Global regularity and solvability of left-invariant differential systems on compact Lie groups
Gabriel Araújo
备注：
28 pages. This is a pre-print of an article published in Annals of Global Analysis and Geometry. The final authenticated version is available online at: https://doi.org/10.1007/s10455-019-09682-9
We are interested in global properties of systems of left-invariant differential operators on compact Lie groups: regularity properties, properties on the closedness of the range and finite dimensionality of their cohomology spaces, when acting on various function spaces e.g. smooth, analytic and Gevrey. Extending the methods of Greenfield and Wallach (1973) to systems, we obtain abstract characterizations for these properties and use them to derive some generalizations of results due to Greenfield (1972), Greenfield and Wallach (1972), as well as global versions of a result of Caetano and Cordaro (2011) for involutive structures.
We are concerned with the initial value problem governed by generalized Rayleigh-Stokes equations, where the nonlinearity depends on history states and takes values in Hilbert scales of negative order. The solvability and Hölder regularity of solutions are proved by using fixed point arguments and embeddings of fractional Sobolev spaces. An application to a related inverse source problem is given.
We provide a self-contained proof of the solvability and regularity of a Hodge-type elliptic system, wherein the divergence and curl of a vector field are prescribed in an open, bounded, Sobolev-class domain, and either the normal component or the tangential components of the vector field are prescribed on the boundary. The proof is based on a regularity theory for vector elliptic equations set on Sobolev-class domains and with Sobolev-class coefficients.
In this article we introduce the notion of fundamental solution in the Colombeau context as an element of the dual $\LL(\Gc(\R^n),\wt{\C})$. After having proved the existence of a fundamental solution for a large class of partial differential operators with constant Colombeau coefficients, we investigate the relationships between fundamental solutions in $\LL(\Gc(\R^n),\wt{\C})$, Colombeau solvability and $\G$- and $\Ginf$-hypoellipticity respectively.
The paper addresses questions of existence and regularity of solutions to linear partial differential equations whose coefficients are generalized functions or generalized constants in the sense of Colombeau. We introduce various new notions of ellipticity and hypoellipticity, study their interrelation, and give a number of new examples and counterexamples. Using the concept of $\G^\infty$-regularity of generalized functions, we derive a general global regularity result in the case of operators with constant generalized coefficients, a more specialized result for second order operators, and a …
We consider the motion of incompressible viscous fluids bounded above by a free surface and below by a solid surface in the $N$-dimensional Euclidean space for $N\geq 2$ when the gravity is not taken into account. The aim of this paper is to show the global solvability of the Naiver-Stokes equations with a free surface, describing the above-mentioned motion, in the maximal $L_p\text{-}L_q$ regularity class. Our approach is based on the maximal $L_p\text{-}L_q$ regularity with exponential stability for the linearized equations, and solutions to the original nonlinear problem are also exponentia…
This is an addendum to a previous article, which aims to provide the proofs of some results in that paper (Theorem 7.5 and Proposition 9.15) which were removed from its final version. The reason for such omission is that these proofs follow quite closely others already present in the literature, with minor modifications. I make them publicly available for the sake of completeness.
We investigate global solvability, in the framework of smooth functions and Schwartz distributions, of certain sums of squares of vector fields defined on a product of compact Riemannian manifolds $T \times G$, where $G$ is further assumed to be a Lie group. As in a recent article due to the authors, our analysis is carried out in terms of a system of left-invariant vector fields on $G$ naturally associated with the operator under study, a simpler object which nevertheless conveys enough information about the original operator so as to fully encode its solvability. As a welcome side effect of …
We use the theory of dual of Fréchet-Schwartz (DFS) spaces to establish a sufficient condition for top-degree solvability for the differential complex associated to a hypocomplex locally integrable structure. As an application, we show that the top-degree cohomology of left-invariant hypocomplex structures on a compact Lie group can be computed only by using left-invariant forms, thus reducing the computation to a purely algebraic one. In the case of left-invariant elliptic involutive structures on compact Lie groups, under certain reasonable conditions, we prove that the cohomology associated…
Let $Ω$ be a Lipschitz domain in $\mathbb R^n,n\geq 3,$ and $L=\divt A\nabla$ be a second order elliptic operator in divergence form. We will establish that the solvability of the Dirichlet regularity problem for boundary data in Hardy-Sobolev space $\HS$ is equivalent to the solvability of the Dirichlet regularity problem for boundary data in $H^{1,p}$ for some $1<p<\infty$. This is a "dual result" to a theorem in \cite{DKP09}, where it has been shown that the solvability of the Dirichlet problem with boundary data in $\text{BMO}$ is equivalent to the solvability for boundary data in $L^p(\pa…
We study classes of variational problems with energy densities of linear growth acting on vector-valued functions. Our energies are strictly convex variants of the TV-regularization model introduced by Rudin, Osher and Fatemi [15] as a powerful tool in the field of image recovery. In contrast to our previous work we here try to figure out conditions under which we can solve these variational problems in classical spaces, e.g. in the Sobolev class $W^{1,1}$.
We continue the development, by reduction to a first order system for the conormal gradient, of $L^2$ \textit{a priori} estimates and solvability for boundary value problems of Dirichlet, regularity, Neumann type for divergence form second order, complex, elliptic systems. We work here on the unit ball and more generally its bi-Lipschitz images, assuming a Carleson condition as introduced by Dahlberg which measures the discrepancy of the coefficients to their boundary trace near the boundary. We sharpen our estimates by proving a general result concerning \textit{a priori} almost everywhere no…
This paper contains two results on the $L^p$ regularity problem on Lipschitz domains. For second order elliptic systems and $1<p<\infty$, we prove that the solvability of the $L^p$ regularity problem is equivalent to that of the $L^{p^\prime}$ Dirichlet problem. For higher order elliptic equations and systems, we show that if $p>2$, the solvability of the $L^p$ regularity problem is equivalent to a weak reverse Hölder condition with exponent $p$.
We prove that the $L^{p'}$-solvability of the homogeneous Dirichlet problem for an elliptic operator $L=-\operatorname{div}A\nabla$ with real and merely bounded coefficients is equivalent to the $L^{p'}$-solvability of the Poisson Dirichlet problem $Lw=H-\operatorname{div} F$, which is defined in terms of an $L^{p'}$ estimate on the non-tangential maximal function, assuming that $\operatorname{dist}(\cdot, \partial Ω) H$ and $F$ lie in certain $L^{p'}$-Carleson-type spaces, and that the domain $Ω\subset\mathbb R^{n+1}$, $n\geq2$, satisfies the corkscrew condition and has $n$-Ahlfors regular bo…
We study the relationship between the solvability of the $L^p$ Dirichlet problem $(D)_p$ and that of the $L^q$ regularity problem $(R)_q$ for second order elliptic equations with bounded measurable coefficients. It is known that the solvability of $(R)_p$ implies that of $(D)_{p^\prime}$. In this note we show that if $(D)_{p^\prime}$ is solvable, then either $(R)_p$ is solvable or $(R)_q$ is not solvable for any $1<q<\infty$.
In the paper arXiv:1708.02289 we have introduced new solvability methods for strongly elliptic second order systems in divergence form on a domains above a Lipschitz graph, satisfying $L^p$-boundary data for $p$ near $2$. The main novel aspect of our result is that it applies to operators with coefficients of limited regularity and applies to operators satisfying a natural Carleson condition that has been first considered in the scalar case. In this paper we extend this result in several directions. We improve the range of solvability of the $L^p$ Dirichlet problem to the interval $2-\varepsil…
We consider a~quasilinear model arising from dynamical magnetization. This model is described by a~magneto-quasistatic (MQS) approximation of Maxwell's equations. Assuming that the medium consists of a~conducting and a~non-conducting part, the derivative with respect to time is not fully entering, whence the system can be described by an abstract differential-algebraic equation. Furthermore, via magnetic induction, the system is coupled with an equation which contains the induced electrical currents along the associated voltages, which form the input of the system. The aim of this paper is to …
The current paper is principally motivated by establishing the global well-posedness to the three-dimensional Boussinesq system with zero diffusivity in the setting of axisymmetric flows without swirling with $v_0\in H^{\frac12}(\mathbb{R}^3)\cap \dot{B}^{0}_{3,1}(\mathbb{R}^3)$ and density $ρ_0\in L^2(\mathbb{R}^3)\cap \dot{B}^{0}_{3,1}(\mathbb{R}^3)$. This respectively enhances the two results recently accomplished in \cite{Danchin-Paicu1, Hmidi-Rousset}. Our formalism is inspired, in particular for the first part from \cite{Abidi} concerning the axisymmetric Navier-Stokes equations once $v_…
We prove that the solvability of the regularity problem in $L^q(\partial Ω)$ is stable under Carleson perturbations. If the perturbation is small, then the solvability is preserved in the same $L^q$, and if the perturbation is large, the regularity problem is solvable in $L^{r}$ for some other $r\in (1,\infty)$. We extend an earlier result from Kenig and Pipher to very general unbounded domains, possibly with lower dimensional boundaries as in the theory developed by Guy David and the last two authors. To be precise, we only need the domain to have non-tangential access to its Ahlfors regular …
We are concerned with the 3D incompressible Hall-magnetohydro-dynamic system (Hall-MHD). Our first aim is to provide the reader with an elementary proof of a global well-posedness result for small data with critical Sobolev regularity, in the spirit of Fujita-Kato's theorem [10] for the Navier-Stokes equations. Next, we investigate the long-time asymptotics of global solutions of the Hall-MHD system that are in the Fujita-Kato regularity class. A weak-strong uniqueness statement is also proven. Finally, we consider the so-called 2 1/2 D flows for the Hall-MHD system, and prove the global exist…
We study the non-existence, existence and multiplicity of positive solutions to the following nonlinear Kirchhoff equation:% \begin{equation*} \left\{ \begin{array}{l} -M\left( \int_{\mathbb{R}^{3}}\left\vert \nabla u\right\vert ^{2}dx\right) Δu+μV\left( x\right) u=Q(x)\left\vert u\right\vert ^{p-2}u+λf\left( x\right) u\text{ in }\mathbb{R}^{N}, \\ u\in H^{1}\left( \mathbb{R}^{N}\right) ,% \end{array}% \right. \end{equation*}% where $N\geq 3,2<p<2^{\ast }:=\frac{2N}{N-2},M\left( t\right) =at+b$ $\left( a,b>0\right) ,$ the potential $V$ is a nonnegative function in $\mathbb{R}% ^{N}$ and the we…
We address the self-similar solvability of a singular parabolic problem and show that solutions to parabolic fully nonlinear equations are not expected to be $C^{2,1}$.
We develop new solvability methods for divergence form second order, real and complex, elliptic systems above Lipschitz graphs, with $L_2$ boundary data. The coefficients $A$ may depend on all variables, but are assumed to be close to coefficients $A_0$ that are independent of the coordinate transversal to the boundary, in the Carleson sense $\|A-A_0\|_C$ defined by Dahlberg. We obtain a number of {\em a priori} estimates and boundary behaviour results under finiteness of $\|A-A_0\|_C$. Our methods yield full characterization of weak solutions, whose gradients have $L_2$ estimates of a non-tan…
We prove the global strong solvability of a quasilinear initial-boundary value problem with fractional time derivative of order less than one. Such problems arise in mathematical physics in the context of anomalous diffusion and the modelling of dynamic processes in materials with memory. The proof relies heavily on a regularity result about the interior Hölder continuity of weak solutions to time fractional diffusion equations, which has been proved recently by the author. We further establish a basic $L_2$ decay estimate for the special case with vanishing external source term and homogeneou…
In this work, we study the Dirichlet problem associated with a strongly coupled system of nonlocal equations. The system of equations comes from a linearization of a model of peridynamics, a nonlocal model of elasticity. It is a nonlocal analogue of the Navier-Lamé system of classical elasticity. The leading operator is an integro-differential operator characterized by a distinctive matrix kernel which is used to couple differences of components of a vector field. The paper's main contributions are proving well-posedness of the system of equations and demonstrating optimal local Sobolev regula…
We study the equation $u_{11}u_{22} = 1$ in $\mathbb{R}^2$. Our results include an interior $C^2$ estimate, classical solvability of the Dirichlet problem, and the existence of non-quadratic entire solutions. We also construct global singular solutions to the analogous equation in higher dimensions. At the end we state some open questions.
The present paper deals with the nonhomogeneous incompressible asymmetric fluids equations in dimension $d= 2,3$. The aim is to prove the unique global solvability of the system with only bounded nonnegative initial density and $H^{1}$ initial velocities. We first construct the global existence of the solution with large data in 2-D. Next, we establish the existence of local in time solution for arbitrary large data and global in time for some smallness conditions in 3-D. Finally, the uniqueness of the solution is proved under quite soft assumptions about its regularity through a Lagrangian ap…
The present paper deals with the Cauchy problem of a compressible generic two-fluid model with capillarity effects in any dimension $N\geq2$. We first study the unique global solvability of the model in spaces with critical regularity indices with respect to the scaling of the associated equations. Due to the presence of the capillary terms, we exploit the parabolic properties of the linearized system for all frequencies which enables us to apply contraction mapping principle to show the unique global solvability of strong solutions close to a stable equilibrium state. Furthermore, under a mil…
We show the existence and the regularity properties of the weak solutions to the two-dimensional stationary incompressible inhomogeneous Navier-Stokes equations with variable viscosity coefficient, by analyzing a fourth-order nonlinear elliptic equation for the stream function. The density function and the viscosity coefficient may have large variations. In addition, we formulate the solutions for the parallel, concentric and radial flows respectively, and as examples we calculate the solutions with piecewise-constant viscosity coefficients explicitly.
The chemotaxis system \[ \left\{ \begin{array}{l} u_t = Δu - χ\nabla \cdot (\frac{u}{v}\nabla v), v_t=Δv - v+u, \end{array} \right. \] is considered in a bounded domain $Ω\subset \mathbb{R}^n$ with smooth boundary, where $χ>0$. An apparently novel type of generalized solution framework is introduced within which an extension of previously known ranges for the key parameter $χ$ with regard to global solvability is achieved. In particular, it is shown that under the hypothesis that\[ χ< \left\{ \begin{array}{ll} \infty \qquad & \mbox{if } n=2, \sqrt{8} \qquad & \mbox{if } n=3, \frac{n}{n-2} \qqu…
We present necessary and sufficient conditions to have global hypoellipticity and global solvability for a class of vector fields defined on a product of compact Lie groups. In view of Greenfield's and Wallach's conjecture, about the non-existence of globally hypoelliptic vector fields on compact manifolds different from tori, we also investigate different notions of regularity weaker than global hypoellipticity and describe completely the global hypoellipticity and global solvability of zero-order perturbations of our vector fields. We also present a class of vector fields with variable coeff…
We present sufficient conditions to have global hypoellipticity for a class of Vekua-type operators defined on a compact Lie group. When the group has the property that every non-trivial representation is not self-dual we show that these sufficient conditions are also necessary. We also present results about the global solvability for this class of operators.
Liouville theorem and sharp solvability for solutions of the parabolic Monge-Ampère equation with periodic data
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education（北京师范大学数学科学学院，教育部数学与复杂系统重点实验室）
We discuss unique existence and microlocal regularity properties of Sobolev space solutions to the fractional Eringen wave equation, initially given in the form of a system of equations in which the classical non-local Eringen constitutive equation is generalized by employing space-fractional derivatives. Numerical examples illustrate the shape of solutions in dependence of the order of the space-fractional derivative.
We derive the solvability and regularity of the Dirichlet problem for fully non-linear elliptic equations possibly with degenerate right-hand side on Hermitian manifolds, through establishing a quantitative version of boundary estimate under a subsolution assumption. In addition, we construct the subsolution when the background manifold is a product of a closed Hermitian manifold with a compact Riemann surface with boundary.
On $T \times G$, where $T$ is a compact real-analytic manifold and $G$ is a compact Lie group, we consider differential operators $P$ which are invariant by left translations on $G$ and are elliptic in $T$. Under a mild technical condition, we prove that global hypoellipticity of $P$ implies its global analytic-hypoellipticity (actually Gevrey of any order $s \geq 1$). We also study the connection between the latter property and the notion of global analytic (resp. Gevrey) solvability, but in a much more general setup.
Introducing a suitable solution concept, we show that in bounded smooth domains $Ω\subset \mathbb{R}^n$, $n\ge 1$, the initial boundary value problem for the chemotaxis system \begin{align*} u_t&=Δu -χ\nabla\cdot\left(\frac{u}{v}\nabla v\right)+κu -μu^2,\\ v_t&=Δv -uv, \end{align*} with homogeneous Neumann boundary conditions and widely arbitrary initial data has a generalized global solution for any $μ, κ, χ>0$.
This paper considers the existence of local and global-in-time strong solutions to the advection-diffusion equation with variable coefficients on an evolving surface with a boundary. We apply both the maximal $L^p$-in-time regularity for Hilbert space-valued functions and the semigroup theory to construct local and global-in-time strong solutions to an evolution equation. Using the approach and our function spaces on the evolving surface, we show the existence of local and global-in-time strong solutions to the advection-diffusion equation. Moreover, we derive the asymptotic stability of the g…
The purpose of this article is to study extrapolation of solvability for boundary value problems of elliptic systems in divergence form on the upper half-space assuming De Giorgi type conditions. We develop a method allowing to treat each boundary value problem independently of the others. We shall base our study on solvability for energy solutions, estimates for boundary layers, equivalence of certain boundary estimates with interior control so that solvability reduces to a one-sided Rellich inequality. Our method then amounts to extrapolating this Rellich inequality using atomic Hardy spaces…
Local solvability and non-solvability are classified for left-invariant differential operators on the Heisenberg group H_1 of the form L=P_n(X,Y)+Q(X,Y) where the P_n are certain homogeneous polynomials of order n greater than or equal to 2 and Q is of lower order with X= \partial_x, Y=\partial_y+x\partial_w on R^3. We extend previous studies of operators of the form P_n(X,Y) via representations involving ordinary differential operators with a parameter.
Consider a free boundary problem of compressible-incompressible two-phase flows with surface tension and phase transition in bounded domains $Ω_{t +}, Ω_{t -} \subset \mathbb{R}^N$, $N \ge 2$, where the domains are separated by a sharp compact interface $Γ_t \subset \mathbb{R}^{N - 1}$. We prove a global in time unique existence theorem for such free boundary problem under the assumption that the initial data are sufficiently small and the initial domain of the incompressible fluid is close to a ball. In particular, we obtain the solution in the maximal $L_p - L_q$-regularity class with $2 < p…
In the present paper we establish the solvability of the Regularity boundary value problem in domains with (flat and Lipschitz) lower dimensional boundaries for operators whose coefficients exhibit small oscillations analogous to the Dahlberg-Kenig-Pipher condition. The proof follows the classical strategy of showing bounds on the square function and the non-tangential maximal function. The key novelty and difficulty of this setting is the presence of multiple non-tangential derivatives. To solve it, we consider a cylindrical system of derivatives and establish new estimates on the "angular de…
