Gagliardo Nirenberg Inequality Research Articles

Combined effects in coupled quasilinear elliptic systems via a “like” Gagliardo–Nirenberg inequality

Combined effects in coupled quasilinear elliptic systems via a “like” Gagliardo–Nirenberg inequality

On the long-time behavior of the continuous and discrete solutions of a nonlocal Cahn–Hilliard type inpainting model

In this paper, we study the analytical and numerical long-time stability of a nonlocal Cahn–Hilliard model with a fidelity term for image inpainting introduced in our previous work (Jiang et al., 2024). First, we establish the uniform boundedness of the continuous problem in both L2 and H1 spaces, which is obtained by using the Gagliardo–Nirenberg inequality and the uniform Grönwall lemma. Then, for the temporal semi-discrete scheme, the uniform estimates in L2 and H1 spaces are derived with the aid of the discrete uniform Grönwall lemma under a suitable assumption on the nonlinear potential. This demonstrates the long-time stability of the proposed scheme in L2 and H1 spaces. Finally, we validate the long-time stability and the applicability of our method in signal reconstruction and image inpainting. These numerical experiments demonstrate the high effectiveness of our proposed model.

Sobolev spaces on arbitrary domains and semigroups generated by the fractional Laplacian

We describe a procedure to introduce on an arbitrary domain of Rd Sobolev spaces of arbitrary order s∈R and with an intrinsic characterization via fractional powers of the Dirichlet Laplacian. In particular, the well-definedness of the spaces of both non-homogeneous and homogeneous type together with their duality properties, embeddings, and Gagliardo-Nirenberg inequalities will be discussed. We also show the continuity and the smoothing property of the semigroup generated by the fractional Dirichlet Laplacian.

The well-posedness of semilinear fractional dissipative equations on [formula omitted]

In this paper, we study the well-posedness of time-space fractional dissipative equations. Combining the Hörmander-multipliers theory and asymptotic property of the Mittag-Leffler functions, we give a useful method to estimate the solution operator which is independent of the C0-semigroup generated by the fractional Laplace operator (−Δ)β2 and the Mainardi's Wright type function. Furthermore, we discuss the global/local well-posedness in some appropriate time-space Lebesgue space and Besov spaces. We also obtain several results about blow-up alternative and asymptotic behavior. The approaches are based on the Hörmander-multipliers theory, Gagliardo-Nirenberg inequalities, fixed point techniques, Sobolev embedding and some methods in harmonic analysis.

Hardy–Sobolev–Rellich, Hardy–Littlewood–Sobolev and Caffarelli–Kohn–Nirenberg Inequalities on General Lie Groups

In this paper, we establish a number of geometrical inequalities such as Hardy, Sobolev, Rellich, Hardy–Littlewood–Sobolev, Caffarelli–Kohn–Nirenberg, Gagliardo-Nirenberg inequalities and their critical versions for an ample class of sub-elliptic differential operators on general connected Lie groups, which include both unimodular and non-unimodular cases in compact and noncompact settings. We also obtain the corresponding uncertainty type principles.

Functional Inequalities on Symmetric Spaces of Noncompact Type and Applications

The aim of this paper is to begin a systematic study of functional inequalities on symmetric spaces of noncompact type of higher rank. Our first main goal of this study is to establish the Stein–Weiss inequality, also known as a weighted Hardy–Littlewood–Sobolev inequality, for the Riesz potential on symmetric spaces of noncompact type. This is achieved by performing delicate estimates of ground spherical function with the use of polyhedral distance on symmetric spaces and by combining the integral Hardy inequality developed by Ruzhansky and Verma with the sharp Bessel-Green-Riesz kernel estimates on symmetric spaces of noncompact type obtained by Anker and Ji. As a consequence of the Stein–Weiss inequality, we deduce Hardy–Sobolev, Hardy–Littlewood–Sobolev, Gagliardo–Nirenberg and Caffarelli–Kohn–Nirenberg inequalities on symmetric spaces of noncompact type. The second main purpose of this paper is to show the applications of aforementioned inequalities for studying nonlinear PDEs on symmetric spaces. Specifically, we show that the Gagliardo-Nirenberg inequality can be used to establish small data global existence results for the semilinear wave equations with damping and mass terms for the Laplace–Beltrami operator on symmetric spaces.

Traveling wave phenomena of inhomogeneous half-wave equation

In this paper, we are concerned with traveling wave phenomena of the inhomogeneous half-wave equation, which models the energy of a spin zero particle in the Coulomb field. We study the Gagliardo-Nirenberg and critical Hardy-Sobolev inequalities with velocity 0<|v|<1 and obtain the estimates for the best constants and optimizers of inequalities. Moreover, we establish the non-scattering results with small traveling wave for energy subcritical and critical cases.

Concentration of blow-up solutions for the Gross-Pitaveskii equation

Abstract We consider the blow-up solutions for the Gross-Pitaveskii equation modeling the attractive Boes-Einstein condensate. First, a new variational characteristic is established by computing the best constant of a generalized Gagliardo-Nirenberg inequality. Then, a lower bound on blow-up rate and a new concentration phenomenon of blow-up solutions are obtained in the L 2 {L}^{2} supercritical case. Finally, in the L 2 {L}^{2} critical case, a delicate limit of blow-up solutions is analyzed.

Singular limit of periodic metric grids

We investigate the asymptotic behavior of nonlinear Schrödinger ground states on d-dimensional periodic metric grids in the limit for the length of the edges going to zero. We prove that suitable piecewise–affine extensions of such states converge strongly in H1(Rd) to the corresponding ground states on Rd. As an application of such convergence results, qualitative properties of ground states and multiplicity results for fixed mass critical points of the energy on grids are derived. Moreover, we compare optimal constants in d-dimensional Gagliardo–Nirenberg inequalities on Rd and on grids. For L2-critical and supercritical powers, we show that the value of such constants on grids is strictly related to that on Rd but, contrary to Rd, constants on grids are not attained. The proofs of these results combine purely variational arguments with new Gagliardo–Nirenberg inequalities on grids.

Sobolev trace-type inequalities via time-space fractional heat equations

Abstract This article aims to establish fractional Sobolev trace inequalities, logarithmic Sobolev trace inequalities, and Hardy trace inequalities associated with time-space fractional heat equations. The key steps involve establishing dedicated estimates for the fractional heat kernel, regularity estimates for the solution of the time-space fractional equations, and characterizing the norm of $\dot {W}^{\nu /2}_p(\mathbb {R}^n)$ in terms of the solution $u(x,t)$ . Additionally, fractional logarithmic Gagliardo–Nirenberg inequalities are proven, leading to $L^p-$ logarithmic Sobolev inequalities for $\dot {W}^{\nu /2}_{p}(\mathbb R^{n})$ . As a byproduct, Sobolev affine trace-type inequalities for $\dot {H}^{-\nu /2}(\mathbb {R}^n)$ and local Sobolev-type trace inequalities for $Q_{\nu /2}(\mathbb {R}^n)$ are established.

On a Generalized Gagliardo–Nirenberg Inequality with Radial Symmetry and Decaying Potentials

We present a generalized version of a Gagliardo–Nirenberg inequality characterized by radial symmetry and involving potentials exhibiting pure power polynomial behavior. As an application of our result, we investigate the existence of extremals for this inequality, which also correspond to stationary solutions for the nonlinear Schrödinger equation with inhomogeneous nonlinearity, competing with Hs-subcritical nonlinearities, either of a local or nonlocal nature.

Logarithmic Sobolev and interpolation inequalities on the sphere: Constructive stability results

We consider Gagliardo–Nirenberg inequalities on the sphere which interpolate between the Poincaré inequality and the Sobolev inequality, and include the logarithmic Sobolev inequality as a special case. We establish explicit stability results in the subcritical regime using spectral decomposition techniques, and entropy and carré du champ methods applied to nonlinear diffusion flows.

Numerical analysis of a hybridized discontinuous Galerkin method for the Cahn–Hilliard problem

Abstract The mixed form of the Cahn–Hilliard equations is discretized by the hybridized discontinuous Galerkin method. For any chemical energy density, existence and uniqueness of the numerical solution is obtained. The scheme is proved to be unconditionally stable. Convergence of the method is obtained by deriving a priori error estimates that are valid for the Ginzburg–Landau chemical energy density and for convex domains. The paper also contains discrete functional tools, namely discrete Agmon and Gagliardo–Nirenberg inequalities, which are proved to be valid in the hybridizable discontinuous Galerkin spaces.

Regularity criterion for 3D generalized Newtonian fluids in BMO

In this paper, we prove that a weak solution of the Cauchy problem for 3D unsteady flows of a generalized Newtonian fluid becomes a strong solution for 53<p<115 provided that the velocity field belongs to the critical space L2p−1(0,T;BMO(R3)). The main key is to apply variants of Gagliardo–Nirenberg's inequality including a BMO-norm. In particular, when 2<p<115, we derive and use a nonlinear variant of Gagliardo–Nirenberg's inequality including ‖u‖BMO and ‖|Du|p−22∇2u‖2.

Fractional Gagliardo–Nirenberg interpolation inequality and bounded mean oscillation

Fractional Gagliardo–Nirenberg interpolation inequality and bounded mean oscillation

On the discrete Sobolev inequalities

Abstract We prove a discrete version of the famous Sobolev inequalities [1] in R d for d ∈ N ∗ , p ∈ [ 1 , + ∞ [ $\mathbb{R}^{d} \text { for } d \in \mathbb{N}^{*}, p \in[1,+\infty[$ for general non orthogonal meshes with possibly non convex cells. We follow closely the proof of the continuous Sobolev inequality based on the embedding of B V R d into L d d − 1 $B V\left(\mathbb{R}^{d}\right) \text { into } \mathrm{L}^{\frac{d}{d-1}}$ [1, theorem 9.9],[12, theorem 1.1] by introducing discrete analogs of the directional total variations. In the case p &gt; d (Gagliardo-Nirenberg inequality), we adapt the proof of the continuous case ( [1, theorem 9.9], [9, theorem 4.8]) and use techniques from [3, 5]. In the case p &gt; d (Morrey’s inequality), we simplify and extend the proof of [12, theorem 1.1] to more general meshes.

Discrete Gagliardo–Nirenberg inequality and application to the finite volume approximation of a convection–diffusion equation with a Joule effect term

Abstract A discrete order-two Gagliardo–Nirenberg inequality is established for piecewise constant functions defined on a two-dimensional structured mesh composed of rectangular cells. As in the continuous framework, this discrete Gagliardo–Nirenberg inequality allows to control in particular the $L^4$ norm of the discrete gradient of the numerical solution by the $L^2$ norm of its discrete Hessian times its $L^\infty $ norm. This result is crucial for the convergence analysis of a finite volume method for the approximation of a convection–diffusion equation involving a Joule effect term on a uniform mesh in each direction. The convergence proof relies on compactness arguments and on a priori estimates under a smallness assumption on the data, which is essential also in the continuous framework.

Stein-Weiss-Adams inequality on Morrey spaces

We establish Adams type Stein-Weiss inequality on global Morrey spaces on general homogeneous groups. Special properties of homogeneous norms and some boundedness results on global Morrey spaces play key roles in our proofs. As consequence, we obtain fractional Hardy, Hardy-Sobolev, Rellich and Gagliardo-Nirenberg inequalities on Morrey spaces on stratified groups. While the results are obtained in the setting of general homogeneous groups, they are new already for the Euclidean space RN.

Ground states for the NLS equation with combined local nonlinearities on noncompact metric graphs

We investigate the existence of ground states for the nonlinear Schrödinger equation with combined local L2-critical and subcritical nonlinearities on noncompact metric graphs. The interplay of the combined local nonlinearities creates some new phenomena. Precisely, we show that the existence (or nonexistence) of ground states mainly depends on the topological and metric properties of the noncompact metric graphs. Our results rely on the Concentration-compactness principle and an application of Gagliardo-Nirenberg inequalities.

Long time dynamics and blow-up for the focusing inhomogeneous nonlinear Schrödinger equation with spatially growing nonlinearity

We investigate the Cauchy problem for the focusing inhomogeneous nonlinear Schrödinger equation i∂tu + Δu = −|x|b|u|p−1u in the radial Sobolev space Hr1(RN), where b &amp;gt; 0 and p &amp;gt; 1. We show the global existence and energy scattering in the intercritical regime, i.e., p&amp;gt;N+4+2bN and p&amp;lt;N+2+2bN−2 if N ≥ 3. We also obtain blowing-up solutions for the mass-critical and mass-supercritical nonlinearities. The main difficulty, coming from the spatial growing nonlinearity, is overcome by refined Gagliardo–Nirenberg-type inequalities. Our proofs are based on improved Gagliardo–Nirenberg inequalities, the Morawetz–Sobolev approach of Dodson and Murphy [Proc. Am. Math. Soc. 145(11), 4859–4867 (2017)], radial Sobolev embeddings, and localized virial estimates.