Embedding Inequalities Research Articles

Interplay of Sobolev Spaces on Compact Manifolds: Embedding Theorems, Inequalities, and Compactness

This research paper explores various properties of Sobolev spaces on compact manifolds, focusing on embedding theorems, compactness, and inequalities. We establish the compact embedding of Sobolev spaces into continuous and Lebesgue spaces, as well as the continuity and compactness of embeddings between different Sobolev spaces. We also derive inequalities involving the Laplacian and gradients of functions, providing insights into their behavior on manifolds. These results contribute to our understanding of the interplay between function smoothness, continuity, and distribution on compact manifolds.

Richardson extrapolation method for solving the Riesz space fractional diffusion problem

AbstractThe Richardson extrapolation is widely utilized in numerically solving the differential equations since it enjoys both high accuracy and ease of implementation. For the Riesz space fractional diffusion equation, the fractional centered difference operator is employed to approximate the fractional derivative, then the Richardson extrapolation methods for two difference schemes are constructed with the help of the asymptotic expansions of the discrete solutions. Specifically, for the Crank–Nicolson difference scheme, the extrapolation method contains two extrapolation formulae that achieve the fourth order and the sixth order both in the temporal and spatial directions, respectively. The extrapolation method for the compact difference scheme involves one extrapolation formula by which the sixth order can be obtained when the time step size is proportional to the squares of the space step size. The maximum norm error estimates of the extrapolation solutions are proved by the discrete fractional Sobolev embedding inequalities. The extension to the high dimensional and nonlinear cases is also demonstrated. Numerical results verify the theoretical convergence orders and efficiency of our methods.

Unconditional convergence analysis of two linearized Galerkin FEMs for the nonlinear time-fractional diffusion-wave equation

In this paper, we present and analyze two linearized Galerkin finite element schemes, which are constructed by employing the H2N2 formula and its fast version in time direction, for solving the nonlinear time-fractional diffusion-wave equation. By utilizing mathematical induction, the optimal error estimates in H1-norm are derived without any ratio restrictions between the time step size τ and the space mesh size h. The key point in our argument is the application of Sobolev’s embedding inequality to the fully discrete solution uhn. On the other hand, additional time-discrete elliptic system and the inverse inequality, which play a vital role in the temporal–spatial error splitting technique, are avoided in our numerical analysis. Finally, two numerical experiments are given to demonstrate the theoretical findings.

Note on a higher order pseudo‐parabolic equation with variable exponents

In this paper, we study a higher order pseudo‐parabolic equation involving ‐Laplacian with the Navier boundary condition. We use the energy method, the Sobolev embedding inequalities and the Galerkin's approximation to show the classification of singular solutions, including the existence and nonexistence of global, blow‐up, and extinction solutions. Extinction rates and time of extinction solutions, blow‐up time of blow‐up solutions, and decay estimates of global solutions are discussed.

High-order numerical algorithm and error analysis for the two-dimensional nonlinear spatial fractional complex Ginzburg–Landau equation

In this paper, we first construct an appropriate new generating function, and then based on this function, we establish a fourth-order numerical differential formula approximating the Riesz derivative with order γ∈(1,2]. Subsequently, we apply the formula to numerically study the two-dimensional nonlinear spatial fractional complex Ginzburg–Landau equation and obtain a difference scheme with convergence order Oτ2+hx4+hy4, where τ denotes the time step size, hx and hy denote the space step sizes, respectively. Furthermore, with the help of some newly derived discrete fractional Sobolev embedding inequalities, the unique solvability, the unconditional stability, and the convergence of the constructed numerical algorithm under different norms are proved by using the discrete energy method. Finally, some numerical results are presented to confirm the correctness of the theoretical results and verify the effectiveness of the proposed scheme.

Unconditional superconvergence analysis of two modified finite element fully discrete schemes for nonlinear Burgers' equation

In this paper, two modified nonconforming finite element fully discrete schemes are considered for solving the nonlinear Burgers' equation. The schemes are constructed by using the nonconforming Wilson element approximation in space combining with the backward Euler(BE) and 2-step backward differentiation formula (BDF2) respectively in time. We present the superconvergence results of the two proposed schemes in modified energy norm of order O(h2+τ) and O(h2+τ2), where h and τ are the space subdivision parameter and time step respectively. The proof involves the time-space error splitting technique, the discrete embedding inequality, the technique of taking difference between two time, the discrete derivative transfer technique and a new modified projection operator. Our analysis not only gets rid of the restriction between temporal and spatial step sizes, but also provides one order higher error estimate results than that of corresponding traditional finite element fully discrete schemes. Finally, numerical experiments are carried out to demonstrate the effective of theoretical analysis.

Configuration-aware model predictive motion planning for Tractor–Trailer Mobile Robot

ABSTRACT This study proposes a motion planning method with strict obstacle avoidance constraints among polygonal-shaped objects for a tractor–trailer mobile robot. The conventional method generally expresses collision-free constraints by approximating a robot as one circle or several circles or ellipsoids. However, accomplishing moving tasks with minimum margins is desirable, with improved efficiency of space availability when multiple robots operate in one area such as a factory, restaurant, office, or warehouse. The target of this study is a tractor–trailer mobile robot that can transport more goods efficiently, while significantly reducing margins during movement. To control the target, multiple convex polygonal objects expressing the mobile robot and surroundings are transformed into linear simultaneous inequalities using ‘Farkas’ lemma.' Embedding inequalities into model predictive control yields a motion planner that allows tractor–trailer mobile robot movement and possesses non-holonomic constraints. The proposed model is validated through simulation and real machine experiments.

Finite difference schemes for the fourth‐order parabolic equations with different boundary value conditions

AbstractIn this paper, the fourth‐order parabolic equations with different boundary value conditions are studied. Six kinds of boundary value conditions are proposed. Several numerical differential formulae for the fourth‐order derivative are established by the quartic interpolation polynomials and their truncation errors are given with the aid of the Taylor expansion with the integral remainders. Effective difference schemes are presented for the third Dirichlet boundary value problem, the first Neumann boundary value problem and the third Neumann boundary value problem, respectively. Some new embedding inequalities on the discrete function spaces are presented and proved. With the method of energy analysis, the unique solvability, unconditional stability and unconditional convergence of the difference schemes are proved. The convergence orders of derived difference schemes are all O(τ2 + h2) in appropriate norms. Finally, some numerical examples are provided to confirm the theoretical results.

Mesh-Robustness of an Energy Stable BDF2 Scheme with Variable Steps for the Cahn–Hilliard Model

The two-step backward differential formula (BDF2) with unequal time-steps is applied to construct an energy stable convex-splitting scheme for the Cahn-Hilliard model. We focus on the numerical influences of time-step variations by using the recent theoretical framework with the discrete orthogonal convolution kernels. Some novel discrete convolution embedding inequalities with respect to the orthogonal convolution kernels are developed such that a concise $L^2$ norm error estimate is established at the first time under an updated step-ratio restriction $0 <r_k:=\tau_k/\tau_{k-1}\leq r_{\mathrm{user}}$, where $r_{\mathrm{user}}$ can be chosen by the user such that $r_{\mathrm{user}}<4.864$. The stabilized convex-splitting BDF2 scheme is shown to be mesh-robustly convergent in the sense that the convergence constant (prefactor) in the error estimate is independent of the adjoint time-step ratios. The suggested method is proved to preserve a modified energy dissipation law at the discrete levels if $0<r_k\le r_{\mathrm{user}}$, such that it is mesh-robustly stable in an energy norm. On the basis of ample tests on random time meshes, a useful adaptive time-stepping strategy is applied to efficiently capture the multi-scale behaviors and to accelerate the long-time simulation approaching the steady state.

A Note on Embedding Inequalities for Weighted Sobolev and Besov Spaces

In this paper, we establish two embedding inequalities for the weighted Sobolev space and the weighted homogeneous endpoint Besov space by using the weighted Hausdorff capacity. To do this, we shall determine the dual spaces of weighted Choquet and weighted homogeneous Besov spaces.

On a critical biharmonic system involving [formula omitted]-Laplacian and Hardy potential

In this paper, I establish a useful embedding inequality in D2,2(Rn,R2). Benefiting from the previous inequality, I also obtain a nontrivial weak solution to a critical biharmonic system involving p-Laplacian and Hardy potential via variational methods.

Global well-posedness of strong solutions to the 2D nonhomogeneous incompressible primitive equations with vacuum

We prove the global well-posedness of the strong solution to the two-dimensional inhomogeneous incompressible primitive equations for initial data with small H12-norm, which also satisfies a natural compatibility condition. A logarithmic type Sobolev embedding inequality for the anisotropic Lx∞Lz2(Ω) norm is established to obtain the global in time a priori H1(Ω)∩W1,6(Ω) estimate of the density, which guarantee the local solution to be a global one.

Asymptotic property of singular solutions in some nonstandard parabolic equation

In this paper, we study the asymptotic properties of singular solutions to some nonstandard parabolic equation involving p(x,t)-Laplacian and nonlocal anisotropic sources. First, we show the existence and uniqueness of weak solutions by using the Galerkin’s approximations in the anisotropic Orlicz–Sobolev spaces. Second, in order to determine extinction rates and time of solutions, we prove some ordinary differential inequalities with the Sobolev embedding inequalities. Third, we combine priori estimates and the embedding theorems to show the rates and time estimates of blow-up solutions.

Some Characterization of the Function Space Type of Lizorkin−Triebel−Morrey

This paper will introduce you to some properties of normed function spaces with many groups variables field of Analysis and it helps me appreciate how normed Lebesgue–Morrey space with many groups of variables that build and studied new normed spaces nowadays. Many of the topics here are important to an Analysis class. By reading this paper, you will discover the “embedding theory” of normed spaces type of Lebesgue–Morrey by introducing few of its “new functions with groups with variables” and along the way you will see to some interesting and article elements of the branch called Analysis. A lot of problems belonging to the characterization of various spaces of differentiability function spaces and relationships between them have been solved using the theory embedding theorems. The purpose of this paper is to review several embedding inequalities of normed spaces that will arise properties of these spaces and again throughout this material. We also give “working definition, notations” of a functions and function spaces. We must note that, the analysis is based on such function spaces to build new space type of Lizorkin–Triebel–Morrey. In addition, throughout this paper we will introduce a working normed function spaces type of Lizorkin–Triebel–Morrey with standard mathematical definitions and terminology. One aspect of this paper involves normed Lebesgue–Morey type spaces that can convert space from one to another.

Global existence and blowup of solutions to semilinear fractional reaction-diffusion equation with singular potential

We consider the following semilinear fractional reaction-diffusion equation with singular potential{Asu=−ut|x|2s+|u|p−2u,(x,t)∈Ω×(0,∞),u=0,(x,t)∈(RN∖Ω)×[0,∞),u(x,0)=u0,x∈Ω. Here As(0<s<1) represents spectral fractional Laplacian operator, 2<p<2s⁎, 2s⁎=2NN−2s is critical exponent of fractional Sobolev trace embedding inequality, N>2s, and Ω is bounded domain in RN with smooth boundary. We first prove that the solution exists globally under appropriate hypotheses. And due to the non-locality of fractional Laplacian operator, we use the Caffarelli-Silvestre extension method to convert the non-local problem into a variable local problem. Being based upon this, by means of potential well, we obtain decay estimate and long time asymptotic behavior of global solution, as well as blow-up behavior of local solution.

A complete classification of initial energy in a [formula omitted]-Laplace pseudo-parabolic equation

This paper deals with a pseudo-parabolic equation involving p(x)-Laplacian, which has been studied in Di et al. (2017), Zhu et al. (2020) and Liao (2019). By the variational methods and embedding inequalities, we determine the complete classification of blow-up and global existence of weak solutions in some Sobolev space with variable exponent by covering all kinds of initial energy characterized with the mountain pass level.

Numerical analysis of a three-species chemotaxis model

A reaction–diffusion system is formulated to describe three interacting species within the Hastings–Powell (HP) food chain structure with chemotaxis produced by three chemicals. We construct a finite volume (FV) scheme for this system, and in combination with the non-negativity and a priori estimates for the discrete solution, the existence of a discrete solution of the FV scheme is proven. It is shown that the scheme converges to the corresponding weak solution of the model. The convergence proof uses two ingredients of interest for various applications, namely the discrete Sobolev embedding inequalities with general boundary conditions and a space–time L1 compactness argument. Finally, numerical tests illustrate the model and the behavior of the FV scheme.

Bounds for the Lifespan of Solutions to Fourth-order Hyperbolic Equations with Initial Data at Arbitrary Energy Level

This paper deals with lower and upper bounds for the lifespan of solutions to a fourth-order nonlinear hyperbolic equation with strong damping: \[ u_{tt} + \Delta^{2} u - \Delta u - \omega \Delta u_t + \alpha(t) u_t = |u|^{p-2} u. \] First of all, the authors construct a new control function and apply the Sobolev embedding inequality to establish some qualitative relationships between initial energy value and the norm of the gradient of the solution for supercritical case ($2(N-2)/(N-4) \lt p \lt 2N/(N-4)$, $N \geq 5$). And then, the concavity argument is used to prove that the solution blows up in finite time for initial data at low energy level, at the same time, an estimate of the upper bound of blow-up time is also obtained. Subsequently, for initial data at high energy level, the authors prove the monotonicity of the $L^{2}$ norm of the solution under suitable assumption of initial data, furthermore, we utilize the concavity argument and energy methods to prove that the solution also blows up in finite time for initial data at high energy level. At last, for the supercritical case, a new control functional with a small dissipative term and an inverse Hölder inequality with correction constants are employed to overcome the difficulties caused by the failure of the embedding inequality ($H^{2}(\Omega) \cap H^{1}_{0}(\Omega) \hookrightarrow L^{2p-2}$ for $2(N-2)/(N-4) \lt p \lt 2N/(N-4)$) and then an explicit lower bound for blow-up time is obtained. Such results extend and improve those of [S. T. Wu, J. Dyn. Control Syst. 24 (2018), no. 2, 287--295].

Global existence and energy decay estimates for weak solutions to the pseudo‐parabolic equation with variable exponents

This paper deals with the following pseudo‐parabolic equation under initial and Dirichlet boundary value conditions. The authors establish some qualitative relationships by constructing a new control function and applying the Sobolev embedding inequality, and then decay estimates are obtained due to a key integral inequality for p−&gt;2. It is shown by the Galerkin's approximation method that weak solutions exist globally for p+ ≤ 2. Furthermore, nonextinction of the global weak solution is also obtained for negative initial energy by using a new differential inequality. These results improve and extend a recent result showed by Di, Shang, Peng (Appl. Math. Lett. 64 (2017) 67‐73).

On the global regularity for the anisotropic dissipative surface quasi-geostrophic equation

In this paper, we consider the two-dimensional surface quasi-geostrophic equation with fractional horizontal dissipation and fractional vertical thermal diffusion. The global existence of classical solutions is established when the dissipation powers are restricted to a suitable range. Due to the nonlocality of these 1D fractional operators, some of the standard energy estimate techniques no longer apply. To overcome this difficulty, we establish several anisotropic embedding and interpolation inequalities involving fractional derivatives. In addition, in order to bypass the unavailability of the classical Gronwall inequality, we establish a new logarithmic-type Gronwall inequality, which may be of independent interest and have potential applications.