Interpolation Inequalities Research Articles

On temporal decay estimates for the compressible nematic liquid crystal flow in

We use a general energy method recently developed by [Guo Y, Wang Y. Decay of dissipative equations and negative sobolev spaces. Commun. Partial Differ. Equ. 2012;37:2165–2208.] to prove the global existence and temporal decay rates of solutions to the three-dimensional compressible nematic liquid crystal flow in the whole space. In particular, the negative Sobolev norms of solutions are shown to be preserved along time evolution, and then the optimal decay rates of the higher order spatial derivatives of solutions are obtained by energy estimates and the interpolation inequalities.

Remarks on asymptotic behaviors of strong solutions to a viscous Boussinesq system

In this paper, we first address the space‐time decay properties for higher‐order derivatives of strong solutions to the Boussinesq system in the usual Sobolev space. The decay rates obtained here are optimal. The proof is based on a parabolic interpolation inequality, bootstrap argument, and some weighted estimates. Secondly, we present a new solution integration formula for the Boussinesq system, which will be employed to establish the existence of strong solutions for small initial data in some scaling invariant function spaces. The smallness conditions are somehow weaker than those presented by Brandolese and Schonbek. We further investigate the asymptotic profiles and decay properties of these strong solutions. Copyright © 2016 John Wiley & Sons, Ltd.

Space–time decay estimates for the incompressible viscous resistive MHD and Hall-MHD equations

In this paper, we address the space–time decay properties for strong solutions to the incompressible viscous resistive Hall-MHD equations. We obtained the same space–time decay rates as those of the heat equation. Based on the temporal decay results in [10] , we find that one can obtain weighted estimates of the magnetic field B by direct weighted energy estimate, and then by regarding the magnetic convection term as a forcing term in the velocity equations, we can obtain the weighted estimates for the vorticity, which yields the corresponding estimates for the velocity field. The higher order derivative estimates will be obtained by using a parabolic interpolation inequality proved in [22] . It should be emphasized that the magnetic field has stronger decay properties than the velocity field in the sense that there is no restriction on the exponent of the weight. The same arguments also yield the sharp space–time decay rates for strong solutions to the usual MHD equations.

On interpolation and curvature via Wasserstein geodesics

AbstractIn this article, a proof of the interpolation inequality along geodesics in p-Wasserstein spaces is given. This interpolation inequality was the main ingredient to prove the Borel–Brascamp–Lieb inequality for general Riemannian and Finsler manifolds and led Lott–Villani and Sturm to define an abstract Ricci curvature condition. Following their ideas, a similar condition can be defined and for positively curved spaces one can prove a Poincaré inequality. Using Gigli’s recently developed calculus on metric measure spaces, even a q-Laplacian comparison theorem holds on q-infinitesimal convex spaces. In the appendix, the theory of Orlicz–Wasserstein spaces is developed and necessary adjustments to prove the interpolation inequality along geodesics in those spaces are given.

Existence and decay estimates of solutions to complex Ginzburg–Landau type equations

This paper deals with the initial-boundary value problem (denoted by (CGL)) for the complex Ginzburg–Landau type equation ∂u∂t−(λ+iα)Δu+(κ+iβ)|u|q−1u−γu=0 with initial data u0∈Lp(Ω) in the case 1<q<1+2p/N, where Ω is bounded or unbounded in RN, λ>0, α,β,γ,κ∈R. There are a lot of studies on local and global existence of solutions to (CGL) including the physically relevant case q=3 and κ>0. This paper gives existence results with precise properties of solutions and rigorous proof from a mathematical point of view. The physically relevant case can be considered as a special case of the results. Moreover, in the case κ<0, local and global existence of solutions with the decay estimate ‖u(t)‖Lp(Ω)≤c1e−c2t (c1,c2 are positive constants) is obtained under some conditions. The key to the local existence is to construct a semigroup {et[(λ+iα)Δ]} and its Lp-Lq estimate. On the other hand, the key to the global existence is to derive estimates for solutions by using a kind of interpolation inequality with Re〈|v|p−2v,−(λ+iα)Δv〉.

Fractional Hardy–Lieb–Thirring and Related Inequalities for Interacting Systems

We prove analogues of the Lieb-Thirring and Hardy-Lieb-Thirring inequalities for many-body quantum systems with fractional kinetic operators and homogeneous interaction potentials, where no anti-symmetry on the wave functions is assumed. These many-body inequalities imply interesting one-body interpolation inequalities, and we show that the corresponding one- and many-body inequalities are actually equivalent in certain cases.

Remarks on the free boundary problem of compressible Euler equations in physical vacuum with general initial densities

In this paper, we establish a priori estimates for three-dimensional compressible Euler equations with the moving physical vacuum boundary, the $\gamma$-gas law equation of state for $\gamma=2$ and the general initial density $\rho_0 \in H^5$. Because of the degeneracy of the initial density, we investigate the estimates of the horizontal spatial and time derivatives and then obtain the estimates of the normal or full derivatives through the elliptic-type estimates. We derive a mixed space-time interpolation inequality which plays a vital role in our energy estimates and obtain some extra estimates for the space-time derivatives of the velocity in $L^3$.

Well-posedness and persistence property for a four-component Novikov system with peakon solutions

In this paper we mainly study the Cauchy problem of a four-component Novikov system. We first show the local well-posedness of the system in Besov spaces $$B^{s+1}_{p,r}\times B^{s+1}_{p,r}\times B^s_{p,r} \times B^s_{p,r}$$ with $$p,r\in [1,\infty ],~s>\max \{\frac{1}{p},\frac{1}{2}\}$$ by using the Littlewood–Paley theory and transport equations theory. Then, by virtue of logarithmic interpolation inequalities and the Osgood lemma, we prove the local well-posedness of the system in the critical Besov space $$B^{\frac{3}{2}}_{2,1}\times B^{\frac{3}{2}}_{2,1} \times B^{\frac{1}{2}}_{2,1}\times B^{\frac{1}{2}}_{2,1}$$ . Next, we establish two blow-up criteria for strong solutions to the system by using the structure of the system. Moreover, we investigate the persistence property for strong solutions to the system. Finally, we verify that the system possesses a special class of peakon solutions.

Blow-up phenomena and local well-posedness for a generalized Camassa–Holm equation in the critical Besov space

In this paper we mainly study the Cauchy problem for a generalized Camassa–Holm equation in a critical Besov space. First, by using the Littlewood–Paley decomposition, transport equations theory, logarithmic interpolation inequalities and Osgood’s lemma, we establish the local well-posedness for the Cauchy problem of the equation in the critical Besov space B2,112. Then we give a new blow-up criterion for the Cauchy problem of the equation. Finally, we present a new blow-up result and the exact blow-up rate of strong solutions to the equation by making use of the conservation law and the obtained blow-up criterion.

Sobolev-Kantorovich Inequalities

Abstract In a recent work, E. Cinti and F. Otto established some new interpolation inequalities in the study of pattern formation, bounding the Lr(μ)-norm of a probability density with respect to the reference measure μ by its Sobolev norm and the Kantorovich-Wasserstein distance to μ. This article emphasizes this family of interpolation inequalities, called Sobolev-Kantorovich inequalities, which may be established in the rather large setting of non-negatively curved (weighted) Riemannian manifolds by means of heat flows and Harnack inequalities.

Sharp Lp-entropy inequalities on manifolds

In 2003, Del Pino and Dolbeault [14] and Gentil [19] investigated, independently, best constants and extremals associated to Euclidean Lp-entropy inequalities for p > 1. In this work, we present some contributions in the Riemannian context. Namely, let (M,g) be a closed Riemannian manifold of dimension n >= 3. For 1 < p <= 2, we establish the validity of the sharp Riemannian Lp-entropy inequality int_M |u|^p log(|u|^p) dv_g <= n/p log ( A_{opt} int_M |Grad_g u|^p dv_g + B ) on all functions u em H^{1,p}(M) such that ||u||_{Lp(M)} = 1 for some constant B. Moreover, we prove that the first best constant A_{opt} is equal to the corresponding Euclidean one. Our approach is inspired on the Bakry, Coulhon, Ledoux and Sallof-Coste's idea [3] of getting Euclidean entropy inequalities as a limit case of suitable subcritical interpolation inequalities. It is conjectured that the inequality sometimes fails for p > 2.

Error estimates for forward Euler shock capturing finite element approximations of the one-dimensional Burgers' equation

We propose an error analysis for a shock capturing finite element method for the Burgers' equation using the duality theory due to Tadmor. The estimates use a one-sided Lipschitz stability (Lip+-stability) estimate on the discrete solution and are obtained in a weak norm, but thanks to a total variation a priori bound on the discrete solution and an interpolation inequality, error estimates in Lp-norms (1 ≤ p &lt; ∞) are deduced. Both first-order artificial viscosity and a nonlinear shock capturing term that formally is of second order are considered. For the discretization in time we use the forward Euler method. In the numerical section we verify the convergence order of the nonlinear scheme using the forward Euler method and a second-order strong stability preserving Runge–Kutta method. We also study the Lip+-stability property numerically and give some examples of when it holds strictly and when it is violated.

Improving interpolated Hardy and trace Hardy inequalities on bounded domains

In this work we study inequalities which interpolate weighted Hardy and trace Hardy inequalities on bounded domains. We establish improvements of these inequalities by adding correction terms of Hardy type with a singular logarithmic weight. We show that this weight is optimal in the sense that the inequality fails for more singular weights. Moreover, we determine the best values of the constants for the remaining terms. We also show that the aforementioned inequalities can be repeatedly improved, obtaining an infinite series of correction terms involving specific Hardy type potentials. In the two borderline cases of these interpolation inequalities we obtain refinements of the weighted Hardy and the trace weighted Hardy inequality respectively. It follows that the trace Hardy and the Hardy weighted inequalities share the same optimal constant for the specific remaining terms. These results constitute an extension of a well known result when the interpolation inequality reduces to the classical nonweighted Hardy inequality.

A Degenerate Fourth-Order Parabolic Equation Modeling Bose-Einstein Condensation Part II: Finite-Time Blow-Up

A degenerate fourth-order parabolic equation modeling condensation phenomena related to Bose-Einstein particles is analyzed. The model can be motivated from the spatially homogeneous and isotropic Boltzmann-Nordheim equation by a formal Taylor expansion of the collision integral. It maintains some of the main features of the kinetic model, namely mass and energy conservation and condensation at zero energy. The existence of local-in-time weak solutions satisfying a certain entropy inequality is proven. The main result asserts that if a weighted L 1 norm of the initial data is sufficiently large and the initial data satisfies some integrability conditions, the solution blows up with respect to the L ∞ norm in finite time. Furthermore, the set of all such blow-up enforcing initial functions is shown to be dense in the set of all admissible initial data. The proofs are based on approximation arguments and interpolation inequalities in weighted Sobolev spaces. By exploiting the entropy inequality, a nonlinear integral inequality is proved which implies the finite-time blow-up property.

One-Dimensional Interpolation Inequalities, Carlson–Landau Inequalities, and Magnetic Schrödinger Operators

In this paper we prove refined first-order interpolation inequalities for periodic functions and give applications to various refinements of the Carlson--Landau-type inequalities and to magnetic Schrodinger operators. We also obtain Lieb-Thirring inequalities for magnetic Schrodinger operators on multi-dimensional cylinders.

Local well-posedness and blow-up criteria for a two-component Novikov system in the critical Besov space

In this paper we mainly investigate the Cauchy problem of a two-component Novikov system. We first prove the local well-posedness of the system in Besov spaces Bp,rs−1×Bp,rs with p,r∈[1,∞],s>max{1+1p,32} by using the Littlewood–Paley theory and transport equations theory. Then, by virtue of logarithmic interpolation inequalities and the Osgood lemma, we establish the local well-posedness of the system in the critical Besov space B2,112×B2,132. Moreover, we present two blow-up criteria for the system by making use of the conservation laws.

The Optimal Decay Estimates on the Framework of Besov Spaces for Generally Dissipative Systems

We give a new decay framework for general dissipative hyperbolic system and hyperbolic-parabolic composite system, which allow us to pay less attention on the traditional spectral analysis in comparison with previous efforts. New ingredients lie in the high-frequency and low-frequency decomposition of a pseudo-differential operator and an interpolation inequality related to homogeneous Besov spaces of negative order. Furthermore, we develop the Littlewood-Paley pointwise energy estimates and new time-weighted energy functionals to establish the optimal decay estimates on the framework of spatially critical Besov spaces for degenerately dissipative hyperbolic system of balance laws. Based on the $L^{p}(\mathbb{R}^{n})$ embedding and improved Gagliardo-Nirenberg inequality, the optimal $L^{p}(\mathbb{R}^{n})$-$L^{2}(\mathbb{R}^{n})(1\leq p<2)$ decay rates and $L^{p}(\mathbb{R}^{n})$-$L^{q}(\mathbb{R}^{n})(1\leq p<2\leq q\leq\infty)$ decay rates are further shown. Finally, as a direct application, the optimal decay rates for 3D damped compressible Euler equations are also obtained.

Sharp interpolation inequalities for discrete operators

We consider the imbeddings of the l 2 sequence spaces defined on ddimensional lattices into the spaces written as interpolation inequalities between the l 2 norm of a sequence and its difference. A general method is developed for finding sharp constants, extremal elements and possible correction terms in this type of inequalities. Applications to the spectral theory of discrete Schrodinger operator are given.

GLOBAL SOLUTIONS OF THE COOPERATIVE CROSS-DIFFUSION SYSTEMS

In this paper the existence of global solutions of the parabolic cross-diffusion systems with cooperative reactions is obtained under certain conditions. The uniform boundedness of <TEX>$W_{1,2}$</TEX> norms of the local maximal solution is obtained by using interpolation inequalities and comparison results on differential inequalities.

Hausdorff Measure of the Singular Set in the Incompressible Magnetohydrodynamic Equations

We derive a local energy inequality for weak solutions of the three dimensional magnetohydrodynamic equations. Combining Biot–Savart law, interpolation inequalities and the local energy inequality, we prove a partial regularity theorem for suitable weak solutions. Furthermore, we obtain an improved estimate for the logarithmic Hausdorff dimension of the singular set of suitable weak solutions.