Fractional Sobolev Spaces Research Articles

On Global Solvability and Regularity for Generalized Rayleigh–Stokes Equations with History-Dependent Nonlinearities

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 using fixed point arguments and embeddings of fractional Sobolev spaces. An application to a related inverse source problem is given.

Stochastic wave equation with Lévy white noise

In this article, we study the stochastic wave equation on the entire space $\mathbb{R}^d$, driven by a space-time L\'evy white noise with possibly infinite variance (such as the $\alpha$-stable L\'evy noise). In this equation, the noise is multiplied by a Lipschitz function $\sigma(u)$ of the solution. We assume that the spatial dimension is $d=1$ or $d=2$. Under general conditions on the L\'evy measure of the noise, we prove the existence of the solution, and we show that, as a function-valued process, the solution has a c\`adl\`ag modification in the local fractional Sobolev space of order $r<1/4$ if $d=1$, respectively $r<-1$ if $d=2$.

On stability and regularity for semilinear anomalous diffusion equations perturbed by weak-valued nonlinearities

Various diffusion processes in media with memory are depicted by anomalous diffusion equations. We are concerned with a class of anomalous diffusion equations with the nonlinearity taking values in Hilbert scales of negative order. The fundamental questions on global solvability, stability and regularity of solutions to the Cauchy problem governed by the mentioned equations are taken into account. We obtain some solvability results under different assumptions on the regularity of the nonlinear function. When the nonlinearity becomes less singular, the asymptotic stability of solutions is proved. Provided that the kernel function is 2-regular and sectorial, we show the Hölder continuity of the obtained solutions. Our approach is based on the resolvent theory, fixed point argument and embeddings of fractional Sobolev spaces. The applicability of our results is presented for some anomalous diffusion problems, where the nonlinear function is of polynomial or convection type.

Dynamics for subcritical fractional nonclassical diffusion equations with nonlinear Wong-Zakai noise and delays

The multi-scale stability of random attractors in the space of continuous functions from the delay interval into the fractional Sobolev space for a class of nonclassical diffusion equations on $ \mathbb{R}^n $ with time-delay and nonlinear Wong-Zakai noise is studied. First, we prove the existence of a unique pullback random attractor, which is a family of compact sets depending on two parameters: the step size of noise and the current time. Secondly, we study the semicontinuity of the pullback random attractor as the current time goes to positive infinity. Thirdly, we consider the semicontinuity of the pullback random attractor as the delay time approach zero. Finally, we prove the semicontinuity of the pullback random attractor as the step size of noise tends to positive infinity. In order to overcome the lack of compact Sobolev embeddings on $ \mathbb{R}^n $ and the weak dissipation of equations, we employ the spectral method and idea of uniform tail-estimates to prove that the solution operator is pullback asymptotic compactness over a time-uniformly tempered attracting universe.

Embedding of fractional Sobolev spaces is equivalent to regularity of the measure

Embedding of fractional Sobolev spaces is equivalent to regularity of the measure

A Hadamard fractional total variation-Gaussian (HFTG) prior for Bayesian inverse problems

This paper studies the infinite-dimensional Bayesian inference method with Hadamard fractional total variation-Gaussian (HFTG) prior for solving inverse problems. First, Hadamard fractional Sobolev space is established and proved to be a separable Banach space under some mild conditions. Afterwards, the HFTG prior is constructed in this separable fractional space, and the proposed novel hybrid prior not only captures the texture details of the region and avoids staircase effects, but also provides a complete theoretical analysis in the infinite dimensional Bayesian inversion. Based on the HFTG prior, the well-posedness and finite-dimensional approximation of the posterior measure of the Bayesian inverse problem are given, and samples are extracted from the posterior distribution using the standard preconditioned Crank-Nicolson (pCN) algorithm. Finally, numerical results under different models indicate that the Bayesian inference method with HFTG prior is effective and accurate.

Bourgain–Brezis–Mironescu–Maz’ya–Shaposhnikova limit formulae for fractional Sobolev spaces via interpolation and extrapolation

The real interpolation spaces between \(L^{p}({{\mathbb {R}}}^{n})\) and \({\dot{H}}^{t,p}({{\mathbb {R}}}^{n})\) (resp. \(H^{t,p}({{\mathbb {R}}}^{n})\)), \(t>0,\) are characterized in terms of fractional moduli of smoothness, and the underlying seminorms are shown to be “the correct” fractional generalization of the classical Gagliardo seminorms. This is confirmed by the fact that, using the new spaces combined with interpolation and extrapolation methods, we are able to extend the Bourgain–Brezis–Mironescu–Maz’ya–Shaposhnikova limit formulae, as well as the Bourgain–Brezis–Mironescu convergence theorem, to fractional Sobolev spaces. On the other hand, we disprove a conjecture of Brazke et al. (Bourgain–Brezis–Mironescu convergence via Triebel–Lizorkin spaces. https://arxiv.org/abs/2109.04159) suggesting fractional convergence results given in terms of classical Gagliardo seminorms. We also solve a problem proposed in Brazke et al. (Bourgain–Brezis–Mironescu convergence via Triebel–Lizorkin spaces. https://arxiv.org/abs/2109.04159) concerning sharp forms of the fractional Sobolev embedding.

John—Nirenberg-Q Spaces via Congruent Cubes

To shed some light on the John—Nirenberg space, the authors of this article introduce the John—Nirenberg-Q space via congruent cubes, \(JNQ_{p,q}^\alpha \left( {{\mathbb{R}^n}} \right)\), which, when p = ∞ and q = 2, coincides with the space Qα (ℝn) introduced by Essén, Janson, Peng and Xiao in [Indiana Univ Math J, 2000, 49(2): 575-615]. Moreover, the authors show that, for some particular indices, \(JNQ_{p,q}^\alpha \left( {{\mathbb{R}^n}} \right)\) coincides with the congruent John—Nirenberg space, or that the (fractional) Sobolev space is continuously embedded into \(JNQ_{p,q}^\alpha \left( {{\mathbb{R}^n}} \right)\). Furthermore, the authors characterize \(JNQ_{p,q}^\alpha \left( {{\mathbb{R}^n}} \right)\) via mean oscillations, and then use this characterization to study the dyadic counterparts. Also, the authors obtain some properties of composition operators on such spaces. The main novelties of this article are twofold: establishing a general equivalence principle for a kind of ‘almost increasing’ set function that is here introduced, and using the fine geometrical properties of dyadic cubes to properly classify any collection of cubes with pairwise disjoint interiors and equal edge length.

New Brezis–Van Schaftingen–Yung–Sobolev type inequalities connected with maximal inequalities and one parameter families of operators

Motivated by the recent characterization of Sobolev spaces due to Brezis–Van Schaftingen–Yung we prove new weak-type inequalities for one parameter families of operators connected with mixed norm inequalities. The novelty of our approach comes from the fact that the underlying measure space incorporates the parameter as a variable. We also show that our framework can be adapted to treat related characterizations of Sobolev spaces obtained earlier by Bourgain–Nguyen. The connection to classical and fractional order Sobolev spaces is shown through the use of generalized Riesz potential spaces and the Caffarelli–Silvestre extension principle. Higher order inequalities are also considered. We indicate many examples and applications to PDE's and different areas of Analysis, suggesting a vast potential for future research. In a different direction, and inspired by methods originally due to Gagliardo and Garsia, we obtain new maximal inequalities which combined with mixed norm inequalities are applied to obtain Brezis–Van Schaftingen–Yung type inequalities in the context of Calderón–Campanato spaces. In particular, Log versions of the Gagliardo–Brezis–Van Schaftingen–Yung spaces are introduced and compared with corresponding limiting versions of Calderón–Campanato spaces, resulting in a sharpening of recent inequalities due to Crippa–De Lellis and Brué–Nguyen.

WEAK SOLUTION FOR NONLINEAR FRACTIONAL P(.)-LAPLACIAN PROBLEM WITH VARIABLE ORDER VIA ROTHE'S TIME-DISCRETIZATION METHOD

In this paper, we prove the existence and uniqueness results of weak solutions to a class of nonlinear fractional parabolic p(.)-Laplacian problem with variable order. The main tool used here is the Rothe’s method combined with the theory of variable-order fractional Sobolev spaces with variable exponent.

A Bourgain-Brezis-Mironescu formula for anisotropic fractional Sobolev spaces and applications to anisotropic fractional differential equations

In this paper we prove Bourgain-Brezis-Mironescu's type results (cf. [4]) (BBM for short) for an energy functional which is strongly related to the pseudo anisotropic p-Laplacian. We also provide with the analogous of Maz'ya-Shaposhnikova (see [23]) type results for these energies and finally we apply these results to analyze the stability of solutions to pseudo anisotropic p-Laplace equations.

Some results of capacity in fractional Sobolev spaces with variable exponents

Some results of capacity in fractional Sobolev spaces with variable exponents

A numerical scheme for stochastic differential equations with distributional drift

In this paper we introduce a scheme for the numerical solution of one-dimensional stochastic differential equations (SDEs) whose drift belongs to a fractional Sobolev space of negative regularity (a subspace of Schwartz distributions). We obtain a convergence rate in a suitable L1-norm and, as a by-product, a convergence rate for a numerical scheme applied to SDEs with drift in Lp-spaces with p∈(1,∞).

A Revisit of the Velocity Averaging Lemma: On the Regularity of Stationary Boltzmann Equation in a Bounded Convex Domain

In the present work, we adopt the idea of velocity averaging lemma to establish regularity for stationary linearized Boltzmann equations in a bounded convex domain. Considering the incoming data, with four iterations, we establish regularity in fractional Sobolev space in space variable up to order \(1^-\).

Existence and multiplicity of solutions for critical nonlocal equations with variable exponents

ABSTRACT In this paper, we are interested in a class of critical nonlocal problems with variable exponents of the form: where M is the Kirchhoff function, λ is a real parameter and f is a continuous function. We also assume that , where is the critical Sobolev exponent for variable exponents. The strategy of the proof for these results is to approach the problem variationally by using the mountain pass theorem and the concentration-compactness principles for fractional Sobolev spaces with variable exponents. In addition, we obtain the existence and multiplicity of nontrivial solutions for the above problem in non-degenerate and degenerate cases.

Robust Monolithic Solvers for the Stokes--Darcy Problem with the Darcy Equation in Primal Form

We construct mesh-independent and parameter-robust monolithic solvers for the coupled primal Stokes-Darcy problem. Three different formulations and their discretizations in terms of conforming and non-conforming finite element methods and finite volume methods are considered. In each case, robust preconditioners are derived using a unified theoretical framework. In particular, the suggested preconditioners utilize operators in fractional Sobolev spaces. Numerical experiments demonstrate the parameter-robustness of the proposed solvers.

Error analysis of a class of semi-discrete schemes for solving the Gross–Pitaevskii equation at low regularity

We analyze a class of time discretizations for solving the nonlinear Schrödinger equation with non-smooth potential and at low-regularity on an arbitrary Lipschitz domain Ω⊂Rd, d≤3. We show that these schemes, together with their optimal local error structure, allow for convergence under lower regularity assumptions on both the solution and the potential than is required by classical methods, such as splitting or exponential integrator methods. Moreover, we show first and second order convergence in the case of periodic boundary conditions, in any fractional positive Sobolev space Hr, r≥0, beyond the more typical L2 or Hσ(σ>d2) -error analysis. Numerical experiments illustrate our results.

Parameter-robust methods for the Biot–Stokes interfacial coupling without Lagrange multipliers

In this paper we advance the analysis of discretizations for a fluid-structure interaction model of the monolithic coupling between the free flow of a viscous Newtonian fluid and a deformable porous medium separated by an interface. A five-field mixed-primal finite element scheme is proposed solving for Stokes velocity-pressure and Biot displacement-total pressure-fluid pressure. Adequate inf-sup conditions are derived, and one of the distinctive features of the formulation is that its stability is established robustly in all material parameters. We propose robust preconditioners for this perturbed saddle-point problem using appropriately weighted operators in fractional Sobolev and metric spaces at the interface. The performance is corroborated by several test cases, including the application to interfacial flow in the brain.

Generalized Navier–Stokes Equations with Non-Homogeneous Boundary Conditions

We study the generalized unsteady Navier–Stokes equations with a memory integral term under non-homogeneous Dirichlet boundary conditions. Using a suitable fractional Sobolev space for the boundary data, we introduce the concept of strong solutions. The global-in-time existence and uniqueness of a small-data strong solution is proved. For the proof of this result, we propose a new approach. Our approach is based on the operator treatment of the problem with the consequent application of a theorem on the local unique solvability of an operator equation involving an isomorphism between Banach spaces with continuously Fréchet differentiable perturbations.

A gradient-based regularization algorithm for the Cauchy problem in steady-state anisotropic heat conduction

We study the numerical reconstruction of the missing thermal boundary data on a portion of the boundary occupied by an anisotropic solid in the case of the steady-state heat conduction equation from the knowledge of both the temperature and the normal heat flux (i.e. Cauchy data) on the remaining and accessible part of the boundary. This inverse problem is known to be ill-posed and hence a regularization procedure is required. Herein we develop a solver for this problem by exploiting two sources of regularization, namely the smoothing nature of the corresponding direct problems and a priori knowledge on the solution to the inverse problem investigated. Consequently, this inverse problem is reformulated as a control one which reduces to minimising a corresponding functional defined on a fractional Sobolev space on the inaccessible part of the boundary. This approach yields a gradient-based iterative algorithm that consists, at each step, of the resolution of two direct problems and three corresponding adjoint problems in accordance with the function space where the control is sought. The theoretical convergence of the algorithm is studied by deriving an iteration-dependent admissible range for the parameter. Numerical experiments are realized for the two-dimensional case by employing the finite-difference method, whilst the numerical solution is stabilised/regularized by stopping the iterative process based on three criteria.