Coercive Estimate Research Articles

A Coercive Estimate for the Nonhomogeneous Differential Operator on the Heisenberg Group

A Coercive Estimate for the Nonhomogeneous Differential Operator on the Heisenberg Group

A Non-Local Problem for the Fractional-Order Rayleigh–Stokes Equation

A nonlocal boundary value problem for the fractional version of the Rayleigh–Stokes equation, well-known in fluid dynamics, is studied. Namely, the condition u(x,T)=βu(x,0)+φ(x), where β is an arbitrary real number, is proposed instead of the initial condition. If β=0, then we have the inverse problem in time, called the backward problem. It is well-known that the backward problem is ill-posed in the sense of Hadamard. If β=1, then the corresponding non-local problem becomes well-posed in the sense of Hadamard, and moreover, in this case a coercive estimate for the solution can be established. The aim of this work is to find values of the parameter β, which separates two types of behavior of the semi-backward problem under consideration. We prove the following statements: if β≥1, or β<0, then the problem is well-posed; if β∈(0,1), then depending on the eigenvalues of the elliptic part of the equation, for the existence of a solution an additional condition on orthogonality of the right-hand side of the equation and the boundary function to some eigenfunctions of the corresponding elliptic operator may emerge.

The Shakhov model near a global Maxwellian

Shakhov model is a relaxation approximation of the Boltzmann equation proposed to overcome the deficiency of the original BGK model, namely, the incorrect production of the Prandtl number. In this paper, we address the existence and the asymptotic stability of the Shakhov model when the initial data is a small perturbation of global equilibrium. We derive a dichotomy in the coercive estimate of the linearized relaxation operator between zero and non-zero Prandtl number, and observe that the linearized relaxation operator is more degenerate in the former case. To remove such degeneracy and recover the full coercivity, we consider a micro–macro system that involves an additional non-conservative quantity related to the heat flux.

Existence and smoothness of solutions of a singular differential equation of hyperbolic type

This paper investigates the question of the existence of solutions to the semiperiodic Dirichlet problem for a class of singular differential equations of hyperbolic type. The problem of smoothness of solutions is also considered, i.e., maximum regularity of solutions. Such a problem will be interesting when the coefficients are strongly growing functions at infinity. For the first time, a weighted coercive estimate was obtained for solutions to a differential equation of hyperbolic type with strongly growing coefficients.

Separability of the third-order differential operator given on the whole plane

In this paper, in the space L_2(R^2), we study a third-order differential operator with continuous coefficients in R(−∞, +∞). Here, these coefficients can be unlimited functions at infinity. In addition under some restrictions on the coefficients, the bounded invertibility of the given operator is proved and a coercive estimate is obtained, i.e. separability is proved.

Solvability of nonlinear problem for some second-order nonstrongly elliptic system

In this work, we investigate the second-order nonlinear elliptic system defined on an unbounded domain and with variable nonconstant coefficients. We establish the existence of the solution of the nonlinear problem for a second-order nonstrongly elliptic system by using a coercive estimate, which is obtained for the linear case. The nonlinear elliptic system is transformed into an equivalent fixed point problem for a suitable nonlinear operator in a convex subset of a Banach function space. Using the topological fixed-point approach and embedding theorems, the conditions for solvability of the semiperiodical nonlinear problem in a weighted Sobolev space are obtained.

COERCIVE SOLVABILITY CONDITIONS OF THE THREE-ORDER DEGENERATE DIFFERENTIAL EQUATIONS

In this paper, we consider one class of the singular nonlinear third-order differential equations given on the entire axis. We show sufficient conditions for the existence of a solution to this equation and the satisfiability of the coercive estimate for solution. The considered equation has the following features. Its intermediate coefficient is not bounded and does not obey to a lower coefficient. In the literature, such equations are called the degenerate differential equations. Further, the corresponding differential operator is not semi-bounded: its energy space may not belong to the Sobolev classes. Previously, the solvability questions of the third-order singular differential equations was studied only in the case that their intermediate coefficients are equal to zero. The main result of this work is proved on the basis of one separability theorem for the linear third-order degenerate differential operators, Schauder's fixed point theorem and some Hardy type weighted integral inequalities.

Analogues of Korn’s inequality on Heisenberg groups

Two analogues of Korn’s inequality on Heisenberg groups are constructed. First, the norm of the horizontal differential is estimated in terms of its symmetric part. Second, Korn’s inequality is treated as a coercive estimate for a differential operator whose kernel coincides with the Lie algebra of the isometry group. For this purpose, we construct a differential operator whose kernel coincides with the Lie algebra of the isometry group on Heisenberg groups and prove a coercive estimate for this operator. Additionally, a coercive estimate is proved for a differential operator whose kernel coincides with the Lie algebra of the group of conformal mappings on Heisenberg groups.

Analogues of Korn’s Inequality on Heisenberg Groups

Two analogues of Korn’s inequality on Heisenberg groups are constructed. First, the norm of the horizontal differential is estimated in terms of its symmetric part. Second, Korn’s inequality is treated as a coercive estimate for a differential operator whose kernel coincides with the Lie algebra of the isometry group. For this purpose, we construct a differential operator whose kernel coincides with the Lie algebra of the isometry group on Heisenberg groups and prove a coercive estimate for this operator. Additionally, a coercive estimate is proved for a differential operator whose kernel coincides with the Lie algebra of the group of conformal mappings on Heisenberg groups.

Decay rates for the quadratic and super-quadratic tilt-excess of integral varifolds

This paper concerns integral varifolds of arbitrary dimension in an open subset of Euclidean space satisfying integrability conditions on their first variation. Firstly, the study of pointwise power decay rates almost everywhere of the quadratic tilt-excess is completed by establishing the precise decay rate for two-dimensional integral varifolds of locally bounded first variation. In order to obtain the exact decay rate, a coercive estimate involving a height-excess quantity measured in Orlicz spaces is established. Moreover, counter-examples to pointwise power decay rates almost everywhere of the super-quadratic tilt-excess are obtained. These examples are optimal in terms of the dimension of the varifold and the exponent of the integrability condition in most cases, for example if the varifold is not two-dimensional. These examples also demonstrate that within the scale of Lebesgue spaces no local higher integrability of the second fundamental form, of an at least two-dimensional curvature varifold, may be deduced from boundedness of its generalised mean curvature vector. Amongst the tools are Cartesian products of curvature varifolds.

The Boltzmann Equation for a Multi-species Mixture Close to Global Equilibrium

We study the Cauchy theory for a multi-species mixture, where the different species can have different masses, in a perturbative setting on the three dimensional torus. The ultimate aim of this work is to obtain the existence, uniqueness and exponential trend to equilibrium of solutions to the multi-species Boltzmann equation in $${L^1_vL^\infty_x(m)}$$ , where $${m\sim (1+ |v|^k)}$$ is a polynomial weight. We prove the existence of a spectral gap for the linear multi-species Boltzmann operator allowing different masses, and then we establish a semigroup property thanks to a new explicit coercive estimate for the Boltzmann operator. Then we develop an $${L^2-L^\infty}$$ theory à la Guo for the linear perturbed equation. Finally, we combine the latter results with a decomposition of the multi-species Boltzmann equation in order to deal with the full equation. We emphasize that dealing with different masses induces a loss of symmetry in the Boltzmann operator which prevents the direct adaptation of standard mono-species methods (for example Carleman representation, Povzner inequality). Of important note is the fact that all methods used and developed in this work are constructive. Moreover, they do not require any Sobolev regularity and the $${L^1_vL^\infty_x}$$ framework is dealt with for any $${k > k_0}$$ , recovering the optimal physical threshold of finite energy $${k_0=2}$$ in the particular case of a multi-species hard spheres mixture with the same masses.

Dimensional Contraction in Wasserstein Distance for Diffusion Semigroups on a Riemannian Manifold

We prove a refined contraction inequality for diffusion semigroups with respect to the Wasserstein distance on a compact Riemannian manifold taking account of the dimension. The result generalizes in a Riemannian context, the dimensional contraction established in Bolley et al. J. Lond. Math. Soc. 90(1), 309–332 (2014) for the Euclidean heat equation. It is proved by using a dimensional coercive estimate for the Hodge-de Rham semigroup on 1-forms.

L 1,p –Coercitivity and Estimates of the Green Function of the Neumann Problem in a Convex Domain

We consider the Neumann problem for the Poisson equation in an arbitrary convex bounded n-dimensional domain. We obtain a coercive estimate for the solution to the Neumann problem and establish the unique solvability of the Neumann problem in the Sobolev space L 1,p . We also obtain sharp pointwise estimates for the Green function and the gradient of the solution. We describe the eigensubspace corresponding to the least positive eigenvalue λ = n − 1 of the Neumann–Laplace operator in a convex subdomain of the unit (n − 1)-dimensional sphere.

Convergence to equilibrium of some kinetic models

We introduce in this paper a new constructive approach to the problem of the convergence to equilibrium for a large class of kinetic equations. The idea of the approach is to prove a ‘weak’ coercive estimate, which implies exponential or polynomial convergence rate. Our method works very well not only for hypocoercive systems in which the coercive parts are degenerate but also for the linearized Boltzmann equation.

Coercive energy estimates for differential forms in semi-convex domains

In this paper, we prove a $H^1$-coercive estimate for differentialforms of arbitrary degrees in semi-convex domains of the Euclidean space.The key result is an integral identity involving a boundary term in whichthe Weingarten matrix of the boundary intervenes, establishedfor any Lipschitz domain $\Omega\subseteq \mathcal{R}^n$ whoseoutward unit normal $\nu$ belongs to $L^{n-1}_1(\partial\Omega)$,the $L^{n-1}$-based Sobolev space of order one on $\partial\Omega$.

The Dirichlet problem for non‐divergence parabolic equations with discontinuous in time coefficients

AbstractWe consider the Dirichlet problem for non‐divergence parabolic equation with discontinuous in t coefficients in a half space. The main result is weighted coercive estimates of solutions in anisotropic Sobolev spaces. We give an application of this result to linear and quasi‐linear parabolic equations in a bounded domain. In particular, if the boundary is of class C1,δ , δ ∈ [0, 1], then we present a coercive estimate of solutions in weighted anisotropic Sobolev spaces, where the weight is a power of the distance to the boundary (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Solution of a model problem related to singularly perturbed, free boundary, Stefan type problems

A model problem of conjunction of two phases with a small parameter κ in front of the principal term of a free boundary condition is considered. A coercive estimate for the solution with a constant independent of κ is derived in terms of a weighted Holder space. Bibliography: 9 titles.

Maximal B-regular integro-differential equation

By using Fourier multiplier theorems, the maximal B-regularity of ordinary integro-differential operator equations is investigated. It is shown that the corresponding differential operator is positive and satisfies coercive estimate. Moreover, these results are used to establish maximal regularity for infinite systems of integro-differential equations.

Coercive estimate for a boundary value problem with noncoordinated degeneration of the data

Coercive estimate for a boundary value problem with noncoordinated degeneration of the data

A One-Dimensional Parabolic Problem Arising in Studies of Some Free Boundary Problems

The paper is concerned with a one-dimensional parabolic problem in a domain bounded by two lines x = 0 and x = kt, k > 0, (x, t) ∈ ℝ2, with the Neumann boundary condition on the line x = 0 and with dynamic boundary condition on the line x = kt. For the solution of this problem, a coercive estimate in a weighted Hölder norm is obtained. It is shown that this estimate can be useful for the analysis of parabolic free boundary problems. Bibliography: 7 titles.