We consider the $L^\infty$-optimal mass transportation problem \[ \min_{Π(μ, ν)} γ-\mathrm{ess\,sup\,} c(x,y), \] for a new class of costs $c(x,y)$ for which we introduce a tentative notion of twist condition. In particular we study the conditions under which the infinitely-motonone minimizers are induced by a transportation map. We also state a uniqueness result for infinitely cyclically monotone Monge minimizers that corresponds to this class of cost functions. We compare the results to previous works.
We establish that for any non-empty, compact set $K\subset\mathbb{R}_{\mathrm{sym}}^{3\times 3}$ the $1$- and $\infty$-symmetric div-quasiconvex hulls $K^{(1)}$ and $K^{(\infty)}$ coincide. This settles a conjecture in a recent work of Conti, Müller and Ortiz (Symmetric Div-Quasiconvexity and the Relaxation of Static Problems. Arch. Ration. Mech. Anal. 235(2):841-880) in the affirmative. As a key novelty, we construct an $\mathrm{L}^{\infty}$-truncation that preserves both symmetry and solenoidality of matrix-valued maps in $\mathrm{L}^{1}$.
We isolate a general condition, that we call "localization principle", on the integrand L:\MM\to[0,\infty], assumed to be continuous, under which W^{1,q}-quasiconvexity with q\in[1,\infty] is a sufficient condition for I(u)=\int_ΩL(\nabla u(x))dx to be sequentially weakly lower semicontinuous on W^{1,p}(Ω;\RR^m) with p\in]1,\infty[. Some applications are given.
We are concerned with the relaxation and existence theories of a general class of geometrical minimisation problems, with action integrals defined via differential forms over fibre bundles. We find natural algebraic and analytic conditions which give rise to a relaxation theory. Moreover, we propose the notion of ``Riemannian quasiconvexity'' for cost functions whose variables are differential forms on Riemannian manifolds, which extends the classical quasiconvexity condition in the Euclidean settings. The existence of minimisers under the Riemannian quasiconvexity condition has been establish…
We establish $\mathrm{C}^{\infty}$-partial regularity results for relaxed minimizers of strongly quasiconvex functionals \begin{align*} \mathscr{F}[u;Ω]:=\int_ΩF(\nabla u)\,\mathrm{d} x,\qquad u\colonΩ\to\mathbb{R}^{N}, \end{align*} subject to a $q$-growth condition $|F(z)|\leq c(1+|z|^{q})$, $z\in\mathbb{R}^{N\times n}$, and natural $p$-mean coercivity conditions on $F\in\mathrm{C}^{\infty}(\mathbb{R}^{N\times n})$ for the basically optimal exponent range $1\leq p\leq q<\min\{\frac{np}{n-1},p+1\}$. With the $p$-mean coercivity condition being stated in terms of a strong quasiconvexity conditi…
We study minimisation problems in $L^\infty$ for general quasiconvex first order functionals, where the class of admissible mappings is constrained by the sublevel sets of another supremal functional and by the zero set of a nonlinear operator. Examples of admissible operators include those expressing pointwise, unilateral, integral isoperimetric, elliptic quasilinear differential, jacobian and null Lagrangian constraints. Via the method of $L^p$ approximations as $p\to \infty$, we illustrate the existence of a special $L^\infty$ minimiser which solves a divergence PDE system involving certain…
We consider conformally invariant energies $W$ on the group $\operatorname{GL}^+(2)$ of $2\times2$-matrices with positive determinant, i.e. $W\colon\operatorname{GL}^+(2)\to\mathbb{R}$ such that \[W(AFB) = W(F) \qquad\text{for all }\; A,B\in\{aR\in\operatorname{GL}^+(2) \,|\, a\in(0,\infty)\,,\; R\in\operatorname{SO}(2)\}\,,\] where $\operatorname{SO}(2)$ denotes the special orthogonal group, and provide an explicit formula for the (notoriously difficult to compute) quasiconvex envelope of these functions. Our results, which are based on the representation $W(F)=h(\frac{λ_1}{λ_2})$ of $W$ in t…
We propose an entropic approximation approach for optimal transportation problems with a supremal cost. We establish $Γ$-convergence for suitably chosen parameters for the entropic penalization and that this procedure selects $\infty$-cyclically monotone plans at the limit. We also present some numerical illustrations performed with Sinkhorn's algorithm.
For cost functions $c(x,y)=h(x-y)$ with $h\in C^2$ homogeneous of degree $p\geq 2$, we show $L^\infty$-estimates of $Tx-x$ on balls, where $T$ is an $h$-monotone map. Estimates for the interpolating mappings $T_t=t(T-I)+I$ are deduced from this.
We show that in any complete metric space the probability measures $μ$ with compact and connected support are the ones having the property that the optimal tranportation distance to any other probability measure $ν$ living on the support of $μ$ is bounded below by a positive function of the $L^\infty$ transportation distance between $μ$ and $ν$. The function giving the lower bound depends only on the lower bound of the $μ$-measures of balls centered at the support of $μ$ and on the cost function used in the optimal transport. We obtain an essentially sharp form of this function. In the case of…
In this article we characterize the $\mathrm{L}^\infty$ eigenvalue problem associated to the Rayleigh quotient $\left.{\|\nabla u\|_{\mathrm{L}^\infty}}\middle/{\|u\|_\infty}\right.$ and relate it to a divergence-form PDE, similarly to what is known for $\mathrm{L}^p$ eigenvalue problems and the $p$-Laplacian for $p<\infty$. Contrary to existing methods, which study $\mathrm{L}^\infty$-problems as limits of $\mathrm{L}^p$-problems for $p\to\infty$, we develop a novel framework for analyzing the limiting problem directly using convex analysis and geometric measure theory. For this, we derive a …
We show that, if $b\in L^1(0,T;L^1_{\mathrm{loc}}(\mathbb{R}))$ has spatial derivative in the John-Nirenberg space $\mathrm{BMO}(\mathbb{R})$, then it generalizes a unique flow $ϕ(t,\cdot)$ which has an $A_\infty(\mathbb R)$ density for each time $t\in [0,T]$. Our condition on the map $b$ is optimal and we also get a sharp quantitative estimate for the density. As a natural application we establish a well-posedness for the Cauchy problem of the transport equation in $\mathrm{BMO}(\mathbb R)$.
We consider a transport-diffusion equation of the form $\partial_t θ+v \cdot \nabla θ+ ν\A θ=0$, where $v$ is a given time-dependent vector field on $\mathbb R^d$. The operator $\A$ represents log-modulated fractional dissipation: $\A=\frac {|\nabla|^γ}{\log^β(λ+|\nabla|)}$ and the parameters $ν\ge 0$, $β\ge 0$, $0\le γ\le 2$, $λ>1$. We introduce a novel nonlocal decomposition of the operator $\A$ in terms of a weighted integral of the usual fractional operators $|\nabla|^{s}$, $0\le s \le γ$ plus a smooth remainder term which corresponds to an $L^1$ kernel. For a general vector field $v$ (pos…
Consider the steady neutron transport equation in 2D convex domains with in-flow boundary condition. In this paper, we establish the diffusive limit while the boundary layers are present. Our contribution relies on a delicate decomposition of boundary data to separate the regular and singular boundary layers, novel weighted $W^{1,\infty}$ estimates for the Milne problem with geometric correction in convex domains, as well as an $L^{2m}-L^{\infty}$ framework which yields stronger remainder estimates.
In this paper, we revisit the notion of temporal intermittency to obtain sharp nonuniqueness results for linear transport equations. We construct divergence-free vector fields with sharp Sobolev regularity $L^1_t W^{1,p}$ for all $p<\infty$ in space dimensions $d\geq 2$ whose transport equations admit nonunique weak solutions belonging to $L^p_tC^k$ for all $p<\infty$ and $k\in \mathbb{N}$. In particular, our result shows that the time-integrability assumption in the uniqueness of the DiPerna-Lions theory is sharp. The same result also holds for transport-diffusion equations with diffusion ope…
We examine the regularity of weak solutions of quasi-geostrophic (QG) type equations with supercritical ($α<1/2$) dissipation $(-Δ)^α$. This study is motivated by a recent work of Caffarelli and Vasseur, in which they study the global regularity issue for the critical ($α= 1/2$) QG equation \cite{CV}. Their approach successively increases the regularity levels of Leray-Hopf weak solutions: from $L^2$ to $L^\infty$, from $L^\infty$ to Hölder ($C^δ$, $δ>0$), and from Hölder to classical solutions. In the supercritical case, Leray-Hopf weak solutions can still be shown to be $L^\infty$, but it do…
In this manuscript we study the following optimization problem with volume constraint: \[ \min\left\{\frac{1}{p}\int_Ω |\nabla v|^pdx- \int_{\partial Ω} gv\,dS \colon v \in W^{1, p} \left(Ω\right), \text{ and } |\{v>0\}| \leq α\right\}. \] Here $g$ is acontinuous function and $α$ is a fixed constant such that $0< α< |Ω|$. Under the assumption that $\displaystyle \int_{\partial Ω} g(x)dS >0$ we prove that a minimizer exists and satisfies $$ \left\{ \begin{array}{ccl} -Δ_p u_p = 0 &\text{in } &\{u_p>0\} \cup \{u_p<0\}, \\[5pt] |\nabla u_p|^{p-2}\frac{\partial u_p}{\partial ν} = g & \text{on } &\…
We face the well-posedness of linear transport Cauchy problems $$\begin{cases}\dfrac{\partial u}{\partial t} + b\cdot\nabla u + c\,u = f&(0,T)\times{\mathbb R}^n\\u(0,\cdot)=u_0\in L^\infty&{\mathbb R}^n\end{cases}$$ under borderline integrability assumptions on the divergence of the velocity field $b$. For $W^{1,1}_{loc}$ vector fields $b$ satisfying $\frac{|b(x,t)|}{1+|x|}\in L^1(0,T; L^1)+L^1(0,T; L^\infty)$ and $$\operatorname{div} b\in L^1(0,T;L^\infty) + L^1\left(0,T; \operatorname{Exp}\left(\frac{L}{\log L}\right)\right),$$ we prove existence and uniqueness of weak solutions. Moreover, …
A linear stochastic transport equation with non-regular coefficients is considered. Under the same assumption of the deterministic theory, all weak $L^\infty$-solutions are renormalized. But then, if the noise is nondegenerate, uniqueness of weak $L^\infty$-solutions does not require essential new assumptions, opposite to the deterministic case where for instance the divergence of the drift is asked to be bounded. The proof gives a new explanation why bilinear multiplicative noise may have a regularizing effect.
We consider the Stokes-transport system, a model for the evolution of an incompressible viscous fluid with inhomogeneous density. This equation was already known to be globally well-posed for any $L^1\cap L^\infty$ initial density with finite first moment in $\mathbb{R}^3$. We show that similar results hold on different domain types. We prove that the system is globally well-posed for $L^\infty$ initial data in bounded domains of $\mathbb{R}^2$ and $\mathbb{R}^3$ as well as in the infinite strip $\mathbb{R}\times(0,1)$. These results contrast with the ill-posedness of a similar problem, the in…
In this paper we study the BV regularity for solutions of variational problems in Optimal Transportation. As an application we recover BV estimates for solutions of some non-linear parabolic PDE by means of optimal transportation techniques. We also prove that the Wasserstein projection of a measure with BV density on the set of measures with density bounded by a given BV function f is of bounded variation as well. In particular, in the case f = 1 (projection onto a set of densities with an L^\infty bound) we precisely prove that the total variation of the projection does not exceed the total …
In this paper we study the transport equation in $\mathbb{R}^n \times (0,T)$, $T >0$, \[ \partial _t f + v\cdot \nabla f = g, \quad f(\cdot ,0)= f_0 \quad \text{in}\quad \mathbb{R}^n \] in generalized Campanato spaces $\mathscr{L}^s_{ q(p, N)}(\mathbb{R}^n)$. The critical case is particularly interesting, and is applied to the local well-posedness problem in a space close to the Lipschitz space in our companion paper\cite{cw}. More specifically, in the critical case $s=q=N=1$ we have the embedding relations, $B^1_{\infty, 1}(\Bbb R^n) \hookrightarrow \mathscr{L}^{ 1}_{ 1(p, 1)}(\mathbb{R}^n) \…
We study in this article the existence and uniqueness of solutions to a class of stochastic transport equations with irregular coefficients and unbounded divergence. In the first result we assume the drift is $L^{2}([0,T] \times \R^{d})\cap L^{\infty}([0,T] \times \R^{d})$ and the divergence is the locally integrable. In the second result we show that the smoothing acts as a selection criterion when the drift is in $L^{2}([0,T] \times \R^{d})\cap L^{\infty}([0,T] \times \R^{d})$ without any condition on the divergence.
We study global existence and uniqueness of solutions to instationary inhomogeneous Navier-Stokes equations on bounded domains of $\R^n, n\geq 3$, with initial velocity in $B^0_{q,\infty}(\Om)$, $q\geq n$, and piecewise constant initial density. \par To this end, first, existence for momentum equations with prescribed density is obtained based on maximal $L^\infty_\ga$-regularity of the Stokes operator in little Nicolskii space $b^{s}_{q,\infty}(\Om)$, $s\in\R$, exploited in \cite{RiZh14} and existence for divergence problem in $b^{-s}_{q,\infty}(\Om)$, $s>0$. Then, we obtain an existence resu…
For $0<p<+\infty$, we prove a global $W^{2,p}$-estimate for potentials of optimal transport maps between convex domains in the plane. Among the tools developed for that purpose are obliqueness in general convex domains and estimates for the growth of eccentricity of sections of the potentials.
We find interpretation using optimal mass transport theory for eigenvalue problems obtained as limits of the eigenvalue problems for the fractional $p-$Laplacian operators as $p\to +\infty$. We deal both with Dirichlet and Neumann boundary conditions.
In this paper we study the limit as $p\to \infty$ in the evolution problem driven by the $p-$Laplacian with dynamical boundary conditions. We prove that the natural energy functional associated with this problem converges to a limit in the sense of Mosco convergence and as a consequence we obtain convergence of the solutions to the evolution problems. For the limit problem we show an interpretation in terms of optimal mass transportation and provide examples of explicit solutions for some particular data.
In this article, we show that prescribing homogeneous Neumann type numerical boundary conditions at an outflow boundary yields a convergent discretization in $\ell^\infty$ for transport equations. We show in particular that the Neumann numerical boundary condition is a stable, local, and absorbing numerical boundary condition for discretized transport equations. Our main result is proved for explicit two time level numerical approximations of transport operators with arbitrarily wide stencils. The proof is based on the energy method and bypasses any normal mode analysis.
We study a 1D transport equation with nonlocal velocity with subcritical or supercritical dissipation. For all data in the weighted Sobolev space $H^{k}(w_{λ,κ}) \cap L^{\infty},$ with $k=\max(0,3/2-α)$ and $w_{λ, κ}$ is a given family of Muckenhoupt weights. We prove a global existence result in the subcritical case $α\in (1,2)$. We also prove a local existence theorem for large data in $H^{2}(w_{λ, κ})\cap L^{\infty}$ in the supercritical case $α\in (0,1)$. The proofs are based on the use of the weighted Littlewood-Paley theory, interpolation along with some new commutator estimates.
Fix a pair of smooth source and target densities $ρ$ and $ρ^*$ of equal mass, supported on bounded domains $Ω, Ω^* \subset \mathbb{R}^n$. Also fix a cost function $c_0 \in C^{4,α}(\overlineΩ \times \overline{Ω^*})$ satisfying the weak regularity criterion of Ma, Trudinger, and Wang, and assume $Ω$ and $Ω^*$ are uniformly $c_0$- and $c_0^*$-convex with respect to each other. We consider a parabolic version of the optimal transport problem between $(Ω,ρ)$ and $(Ω^*,ρ^*)$ when the cost function $c$ is a sufficiently small $C^4$ perturbation of $c_0$, and where the size of the perturbation depends…
Consider the unsteady neutron transport equation with diffusive boundary condition in 2D convex domains. We establish the diffusive limit with both initial layer and boundary layer corrections. The major difficulty is the lack of regularity in the boundary layer with geometric correction. Our contribution relies on a detailed analysis of asymptotic expansions inspired by the compatibility condition and an intricate $L^{2m}-L^{\infty}$ framework which yields stronger remainder estimates.
We consider the diffusive limit of a steady neutron transport equation with one-speed velocity in a two-dimensional annulus. A classical theorem states that the solution can be approximated in $L^{\infty}$ by the leading order interior solution plus the corresponding Knudsen layers in the diffusive limit. In this paper, we construct a counterexample of this result via a different boundary layer expansion with geometric correction.
We investigate the convergence rate of the optimal entropic cost $v_\varepsilon$ to the optimal transport cost as the noise parameter $\varepsilon \downarrow 0$. We show that for a large class of cost functions $c$ on $\mathbb{R}^d\times \mathbb{R}^d$ (for which optimal plans are not necessarily unique or induced by a transport map) and compactly supported and $L^{\infty}$ marginals, one has $v_\varepsilon-v_0= \frac{d}{2} \varepsilon \log(1/\varepsilon)+ O(\varepsilon)$. Upper bounds are obtained by a block approximation strategy and an integral variant of Alexandrov's theorem. Under an infin…
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.
We study the existence and uniqueness of solution of a evolutionary partial differential equation originating from the continuum limit of a coupled process of totally asymmetric simple exclusion process (TASEP) and Langmuir kinetics (LK). In the fields of physics and biology, the TASEP-LK coupled process has been extensively studied by Monte Carlo simulations, numerical computations, and detailed experiments. However, no rigorous mathematical analysis so far has been given for the corresponding differential equations, especially the existence and uniqueness of their solutions. In this paper, w…
The present paper is the continuation of work [14], devoted to the study of an inviscid zero-Mach number system in the framework of \emph{endpoint} Besov spaces of type $B^s_{\infty,r}(\mathbb{R}^d)$, $r\in [1,\infty]$, $d\geq 2$, which can be embedded in the Lipschitz class $C^{0,1}$. In particular, the largest case $B^1_{\infty,1}$ and the case of Hölder spaces $C^{1,α}$ are permitted. The local in time well-posedness result is proved, under an additional $L^2$ hypothesis on the initial inhomogeneity and velocity field. A new a priori estimate for parabolic equations in endpoint spaces $B^s_…
In this work, we demonstrate well-posedness and regularisation by noise results for a class of geometric transport equations that contains, among others, the linear transport and continuity equations. This class is known as linear advection of $k$-forms. In particular, we prove global existence and uniqueness of $L^p$-solutions to the stochastic equation, driven by a spatially $α$-Hölder drift $b$, uniformly bounded in time, with an integrability condition on the distributional derivative of $b$, and sufficiently regular diffusion vector fields. Furthermore, we prove that all our solutions are…
In this paper we consider the Cauchy problem for 2D viscous shallow water system in Besov spaces. We firstly prove the local well-posedness of this problem in $B^s_{p,r}(\mathbb{R}^2)$, $s>max\{1,\frac{2}{p}\}$, $1\leq p,r\leq \infty$ by using the Littlewood-Paley theory, the Bony decomposition and the theories of transport equations and transport diffusion equations. Then we can prove the global existence of the system with small enough initial data in $B^s_{p,r}(\mathbb{R}^2)$, $1\leq p\leq2$, $1\leq r<\infty$ and $s>\frac{2}{p}$. Our obtained results generalize and cover the recent results …
In this contribution we prove the existence of weak solutions to degenerate parabolic systems arising from the coupled moisture movement, transport of dissolved species and heat transfer through partially saturated porous materials. Physically motivated mixed Dirichlet-Neumann boundary conditions and initial conditions are considered. Existence of a global weak solution of the problem is proved by means of semidiscretization in time and by passing to the limit from discrete approximations. Degeneration occurs in the nonlinear transport coefficients which are not assumed to be bounded below and…
In this paper we prove the existence of weak solutions to degenerate parabolic systems arising from the fully coupled moisture movement, solute transport of dissolved species and heat transfer through porous materials. Physically relevant mixed Dirichlet-Neumann boundary conditions and initial conditions are considered. Existence of a global weak solution of the problem is proved by means of semidiscretization in time, proving necessary uniform estimates and by passing to the limit from discrete approximations. Degeneration occurs in the nonlinear transport coefficients which are not assumed t…
This paper considers the neutron transport equation in bounded domain with a combination of the diffusive boundary condition and the in-flow boundary condition. We firstly study the existence of solution in any fixed time by $L^2-L^{\infty}$ method, which was established to study Boltzmann equation in \cite{[Guo2]}. Based on the uniform estimates of the solution, we also consider the BV-regularity of the solution in non-convex domain. A cut-off function, which aims to exclude all the characteristics emanating from the grazing set $\mathfrak{S}_B$, has been constructed precisely.
We investigate the spectral properties of the time-dependent linear transport equation with bounce-back boundary conditions. A fine analysis of the spectrum of the streaming operator is given and the explicit expression of the strongly continuous streaming semigroup is derived. Next, making use of a recent result from M. Sbihi \cite{Sbihi}, we prove, via a compactness argument, that the essential spectrum of the transport semigroup and that of the streaming semigroup coincide on all $L^p$-spaces with $1<p<\infty$.
In this paper we develop a time reversal method for the radiative transport equation to solve two problems: an inverse problem for the recovery of an initial condition from boundary measurements, and the exact boundary controllability of the transport field with finite steering time. Absorbing and scattering effects, modeled by coefficients with low regularity, are incorporated in the formulation of these problems. This time reversal approach leads to a convergent iterative procedure to reconstruct the initial condition provided that the scattering coefficient is sufficiently small in the $L^{…
We consider the linear transport equations driven by an incompressible flow in dimensions $d\geq 3$. For divergence-free vector fields $u \in L^1_t W^{1,q}$, the celebrated DiPerna-Lions theory of the renormalized solutions established the uniqueness of the weak solution in the class $L^\infty_t L^p$ when $\frac{1}{p} + \frac{1}{q} \leq 1$. For such vector fields, we show that in the regime $\frac{1}{p} + \frac{1}{q} > 1$, weak solutions are not unique in the class $ L^1_t L^p$. One crucial ingredient in the proof is the use of both temporal intermittency and oscillation in the convex integrat…
We revisit our study of general transport operator with general force field and general invariant measure by considering, in the $L^1$ setting, the linear transport operator $\T_H$ associated to a linear and positive boundary operator $H$ of unit norm. It is known that in this case an extension of $\T_H$ generates a substochastic (i.e. positive contraction) $C_0$-semigroup $(V_H(t))_{t\geq 0}$. We show here that $(V_H(t))_{t\geq 0}$ is the smallest substochastic $C_0$-semigroup with the above mentioned property and provides a representation of $(V_H(t))_{t \geq 0}$ as the sum of an expansion s…
A recent paper by Cordero-Erausquin and Klartag provides a characterization of the measures $μ$ on $\R^d$ which can be expressed as the moment measures of suitable convex functions $u$, i.e. are of the form $(\nabla u)\_\\#e^{- u}$ for $u:\R^d\to\R\cup\{+\infty\}$ and finds the corresponding $u$ by a variational method in the class of convex functions. Here we propose a purely optimal-transport-based method to retrieve the same result. The variational problem becomes the minimization of an entropy and a transport cost among densities $ρ$ and the optimizer $ρ$ turns out to be $e^{-u}$. This req…
This article addresses regularity of optimal transport maps for cost="squared distance" on Riemannian manifolds that are products of arbitrarily many round spheres with arbitrary sizes and dimensions. Such manifolds are known to be non-negatively cross-curved [KM2]. Under boundedness and non-vanishing assumptions on the transfered source and target densities we show that optimal maps stay away from the cut-locus (where the cost exhibits singularity), and obtain injectivity and continuity of optimal maps. This together with the result of Liu, Trudinger and Wang [LTW] also implies higher regular…
