Our main result asserts sharp $C^{1,α}$-regularity of transport maps at the boundary in form of a linear estimate under certain assumptions: The main quantitative assumptions are that the local nondimensionalized transport cost is small and that the boundaries are locally almost flat in $C^{1,α}$. Our method is completely variational and builds on the recently developed interior regularity theory.
We consider reaction-diffusion systems where components diffuse inside the domain and react on the surface through mass transport type boundary conditions on an evolving domain. Using Lyapunov functional and duality arguments, we establish the existence of component-wise non-negative global solutions.
This work studies the inverse boundary problem for the two photon absorption radiative transport equation. We show that the absorption coefficients and scattering coefficients can be uniquely determined from the \emph{albedo} operator. If scattering is absent, we do not require smallness of the incoming source and the reconstructions of the absorption coefficients are explicit.
This paper provides a new tauberian approach to the study of quantitative time asymptotics of collisionless transport semigroups with general diffuse boundary operators. We obtain an (almost) optimal algebraic rate of convergence to equilibrium under very general assumptions on the initial datum and the boundary operator. The rate is prescribed by the maximal gain of integrability that the boundary operator is able to induce. The proof relies on a representation of the collisionless transport semigroups by a (kind of) Dyson-Phillips series and on a fine analysis of the trace on the imaginary a…
We study the continuous version of a hyperbolic rescaling of a discrete game, called open mancala. The resulting PDE turns out to be a singular transport equation, with a forcing term taking values in $\{0,1\}$, and discontinuous in the solution itself. We prove existence and uniqueness of a certain formulation of the problem, based on a nonlocal equation satisfied by the free boundary dividing the region where the forcing is one (active region) and the region where there is no forcing (tail region). Several examples, most notably the Riemann problem, are provided, related to singularity forma…
We prove that exponential moments of a fluctuation of the pure transport equation decay pointwisely almost as fast as $t^{-3}$ when the domain is any general strictly convex subset of $\mathbb{R}^3$ with the smooth boundary of the diffuse boundary condition. We prove the theorem by establishing a novel $L^1$-$L^\infty$ framework via stochastic cycles.
The free transport operator of probability density function $f(t,x,v)$ is one the most fundamental operator which is widely used in many areas of PDE theory including kinetic theory, in particular. When it comes to general boundary problems in kinetic theory, however, it is well-known that high order regularity is very hard to obtain in general. In this paper, we study the free transport equation in a disk with the specular boundary condition. We obtain initial-boundary compatibility conditions for $C^1_{t,x,v}$ and $C^2_{t,x,v}$ regularity of the solution. We also provide regularity estimates…
We prove existence and uniqueness of solutions to the initial-boundary value problem for the Lifshitz--Slyozov equation (a nonlinear transport equation on the half-line), focusing on the case of kinetic rates with unbounded derivative at the origin. Our theory covers in particular those cases with rates behaving as power laws at the origin, for which an inflow behavior is expected and a boundary condition describing nucleation phenomena needs to be imposed. The method we introduce here to prove existence is based on a formulation in terms of characteristics, with a careful analysis on the beha…
The justification of hydrodynamic limits in non-convex domains has long been an open problem due to the singularity at the grazing set. In this paper, we investigate the unsteady neutron transport equation in a general bounded domain with the in-flow, diffuse-reflection, or specular-reflection boundary condition. Using a novel kernel estimate, we demonstrate the optimal $L^2$ diffusive limit in the presence of both initial and boundary layers. Previously, this result was only proved for convex domains when the time variable is involved. Our approach is highly robust, making it applicable to al…
We show that that the stochastic 3D primitive equations with either the physical boundary conditions or Neumann boundary conditions on the top and bottom and Dirichlet boundary condition on the sides driven by multiplicative gradient-dependent white noise have unique maximal strong solutions both in stochastic and PDE sense under certain assumptions on the growth of the noise. For the latter boundary conditions global existence is established using an argument based on decomposition of vertical velocity to barotropic and baroclinic modes and an iterated stopping time argument. An explicit exam…
We consider an inverse source problem in the stationary radiating transport through a two dimensional absorbing and scattering medium. Of specific interest, the exiting radiation is measured on an arc. The attenuation and scattering properties of the medium are assumed known. For scattering kernels of finite Fourier content in the angular variable, we show how to quantitatively recover the part of the isotropic sources restricted to the convex hull of the measurement arc. The approach is based on the Cauchy problem with partial data for a Beltrami-like equation associated with $A$-analytic map…
In this paper, we prove a maximum principle for the general multi-term space-time-fractional transport equation and apply it for establishing uniqueness of solution to an initial-boundary-value problem for this equation. We also derive some comparison principles for solutions to the initial-boundary-value problems with different problem data. Finally, we present a maximum principle for the Cauchy problem for a time-fractional transport equation on an unbounded domain.
Consider neutron transport equations in 3D convex domains with in-flow boundary. We mainly study the asymptotic limits as the Knudsen number $ε\rightarrow 0^+$. Using Hilbert expansion, we rigorously justify that the solution of steady problem converges to that of the Laplace's equation, and the solution of unsteady problem converges to that of the heat equation. The proof relies on a detailed analysis on the boundary layer effect with geometric correction. This problem can be formulated in many different settings, and the above one is probably the most physically significant and most mathemat…
We provide a new perturbation theorem for substochastic semigroups on abstract AL spaces extending Kato's perturbation theorem to non-densely defined operators. We show how it can be applied to piecewise deterministic Markov processes and transport equations with abstract boundary conditions. We give particular examples to illustrate our results.
This paper studies a variant of ramified/branched optimal transportation problems. Given the distributions of production capacities and market sizes, a firm looks for an allocation of productions over factories, a distribution of sales across markets, and a transport path that delivers the product to maximize its profit. Mathematically, given any two measures $μ$ and $ν$ on $X$, and a payoff function $h$, the planner wants to minimize $\mathbf{M}_{α}(T)-\int_{X}hd(\partial T)$ among all transport paths $T$ from $\tildeμ$ to $\tildeν$ with $\tildeμ\leq μ$ and $\tildeν\leq ν$, where $\mathbf{M}_…
In most results concerning bounds on the heat transport in the Rayleigh-Bénard convection problem no-slip boundary conditions for the velocity field are assumed. Nevertheless it is debatable, whether these boundary conditions reflect the behavior of the fluid at the boundary. This problem is important in theoretical fluid mechanics as well as in industrial applications, as the choice of boundary conditions has effects in the description of the boundary layers and its properties. In fact, different boundary conditions may inhibit or enhance heat transport. This review presents a selection of co…
Summability estimates on transport densities with dirichlet regions on the boundary via symmetrization techniques
本文考虑有界区域$Ω$中的质量运输问题，其中内部的正质量f +被发送到边界$\partialΩ$，例如出现在一些形状最优化问题中，并且我们证明了相关的输运密度$σ$的可和性估计，它是从扩散测度到边界上测度的输运密度f -- = P f +（P是边界上的投影），因此是奇异的。通过对称化技巧，只要$Ω$是凸的或满足一致的外球条件，我们就证明L p估计（如果f + $\in$ L p，则$σ$ $\in$ L p）。最后，通过反例证明了如果f + $\in$ L $\infty$ $(Ω)$和f --有界密度w.r.在$\partialΩ$上的表面测量，f +和f -之间的输运密度$σ$不一定在L $\infty$ $(Ω)$中，这意味着f - = P \# f +的事实是至关重要的。
An initial-boundary value problem describing moisture transport in porous media: existence of strong solutions and an error estimate for a finite volume scheme
Transport-collapse scheme for scalar conservation laws -- initial and boundary value problems
University of Zagreb, Faculty of Mathematics and Natural Sciences（萨格勒布大学数学与自然科学学院）；University of Zagreb, Faculty of Electrical Engineering（萨格勒布大学电气工程学院）
We study the consequences of the equivalence between the least gradient problem and a boundary-to-boundary optimal transport problem in two dimensions. We extend the relationship between the two problems to their respective dual problems, as well as prove several regularity and stability results for the least gradient problem using optimal transport techniques.
