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This is an expository note to give a brief review of classical elastica theory, mainly prepared for giving a more detailed proof of the author's Li--Yau type inequality for self-intersecting curves in Euclidean space. We also discuss some open problems in related topics. |
This paper extends the theory of regular solutions ($C^1$ in a suitable sense) for a class of semilinear elliptic equations in Hilbert spaces. The notion of regularity is based on the concept of $G$-derivative, which is introduced and discussed. |
We obtain improved remainder estimates in the Kuznecov sum formula for period integrals of Laplace eigenfunctions on a Riemannian manifold $M$. Building upon the two-term asymptotic expansion established in [<a href="https://arxiv.org/abs/2204.13525" data-arxiv-id="2204.13525" class="link-https">arXiv:2204.13525</a>],... |
We show a general stability result in the framework of strong solutions of the Navier-Stokes-Fourier system describing the motion of a compressible viscous and heat conducting gas. As a corollary, we develop a concept of statistical solution in the class of regular solutions "beyond the blow up time". |
We consider problems of static equilibrium in which the primary unknown is the stress field and the solutions maximize a complementary energy subject to equilibrium constraints. A necessary and sufficient condition for the sequential lower-semicontinuity of such functionals is symmetric ${\rm div}$-quasiconvexity, a s... |
This paper is about the evolution of a temperature front governed by the Surface quasi-geostrophic equation. The existence part of that program within the scale of Sobolev spaces was obtained by one of the authors [10]. |
We consider the equation $- \e^2 \D u + u= u^p$ in $\Omega \subseteq \R^N$, where $\Omega$ is open, smooth and bounded, and we prove concentration of solutions along $k$-dimensional minimal submanifolds of $\partial Ø$, for $N \geq 3$ and for $k \in \{1, ..., N-2\}$. We impose Neumann boundary conditions, assuming $1&... |
In this paper, as a step towards a unified mathematical treatment of the gauge functionals from quantum field theory that have found profound applications in mathematics, we generalize the Seiberg-Witten functional that in particular includes the Kapustin-Witten functional as a special case. We first demonstrate the s... |
This paper is devoted to the Lin-Ni conjecture for a semi-linear elliptic equation with a super-linear, sub-critical nonlinearity and homogeneous Neumann boundary conditions. We establish a new rigidity result, that is, we prove that the unique positive solution is a constant if the parameter of the problem is below a... |
The recently developed theory of Lagrangian flows for transport equations with low regularity coefficients enables to consider non BV vector fields. We apply this theory to prove existence and stability of global Lagrangian solutions to the repulsive Vlasov-Poisson system with only integrable initial distribution func... |
We study the uniqueness of non-negative solutions of the equation \begin{align*} <br>\partial_t\left(|u|^{p-2}u\right)\,=\, \operatorname{div}(|\nabla u|^{p-2}\nabla u). <br>\end{align*} Basic estimates are derived with the Galerkin Method. |
We consider the semilinear heat equation $$u_t-\Delta u=f(u) $$ <br>for a large class of non scale invariant nonlinearities of the form <br>$f(u)=u^pL(u)$, where $p>1$ is Sobolev subcritical and $L$ is a slowly varying function (which includes for instance logarithms and their powers and iterates, as well as some st... |
We study a nonlinear equation in the half-space $\{x_1>0\}$ with a Hardy potential, specifically \[-\Delta u -\frac{\mu}{x_1^2}u+u^p=0\quad\text{in}\quad \mathbb R^n_+,\] where $p>1$ and $-\infty<\mu<1/4$. The admissible boundary behavior of the positive solutions is either $O(x_1^{-2/(p-1)})$ as $x_1\to 0... |
Fix $s>1$. Colliander, Keel, Staffilani, Tao and Takaoka proved in \cite{CollianderKSTT10} the existence of solutions of the cubic defocusing nonlinear Schrödinger equation in the two torus with $s$-Sobolev norm growing in time. |
This paper is devoted to studying the local well-posedness (existence,uniqueness and continuous dependence) for the generalized Camassa-Holm equations in critial Besov spaces $B^{\frac{1}{p}}_{p,1}$ with $1\leq p<+\infty$, which improves the previous index $s> \max\{\frac{1}{2},\frac{1}{p}\}$ or $s=\frac{1}{p},\ ... |
The quantitative analysis of bubbling phenomena for almost constant mean curvature boundaries is an important question having significant applications in various fields including capillarity theory and the study of mean curvature flows. Such a quantitative analysis was initiated in [G. Ciraolo and F. Maggi, Comm. Pure... |
We are interested in a kinetic equation intended to describe the interactions of particles with their environment. We focus on the long time behaviour. |
In this paper, we study the set of stationary solutions of the Vlasov-Fokker-Planck (VFP) equation. This equation describes the time evolution of the probability distribution of a particle moving under the influence of a double-well potential, an interaction potential, a friction force and a stochastic force. |
In this article, we show how to embed the so-called CH2 equations into the geodesic flow of the Hdiv metric in 2D, which, itself, can be embedded in the incompressible Euler equation of a non compact Riemannian manifold. The method consists in embedding the incompressible Euler equation with a potential term coming fr... |
This paper studies the classical water wave problem with vorticity described by the Euler equations with a free surface under the influence of gravity over a flat bottom. Based on fundamental work \cite{ConstantinStrauss}, we first obtain two continuous bifurcation curves which meet the laminar flow only one time by u... |
We consider a many-body Boson system with pairwise particle interaction given by $N^{3\beta-1}v(N^\beta x)$ for $0<\beta<1$ and $v$ a non-negative spherically-symmetric function. Our main result is the extension of the local-in-time Fock space (norm) approximation of the exact dynamics of squeezed states proved ... |
In this paper, we showed that for some given suitable density and pressure, there exist infinitely many compactly supported solutions with prescribed energy profile. The proof is mainly based on the convex integration scheme. |
This paper is concerned with positive solutions of boundary value problems \begin{equation*} \left\{\begin{array}{ll} {\rm div}\left(d(v)\nabla u-u\chi(v)\nabla v\right)+\lambda u-u^2 +\gamma u F(v)=0, & x \in \Omega,\\[1mm] D \Delta v+\mu v-v^2-u F(v)=0, & x \in \Omega,\\[1mm] u=v=0, & x \in \partial \Omeg... |
Motivated by the recent work of Hassainia and Hmidi [Z. Hassainia, T. Hmidi - On the {V}-states for the generalized quasi-geostrophic equations,arXiv preprint <a href="https://arxiv.org/abs/1405.0858" data-arxiv-id="1405.0858" class="link-https">arXiv:1405.0858</a>], we close the question of the existence of convex glo... |
In this work, we establish the so-called backward uniqiueness property for a coupled system of partial differential equations (PDEs) which governs a certain fluid-structure interaction. In particular, a three-dimensional Stokes flow interacts across a boundary interface with a two-dimensional mechanical plate equation... |
Multi-soliton solutions, i.e. solutions behaving as the sum of N given solitons as $t\to +\infty$, were constructed in previous works for the L2 critical and subcritical (NLS) and (gKdV) equations. <br>In this paper, we extend the construction of multi-soliton solutions to the L2 supercritical case both for (gKdV) and... |
The study focuses on propagation of longitudinal waves in an elastic pipe partly embedded in a medium with dry friction. Mathematical formulation of the problem on the impact pipe driving into the soil is based on the model of longitudinal vibration of an elastic rod with taking into account lateral resistance. |
We begin by proving a local existence result for a fractional Caputo nonlocal thermistor problem. Then, additional existence and continuation theorems are obtained, ensuring global existence of solutions. |
We define the Ladyzhenskaya-Lions exponent $\alpha_{\rm {\tiny <br>\sc l}} (n)=({2+n})/4$ for Navier-Stokes equations with dissipation $-(-\Delta)^{\alpha}$ in ${\Bbb R}^n$, for all $n\geq 2$. We review the proof of strong global solvability when $\alpha\geq \alpha_{\rm <br>{\tiny \sc l}} (n)$, given smooth initial da... |
We consider the relationship between three continuum liquid crystal theories: Oseen-Frank, Ericksen and Landau-de Gennes. It is known that the function space is an important part of the mathematical model and by considering various function space choices for the order parameters $s$, ${\bf n}$, and ${\bf Q}$, we estab... |
A 3D-2D dimension reduction for $-\Delta_1$ is obtained. A power law approximation from $-\Delta_p$ as $p \to 1$ in terms of $\Gamma$- convergence, duality and asymptotics for least gradient functions has also been provided. |
This is a continuation of our work \cite{dns-part1} to investigate the long-time dynamics of a two species competition model of Lotka-Volterra type with nonlocal diffusions, where the territory (represented by the real line $\R$) of a native species with density $v(t,x)$, is invaded by a competitor with density $u(t,x)... |
In this article, we establish embeddings {à} la Lions and transfer of regularity {à} la Bouchut for a large scale of kinetic spaces. We use them to identify a notion of weak solutions to Kolmogorov-Fokker-Planck equations with (local or integral) diffusion and rough (measurable) coefficients under minimal requirements... |
The rigorous derivation of the Uehling-Uhlenbeck equation from more fundamental quantum many-particle systems is a challenging open problem in mathematics. In this paper, we exam the weak coupling limit of quantum N-particle dynamics. |
A mathematical continuum limit of the interaction energy of a random particle chain is shown to yield new insight into the effect of microscopic heterogeneities on macroscopic fracture laws in brittle materials. We derive a formula which yields that either elastic behaviour or a crack is energetically preferred. |
We investigate the asymptotic behaviour of nonlinear Schrödinger ground states on $d$-dimensional periodic metric grids in the limit for the length of the edges going to zero. We prove that suitable piecewise-affine extensions of such states converge strongly in $H^1(\mathbb{R}^d)$ to the corresponding ground states o... |
We derive conditions that ensure the existence of a bounded $H_\infty$-calculus in weighted $L_p$-Sobolev spaces for closed extensions $\underline{A}_T$ of a differential operator $A$ on a conic manifold with boundary, subject to differential boundary conditions $T$. In general, these conditions ask for a particular p... |
We discuss some estimates of subelliptic type related with vector fields satisfying the Hörmander condition. Our approach makes use of a class of approximate exponentials maps. |
We consider the three-dimensional time-dependent Gross-Pitaevskii equation arising in the description of rotating Bose-Einstein condensates and study the corresponding scaling limit of strongly anisotropic confinement potentials. The resulting effective equations in one or two spatial dimensions, respectively, are rig... |
In this paper, we first establish decay estimates for the fractional and higher-order fractional Hénon-Lane-Emden systems by using a nonlocal average and integral estimates, which deduce a result of non-existence. Next, we apply the method of scaling spheres introduced in \cite{DQ2} to derive a Liouville type theorem.... |
We provide a new and simple proof based on Harnack's inequality to the Lipschitz continuity of the solutions of a class of free boundary problems. |
We show that weak solutions to conormal derivative problem for elliptic equations in divergence form are continuously differentiable up to the boundary provided that the mean oscillations of the leading coefficients satisfy the Dini condition, the lower order coefficients satisfy certain suitable conditions, and the bo... |
In this paper, we develop an operator splitting scheme for the fractional kinetic Fokker-Planck equation (FKFPE). The scheme consists of two phases: a fractional diffusion phase and a kinetic transport phase. |
A representation for the sharp coefficient in a pointwise estimate for the gradient of a generalized Poisson integral of a function $f$ on ${\mathbb R}^{n-1}$ is obtained under the assumption that $f$ belongs to $L^p$. It is assumed that the kernel of the integral depends on the parameters $\alpha$ and $\beta$. |
Every homeomorphism h : X -> Y between planar open sets that belongs to the Sobolev class W^{1,p}(X,Y), 1<p<\infty, can be approximated in the Sobolev norm by diffeomorphisms. |
This paper is devoted to the study of the nonlinear Schrödinger-Poisson system with a doping profile. We are interested in the strong instability of standing waves associated with ground state solutions in the $L^2$-supercritical case. |
In this note we revisit a result in [9], where we established nonlocal isoperimetric inequalities and the related embeddings for Besov spaces adapted to a class of Hörmander operators of Kolmogorov-type. We provide here a new proof which exploits a weak-type Sobolev embedding established in [11]. |
The fractional Caffarelli-Kohn-Nirenberg inequality states that $$ \int_{\mathbb{R}^n}\int_{\mathbb{R}^n} \frac{(u (x)-u |
By assuming a certain localized energy estimate, we prove the existence portion of the Strauss conjecture on asymptotically flat manifolds, possibly exterior to a compact domain, when the spatial dimension is 3 or 4. In particular, this result applies to the 3 and 4-dimensional Schwarzschild and Kerr (with small angul... |
We introduce a large class of concentrated $p$-Lévy integrable functions approximating the unity, which serves as the core tool from which we provide a nonlocal characterization of Sobolev spaces and the space of functions of bounded variation via nonlocal energies forms. It turns out that this nonlocal characterizati... |
In this paper, we are interested in the nonlinear Rayleigh-Taylor instability for the gravity-driven incompressible Navier-Stokes equations with Navier-slip boundary conditions around a smooth increasing density profile $\rho_0(x_2)$ in a slab domain $2\pi L\mathbb{T} \times (-1,1)$ ($L>0$, $\mathbb{T}$ is the usual... |
In this paper, we prove a Carleman estimate for fully-discrete approximations of parabolic operators in which the discrete parameters $h$ and $\triangle t$ are connected to the large Carleman parameter. We use this estimate to obtain relaxed observability inequalities which yield, by duality, controllability results f... |
We present a time dependent quantum perturbation result, uniform in the Planck constant, for perturbations of potentials whose gradients are Lipschitz continuous by potentials whose gradients are only bounded a.e.. Though this low regularity of the full potential is not enough to provide the existence of the classical... |
This study delves into a comprehensive examination of the three-dimensional $(3D)$ incompressible magneto-hydrodynamic $(MHD)$ equations in $H^{1}(\R^{3})$. The modification involves incorporating a power term in the nonlinear convection component, a particularly relevant adjustment in porous media scenarios, especial... |
We provide a sufficient condition for a linear differential operator with constant coefficients $P(D)$ to be surjective on $C^\infty(X)$ and $\mathscr{D}'(X)$, respectively, where $X\subseteq\mathbb{R}^d$ is open. Moreover, for certain differential operators this sufficient condition is also necessary and thus a c... |
Let $f\in L^1(\R^d)$ be real. The Rudin-Osher-Fatemi model is to minimize $\|u\|_{\dot{BV}}+\lambda\|f-u\|_{L^2}^2$, in which one thinks of $f$ as a given image, $\lambda > 0$ as a "tuning parameter", $u$ as an optimal "cartoon" approximation to $f$, and $f-u$ as "noise" or "texture"... |
In this work, we prove global existence of solutions for second order differential problems in a general framework. More precisely, we consider second order differential inclusions involving proximal normal cone to a set-valued map. |
We continue the rigorous study of classical effective field theories (EFTs) that was recently initiated in the work of Reall and Warnick [RW22]. We study a system with one light and one heavy field with cubic coupling and prove global existence (of the UV solution) under an effective norm in the high mass limit. |
The understanding of some large energy, negative specific heat states in the Onsager description of 2D turbulence, seems to require the analysis of a subtle open problem about bubbling solutions of the mean field equation. Motivated by this application we prove that, under suitable non degeneracy assumptions on the as... |
We study the behavior at infinity in time of the global solution of the anisotropic quasi-geostrophic equation $\theta\in C_b(\mathbb{R}^+,H^s( \mathbb{R}^2))$. We prove that this solution decays to zero as time goes to infinity in $L^p(\mathbb{R}^2)$, $p\in [2,+\infty],$ moreover, we prove also that $\lim\limits_{t\r... |
We study the incompressible limit of the porous medium equation with a right hand side representing either a source or a sink term, and an injection boundary condition. This model can be seen as a simplified description of non-monotone motions in tumor growth and crowd motion, generalizing the congestion-only motions ... |
We investigate inverse diffraction problems for penetrable gratings in a piecewise constant medium. In the TE polarization case, it is proved that a binary grating profile together with the refractive index beneath it can be uniquely determined by the near-field observation data incited by a single plane wave and meas... |
This article introduces the degenerate special Lagrangian equation (DSL) and develops the basic analytic tools to construct and study its solutions. The DSL governs geodesics in the space of positive graph Lagrangians in $\mathbb{C}^n. |
In this work we establish conditions which guarantee the existence of (strictly) positive steady states of a nonlinear structured population model. In our framework the steady state formulation amounts to recasting the nonlinear problem as a family of eigenvalue problems combined with a fixed point problem. |
S. Zelditch introduced an equivariant version of a pseudo-differential calculus on a hyperbolic Riemann surface. We recast his construction in terms of trilinear invariant functionals on irreducible unitary representations of PGL(2,R). |
In this paper we rigorously justify the convergence of smooth solutions of the Navier-Stokes-Maxwell equations towards smooth solutions of the classical $2D$ parabolic MHD equations in the case of vanishing dielectric constant . The result is achieved by means of higher-order energy estimates. |
In this paper, we consider forward and inverse problems for subdiffusion equations with time-dependent coefficients. The fractional derivative is taken in the sense of Riemann-Liouville. |
We consider the initial boundary problem of 2D non-homogeneous incompressible heat conducting Navier-Stokes equations with vacuum, where the viscosity and heat conductivity depend on temperature in a power law of Chapman-Enskog. We derive the global existence of strong solution to the initial-boundary value problem, w... |
In this paper we consider a family of active scalars with a velocity field given by $u = \Lambda^{-1+\alpha}\nabla^{\perp} \theta$, for $\alpha \in (0,1)$. This family of equations is a more singular version of the two-dimensional Surface Quasi-Geostrophic (SQG) equation, which would correspond to $\alpha=0$. |
We prove global wellposedness of the Klein-Gordon equation with power nonlinearity $|u|^{\alpha-1}u$, where $\alpha\in\left[1,\frac{d}{d-2}\right]$, in dimension $d\geq3$ with initial data in $M_{p, p'}^{1}(\mathbb{R}^d)\times M_{p,p'}(\mathbb{R}^d)$ for $p$ sufficiently close to $2$. The proof is an applicati... |
In this paper we study a gradient flow approach to the problem of quantization of measures in one dimension. By embedding our problem in $L^2$, we find a continuous version of it that corresponds to the limit as the number of particles tends to infinity. |
Let $\Omega \subset \mathbb{R}^n$ be a bounded smooth domain (open and connected) in $\mathbb{R}^n$. Given $u_0\in L^2(\Omega)$, $g\in L^\infty(\Omega)$ and $\lambda \in \mathbb{R}$, our purpose is to describe the asymptotic behavior of weak solutions of the family of problems \begin{equation*} \left\{ \begin{array}{r... |
We review some recent results for a class of fluid mechanics equations called active scalars, with fractional dissipation. Our main examples are the surface quasi-geostrophic equation, the Burgers equation, and the Cordoba-Cordoba-Fontelos model. |
The regularized determinant of the Paneitz operator arises in quantum gravity (see Connes 1994, IV.4.$\gamma$). An explicit formula for the relative determinant of two conformally related metrics was computed by Branson in Branson (1996). |
We consider a family of quasilinear second order elliptic differential operators which are not coercive and are defined by functions in Marcinkiewicz spaces. We prove the existence of a solution to the corresponding Dirichlet problem. |
We solve a class of isoperimetric problems on $\mathbb{R}^2_+ :=\left\{ (x,y)\in \mathbb{R} ^2 : y>0 \right\}$ with respect to monomial weights. Let $\alpha $ and $\beta $ be real numbers such that $0\le \alpha <\beta+1$, $\beta\le 2 \alpha$. |
The processes of interplant competition within a field are still poorly understood. However, they explain a large part of the heterogeneity in a field and may have longer-term consequences, especially in mixed stands. |
We study the expected transition frequency between the two metastable states of a stochastic wave equation with double-well potential. By transition state theory, the frequency factorizes into two components: one depends only on the invariant measure, given by the $\phi^4_d$ quantum field theory, and the other takes t... |
In a recent paper the author introduced a new method based on viscosity techniques for producing minimal surfaces by minmax arguments. The present work corresponds to the regularity part of the method. |
This note presents a proof that the non-tangential maximal function of the Ornstein-Uhlenbeck semigroup is bounded almost surely by the Gaussian Hardy-Littlewood maximal function. In particular this entails improvement on a result by Pineda and Urbina who proved a similar result for a `trunctated' version of the n... |
Near every point of a real-analytic set in $\mathbb R^n$, we make use of Hironaka's resolution of singularity theorem to construct a family of continuous functions in $W^{1, 1}_{loc}$ such that their weak derivatives have (removable) singularity precisely on that set. |
We carry on an analysis of the size of the contact surface of a Cheeger set $E$ with the boundary of its ambient space $\Omega$. We show that this size is strongly related to the regularity of $\partial \Omega$ by providing bounds on the Hausdorff dimension of $\partial E\cap \partial\Omega$. |
Let $n,N\in \mathbb{N}$ with $\Omega \subseteq \mathbb{R}^n$ open. Given $H \in C^2(\Omega \times \mathbb{R}^N\times \mathbb{R}^{Nn}),$ we consider the functional \[ \tag{1} \label{1} <br>E_\infty (u,\mathcal{O})\, :=\, \underset{\mathcal{O}}{\mathrm{ess}\,\sup}\, H (\cdot,u,\mathrm{D} u) ,\ \ \ u\in W^{1,\infty}_\tex... |
This paper addresses the existence of codimension one stable manifolds for the pseudo-conformal blow-up solution for critical one-dimensional NLS. By the work of Perelman and Merle, Raphael, the blow-up rate of these solutions is far from the generic one. |
Let $G=(V,E)$ be a locally finite graph. Firstly, using calculus of variations, including a direct method of variation and the mountain-pass theory, we get sequences of solutions to several local equations on $G$ (the Schrödinger equation, the mean field equation, and the Yamabe equation). |
A method is presented for approximating the effective conductivity of composite media with thin interphase regions, which is exact to first order in the interphase thickness. The approximations are computationally efficient in the sense the fields need to be computed only in a reference composite in which the interpha... |
We study the adjoint of the double layer potential associated with the Laplacian (the adjoint of the Neumann-Poincaré operator), as a map on the boundary surface $\Gamma$ of a domain in $\mathbb{R}^3$ with conical points. The spectrum of this operator directly reflects the well-posedness of related transmission proble... |
We give a survey of author's results on the inverse hyperbolic problems with time-dependent and time-independent coefficients. We consider the case of hyperbolic equations with Yang-Mills potentials and the case of domains with obstacles. |
Let $G:\mathbb{R\rightarrow R}$ be a continuous function. In the first part of this paper, we investigate sufficient conditions on $G$ such that <br>\begin{equation*} <br>\{G(f):f\in \dot{K}_{p,q}^{\alpha }F_{\beta }^{s}\}\subset \dot{K}_{p,q}^{\alpha }F_{\beta }^{s} <br>\end{equation*} holds. |
We consider relaxation systems of transport equations with heterogeneous source terms and with boundary conditions, which limits are scalar conservation laws. Classical bounds fail in this context and in particular BV estimates. |
Four quantities are fundamental in homogenization of elliptic systems in divergence form and in its applications: the field and the flux of the solution operator (applied to a general deterministic right-hand side), and the field and the flux of the corrector. Homogenization is the study of the large-scale properties ... |
This paper continues our study of the interconnection between controllability and mixing properties of random dynamical systems. We begin with an abstract result showing that the approximate controllability to a point and a local stabilisation property imply the uniqueness of a stationary measure and exponential mixin... |
Complexity of geometric symmetry for differential operators with mixed homogeniety is examined here. Sharp Sobolev estimates are calculated for the Grushin operator in low dimensions using hyperbolic symmetry and conformal geometry. |
We establish that the $C^{1,\gamma}$ regularity theory for translation invariant fractional order parabolic integro-differential equations (via Krylov-Safonov estimates) gives an improvement of regularity mechanism for solutions to a special case of a two-phase free boundary flow related to Hele-Shaw. The special case... |
We consider nonlocal minimal surfaces in a cylinder with prescribed datum given by the complement of a slab. We show that when the width of the slab is large the minimizers are disconnected and when the width of the slab is small the minimizers are connected. |
We prove the global well-posedness of three dimensional compressible Navier-Stokes equations for some classes of large initial data, which is of large oscillation for the density and large energy for the velocity. The structure of the system (especially, the effective viscous flux) is fully exploited. |
We prove that the Stokes semigroup is a bounded analytic semigroup on $L^{\infty}_{\sigma}$ of angle $\pi/2$ for two-dimensional exterior domains. This result is an end point case of the $L^{p}$-boundedness of the semigroup for $p\in (1,\infty)$, established by Borchers and Varnhorn (1993) and an extension of finite t... |
In 1983, Antontsev, Kazhikhov, and Monakhov published a proof of the existence and uniqueness of solutions to the 3D Euler equations in which on certain inflow boundary components fluid is forced into the domain while on other outflow components fluid is drawn out of the domain. A key tool they used was the linearized... |
In this article we set up a splitting variant of the JKO scheme in order to handle gradient flows with respect to the Kantorovich-Fisher-Rao metric, recently introduced and defined on the space of positive Radon measure with varying masses. We perform successively a time step for the quadratic Wasserstein/Monge-Kantor... |
The main aim of this note is to prove sharp weighted integral Hardy inequality and conjugate integral Hardy inequality on homogeneous Lie groups with any quasi-norm for the range $1<p\leq q<\infty. $ We also calculate the precise value of sharp constants in respective inequalities, improving the result of $[19]$ ... |
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