Sobolev Spaces Of Order Research Articles

On the weak solutions for the rotation-two-component Camassa–Holm equation

This paper deals with a model the equatorial water waves with the Coriolis effect in the rotating fluid, which is called rotation-two-component Camassa–Holm system. The purpose of this work is to utilize the pseudo-parabolic regularization to establish the existence and uniqueness of weak solutions in a lower order Sobolev space Hs(R)×Hs−1(R) with 1<s≤32.

The sharp Sobolev type inequalities in the Lorentz–Sobolev spaces in the hyperbolic spaces

Let W1Lp,q(Hn), 1≤q,p<∞ denote the Lorentz–Sobolev spaces of order one in the hyperbolic spaces Hn. Our aim in this paper is three-fold. First of all, we establish a sharp Poincaré inequality in W1Lp,q(Hn) with 1≤q≤p which generalizes the result in [41] to the setting of Lorentz–Sobolev spaces. Second, we prove several sharp Poincaré–Sobolev type inequalities in W1Lp,q(Hn) with 1≤q≤p<n which generalize the results in [45] to the setting of Lorentz–Sobolev spaces. Finally, we provide the improved Moser–Trudinger type inequalities in W1Ln,q(Hn) in the critical case p=n with 1≤q≤n which generalize the results in [43] and improve the results in [57]. In the proof of the main results, we shall prove a Pólya–Szegö type principle in W1Lp,q(Hn) with 1≤q≤p which maybe is of independent interest.

The conditional law of the Bacry–Muzy and Riemann–Liouville log correlated Gaussian fields and their GMC, via Gaussian Hilbert and fractional Sobolev spaces

We compute E(Xt|(Xs)0≤s≤L) for the standard Bacry–Muzy log-correlated Gaussian field X with covariance log+Tt−s, which corrects the finite-horizon prediction formula in Vargas et al. (Duchon et al., 0000). The problem can be viewed as a linear filtering problem, and we solve the problem by showing that the L2(P) closure of {∫[0,L]ϕ(s)Xsds:ϕ∈S,supp(ϕ)⊆[0,L]} is equal to {X(ϕ):ϕ∈H−12,supp(ϕ)⊆[0,L]}, where X(ϕ) is defined as a continuous linear extension of X acting on S⊂Hs, Hs denotes the fractional Sobolev space of order s and P is the law of the field X on the space of tempered distributions. The explicit formula for the filter is obtained as the solution to a Fredholm integral equation of the first kind with logarithmic kernel. From this we characterize the conditional law of the Gaussian multiplicative chaos (GMC) Mγ generated by X, using that Mγ is measurable with respect to X. We also outline how one can adapt this result for the Riemann–Liouville GMC introduced in Forde et al. (2019), which has a natural application to the Rough Bergomi volatility model in the H→0 limit.11We would like to thank Juhan Aru, Janne Junilla, Vincent Vargas and Lauri Vitasaari for helpful discussions.

Local well-posedness of the Boltzmann equation with polynomially decaying initial data

<p style='text-indent:20px;'>We consider the Cauchy problem for the spatially inhomogeneous non-cutoff Boltzmann equation with polynomially decaying initial data in the velocity variable. We establish short-time existence for any initial data with this decay in a fifth order Sobolev space by working in a mixed <inline-formula><tex-math id="M1">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ L^\infty $\end{document}</tex-math></inline-formula> space that allows to compensate for potential moment generation and obtaining new estimates on the collision operator that are well-adapted to this space. Our results improve the range of parameters for which the Boltzmann equation is well-posed in this decay regime, as well as relax the restrictions on the initial regularity. As an application, we can combine our existence result with the recent conditional regularity estimates of Imbert-Silvestre (arXiv:1909.12729 [math.AP]) to conclude solutions can be continued for as long as the mass, energy, and entropy densities remain under control. This continuation criterion was previously only available in the restricted range of parameters of previous well-posedness results for polynomially decaying initial data.

Remarks on a limiting case of Hardy type inequalities

The classical Hardy inequality holds in Sobolev spaces $W_0^{1,p}$ when $1\le p< N$. In the limiting case where $p=N$, it is known that by adding a logarithmic function to the Hardy potential, some inequality which is called the critical Hardy inequality holds in $W_0^{1,N}$. In this note, in order to give an explanation of appearance of the logarithmic function at the potential, we derive the logarithmic function from the classical Hardy inequality with the best constant via some limiting procedure as $p \nearrow N$. And we show that our limiting procedure is also available for the classical Rellich inequality in second order Sobolev spaces $W_0^{2,p}$ with $p \in (1, \frac{N}{2})$ and the Poincare inequality.

Null controllability of a coupled degenerate system with the first and zero order terms by a single control

This paper concerns the null controllability of a system ofmlinear degenerate parabolic equations with coupling terms of first and zero order, and only one control force localized in some arbitrary nonempty open subsetωof Ω. The key ingredient for proving the null controllability is to obtain the observability inequality for the corresponding adjoint system. Due to the degeneracy, we transfer to study an approximate nondegenerate adjoint system. In order to deal with the coupling first order terms, we first prove a new Carleman estimate for a degenerate parabolic equation in Sobolev spaces of negative order. Based on this Carleman estimate, we obtain a uniform Carleman estimate and then an observation inequality for this approximate adjoint system.

A trace theorem for Sobolev spaces on the Sierpinski gasket

We give a discrete characterization of the trace of a class of Sobolev spaces on the Sierpinski gasket to the bottom line. This includes the L2 domain of the Laplacian as a special case. In addition, for Sobolev spaces of low orders, including the domain of the Dirichlet form, the trace spaces are Besov spaces on the line.

On the Approximation of Electromagnetic Fields by Edge Finite Elements. Part 3: Sensitivity to Coefficients

In bounded domains, the regularity of the solutions to boundary value problems depends on the gometry, and on the coefficients that enter into the definition of the model. This is in particular the case for the time-harmonic Maxwell equations, whose solutions are the electromagnetic fields. In this paper, emphasis is put on the electric field. We study the regularity in terms of the fractional order Sobolev spaces $H^s$, $s\in[0,1]$. Precisely, our first goal is to determine the regularity of the electric field and of its curl, that is to find some regularity exponent $\tau\in(0,1)$, such that they both belong to $H^s$, for all $s\in[0,\tau)$. After that, one can derive error estimates. Here, the error is defined as the difference between the exact field and its approximation, where the latter is built with N\'ed\'elec's first family of finite elements. In addition to the regularity exponent, one needs to derive a stability constant that relates the norm of the error to the norm of the data: this is our second goal. We provide explicit expressions for both the regularity exponent and the stability constant with respect to the coefficients. We also discuss the accuracy of these expressions, and we provide some numerical illustrations.

On approximate solutions of the equations of incompressible magnetohydrodynamics

Inspired by an approach proposed previously for the incompressible Navier–Stokes (NS) equations, we present a general framework for the a posteriori analysis of the equations of incompressible magnetohydrodynamics (MHD) on a torus of arbitrary dimension d; this setting involves a Sobolev space of infinite order, made of C∞ vector fields (with vanishing divergence and mean) on the torus. Given any approximate solution of the MHD Cauchy problem, its a posteriori analysis with the method of the present work allows one to infer a lower bound on the time of existence of the exact solution, and to bound from above the Sobolev distance of any order between the exact and the approximate solution. In certain cases the above mentioned lower bound on the time of existence is found to be infinite, so one infers the global existence of the exact MHD solution. We present some applications of this general scheme; the most sophisticated one lives in dimension d=3, with the ABC flow (perturbed magnetically) as an initial datum, and uses for the Cauchy problem a Galerkin approximate solution in 124 Fourier modes. We illustrate the conclusions arising in this case from the a posteriori analysis of the Galerkin approximant; these include the derivation of global existence of the exact MHD solution with the ABC datum, when the dimensionless viscosity and resistivity are equal and stay above a critical value.

Characterization of the function spaces associated with weighted Sobolev spaces of the first order on the real line

A brief survey of results on the characterization of the spaces associated with given classes of function spaces is presented. It is shown that the situation differs in general for ideal and non-ideal spaces. In the second case the notion of associated space splits into two. In the main body of the text a complete description is given of the function spaces associated with weighted Sobolev spaces of the first order on the real line. Bibliography: 54 titles.

A supercritical Sobolev type inequality in higher order Sobolev spaces and related higher order elliptic problems

A Sobolev type embedding for radially symmetric functions on the unit ball B in Rn, n≥3, into the variable exponent Lebesgue space L2⋆+|x|α(B), 2⋆=2n/(n−2), α>0, is known due to J.M. do Ó, B. Ruf, and P. Ubilla, namely, the inequalitysup⁡{∫B|u(x)|2⋆+|x|αdx:u∈H0,rad1(B),‖∇u‖L2(B)=1}<+∞ holds. In this work, we generalize the above inequality for higher order Sobolev spaces of radially symmetric functions on B, namely, the embeddingH0,radm(B)↪L2m⋆+|x|α(B) with 2≤m<n/2, 2m⁎=2n/(n−2m), and α>0 holds. Questions concerning the sharp constant for the inequality including the existence of the optimal functions are also studied. To illustrate the finding, an application to a boundary value problem on balls driven by polyharmonic operators is presented. This is the first in a set of our works concerning functional inequalities in the supercritical regime.

Dyadic Norm Besov-Type Spaces as Trace Spaces on Regular Trees

In this paper, we study function spaces defined via dyadic energies on the boundaries of regular trees. We show that correct choices of dyadic energies result in Besov-type spaces that are trace spaces of (weighted) first order Sobolev spaces.

Characterization of the function spaces associated with weighted Sobolev spaces of the first order on the real line

В работе дается краткий обзор результатов по проблеме характеризации пространств, ассоциированных с заданными функциональными классами. Показано, что для идеальных и неидеальных структур ситуация, вообще говоря, различна. В последнем случае понятие ассоциированного пространства раздваивается. В основной части статьи представлено полное описание функциональных пространств, ассоциированных с весовыми пространствами Соболева первого порядка на действительной оси. Библиография: 54 названия.

Well-posedness for a perturbation of the KdV equation

This is a continuation of Carvajal and Panthee (Q Appl Math 74:571–594, 2016). In the present paper we study a dissipative versions of the Korteweg de Vries equation with rough initial data. By working in Bourgain’s type spaces we prove local well posedness results in Sobolev spaces of negative order.

Petrov–Galerkin space-time hp-approximation of parabolic equations in H1/2

Abstract We analyse a class of variational space-time discretizations for a broad class of initial boundary value problems for linear, parabolic evolution equations. The space-time variational formulation is based on fractional Sobolev spaces of order $1/2$ and the Riemann–Liouville derivative of order $1/2$ with respect to the temporal variable. It accommodates general, conforming space discretizations and naturally accommodates discretization of infinite horizon evolution problems. We prove an inf-sup condition for $hp$-time semidiscretizations with an explicit expression of stable test functions given in terms of Hilbert transforms of the corresponding trial functions; inf-sup constants are independent of temporal order and the time-step sequences, allowing quasi-optimal, high-order discretizations on graded time-step sequences, and also $hp$-time discretizations. For solutions exhibiting Gevrey regularity in time and taking values in certain weighted Bochner spaces, we establish novel exponential convergence estimates in terms of $N_t$, the number of (elliptic) spatial problems to be solved. The space-time variational setting allows general space discretizations and, in particular, for spatial $hp$-FEM discretizations. We report numerical tests of the method for model problems in one space dimension with typical singular solutions in the spatial and temporal variable. $hp$-discretizations in both spatial and temporal variables are used without any loss of stability, resulting in overall exponential convergence of the space-time discretization.

Null-controllability properties of a fractional wave equation with a memory term

We study the null-controllability properties of a one-dimensional wave equation with memory associated with the fractional Laplace operator. The goal is not only to drive the displacement and the velocity to rest at some time-instant but also to require the memory term to vanish at the same time, ensuring that the whole process reaches the equilibrium. The problem being equivalent to a coupled nonlocal PDE-ODE system, in which the ODE component has zero velocity of propagation, we are required to use a moving control strategy. Assuming that the control is acting on an open subset $ \omega(t) $ which is moving with a constant velocity $ c\in\mathbb{R} $, the main result of the paper states that the equation is null controllable in a sufficiently large time $ T $ and for initial data belonging to suitable fractional order Sobolev spaces. The proof will use a careful analysis of the spectrum of the operator associated with the system and an application of a classical moment method.

Formula omitted]-solutions for stochastic Navier–Stokes equations with jump noise

We study the existence and uniqueness of solutions of 2D Stochastic Navier–Stokes equation with space irregular jump noise for initial data in certain Sobolev spaces of negative order. Comparing with the Galerkin approximation method, the main advantage of this work is to use an Lp-setting to obtain the solution under much weaker assumptions on the noise and the initial condition.

Normal approximation of the solution to the stochastic heat equation with Lévy noise

Given a sequence $\dot{L}^{\varepsilon}$ of L\'evy noises, we derive necessary and sufficient conditions in terms of their variances $\sigma^2(\varepsilon)$ such that the solution to the stochastic heat equation with noise $\sigma(\varepsilon)^{-1} \dot{L}^\varepsilon$ converges in law to the solution to the same equation with Gaussian noise. Our results apply to both equations with additive and multiplicative noise and hence lift the findings of S. Asmussen and J. Rosi\'nski [J. Appl. Probab. 38 (2001) 482-493] and S. Cohen and J. Rosi\'nski [Bernoulli 13 (2007) 195-210] for finite-dimensional L\'evy processes to the infinite-dimensional setting without making distributional assumptions on the solutions such as infinite divisibility. One important ingredient of our proof is to characterize the solution to the limit equation by a sequence of martingale problems. To this end, it is crucial to view the solution processes both as random fields and as c\`adl\`ag processes with values in a Sobolev space of negative real order.

Density Problems for second order Sobolev Spaces and Cut-off Functions on Manifolds With Unbounded Geometry

AbstractWe consider complete non-compact manifolds with either a sub-quadratic growth of the norm of the Riemann curvature, or a sub-quadratic growth of both the norm of the Ricci curvature and the squared inverse of the injectivity radius. We show the existence on such a manifold of a distance-like function with bounded gradient and mild growth of the Hessian. As a main application, we prove that smooth compactly supported functions are dense in $W^{2,p}$. The result is improved for $p=2$ avoiding both the upper bound on the Ricci tensor, and the injectivity radius assumption. As further applications we prove new disturbed Sobolev and Calderón–Zygmund inequalities on manifolds with possibly unbounded curvature and highlight consequences about the validity of the full Omori–Yau maximum principle for the Hessian.

The fractional Calderón problem: Low regularity and stability

The Calderón problem for the fractional Schrödinger equation was introduced in the work Ghosh et al. (to appear) which gave a global uniqueness result also in the partial data case. This article improves this result in two ways. First, we prove a quantitative uniqueness result showing that this inverse problem enjoys logarithmic stability under suitable a priori bounds. Second, we show that the results are valid for potentials in scale-invariant Lp or negative order Sobolev spaces. A key point is a quantitative approximation property for solutions of fractional equations, obtained by combining a careful propagation of smallness analysis for the Caffarelli–Silvestre extension and a duality argument.