Interpolation Inequalities Research Articles

Global Existence of the Three Dimensional Heat-conductive Incompressible Viscous Fluids

In this paper, we consider the Cauchy problem of non-stationary motion of heatconducting incompressible viscous fluids in ℝ3. About the heat-conducting incompressible viscous fluids, there are many mathematical researchers study the variants systems when the viscosity and heat-conductivity coefficient are positive. For the heat-conductive system, it is difficulty to get the better regularity due to the gradient of velocity of fluid own the higher order term. It is hard to control it. In order to get its global solutions, we must obtain the a priori estimates at first, then using fixed point theorem, it need the mapping is contracted. We can get a local solution, then applying the criteria extension. We can extend the local solution to the global solutions. For the two dimensional case, the Gagliardo-Nirenberg interpolation inequality makes use of better than the three dimensional situation. Thus, our problem will become more difficulty to handle. In this paper, we assume the coefficient of viscosity is a constant and the coefficient of heat-conductivity satisfying some suitable conditions. We show that the Cauchy problem has a global-in-time strong solution (u,θ) on ℝ3 ×(0, ∞).

Lower bounds for the blow-up time to a nonlinear viscoelastic wave equation with strong damping

This paper deals with a nonlinear viscoelastic wave equation with strong damping. By the means of the interpolation inequalities and di erential inequality technique, we obtain a lower bound for blow-up time of the solution. This result extends our earlier work Peng et al. [Appl. Math. Lett., 76, 2018].

Well-posedness, blow-up criteria and gevrey regularity for a rotation-two-component camassa-holm system

In this paper, we are concerned with the Cauchy problem for a new two-component Camassa-Holm system with the effect of the Coriolis force in the rotating fluid, which is a model in the equatorial water waves. We first investigate the local well-posedness of the system in \begin{document}$ B_{p,r}^s× B_{p,r}^{s-1}$\end{document} with \begin{document}$s>\max\{1+\frac{1}{p},\frac{3}{2},2-\frac{1}{p}\}$\end{document} , \begin{document}$p,r∈ [1,∞]$\end{document} by using the transport theory in Besov space. Then by means of the logarithmic interpolation inequality and the Osgood's lemma, we establish the local well-posedness in the critical Besov space \begin{document}$ B_{2,1}^{3/2}× B_{2,1}^{1/2}$\end{document} , and we present a blow-up result with the initial data in critical Besov space by virtue of the conservation law. Finally, we study the Gevrey regularity and analyticity of solutions to the system in a range of Gevrey-Sobolev spaces in the sense of Hardamard. Moreover, a precise lower bound of the lifespan is obtained.

Asymptotic behavior of a delayed wave equation without displacement term

This paper is dedicated to the investigation of the asymptotic behavior of a delayed wave equation without the presence of any position term. First, it is shown that the problem is well-posed in the sense of semigroups theory. Thereafter, LaSalle's invariance principle is invoked in order to establish the asymptotic convergence for the solutions of the system to a stationary position which depends on the initial data. More importantly, without any geometric condition such as BLR condition in the control zone, the logarithmic convergence is proved by using an interpolation inequality combined with a resolvent method.

Sharp uncertainty principles on Riemannian manifolds: the influence of curvature

We present a rigidity scenario for complete Riemannian manifolds supporting the Heisenberg–Pauli–Weyl uncertainty principle with the sharp constant in Rn (shortly, sharp HPW principle). Our results deeply depend on the curvature of the Riemannian manifold which can be roughly formulated as follows:(a)When (M,g) has non-positive sectional curvature, the sharp HPW principle holds on (M,g). However, positive extremals exist in the sharp HPW principle if and only if (M,g) is isometric to Rn, n=dim(M).(b)When (M,g) has non-negative Ricci curvature, the sharp HPW principle holds on (M,g) if and only if (M,g) is isometric to Rn.Since the sharp HPW principle and the Hardy–Poincaré inequality are endpoints of the Caffarelli–Kohn–Nirenberg interpolation inequality, we establish further quantitative results for the latter inequalities in terms of the curvature on Cartan–Hadamard manifolds.

Introduction to Lp Sobolev Spaces via Muramatu\u2019s Integral Formula

Muramatu’s integral formula is a very useful tool for the study of Sobolev spaces, although this does not seem to be widely recognized. Most theorems in Sobolev spaces can be proved by this formula combined with basic inequalities in analysis, and it is possible to directly treat not only the whole space but also a special Lipschitz domain. In this paper, we present an introduction to Lp-based Sobolev spaces of integer order by making Muramatu’s integral formula play a central role, as Cauchy’s integral formula does in complex analysis. The topics we take up are approximation by smooth functions, the interpolation inequality, the Sobolev embedding theorems, the trace theorem, construction of an extension operator, complex interpolation of Sobolev spaces and real interpolation of Sobolev spaces.

Uniqueness and rigidity in nonlinear elliptic equations, interpolation inequalities, and spectral estimates

This paper is devoted to the Lin-Ni conjecture for a semi-linear elliptic equation with a super-linear, sub-critical nonlinearity and homogeneous Neumann boundary conditions. We establish a new rigidity result, that is, we prove that the unique positive solution is a constant if the parameter of the problem is below an explicit bound that we relate with an optimal constant for a Gagliardo-Nirenberg-Sobolev interpolation inequality and also with an optimal Keller-Lieb-Thirring inequality. Our results are valid in a sub-linear regime as well. The rigidity bound is obtained by nonlinear flow methods inspired by recent results on compact manifolds, which unify nonlinear elliptic techniques and the carre du champ method in semi-group theory. Our method requires the convexity of the domain. It relies on integral quantities, takes into account spectral estimates and provides improved functional inequalities.

Local and global existence of solutions to a fourth-order parabolic equation modeling kinetic roughening and coarsening in thin films

In this paper we study both the Cauchy problem and the initial boundary value problem for the equation $\partial_tu+\mbox{div}\left(\nabla\Delta u-{\bf g}(\nabla u)\right)=0$. This equation has been proposed as a continuum model for kinetic roughening and coarsening in thin films. In the Cauchy problem, we obtain that local existence of a weak solution is guaranteed as long as the vector-valued function ${\bf g}$ is continuous and the initial datum $u_0$ lies in $C^1(\mathbb{R}^N)$ with $\nabla u_0(x)$ being uniformly continuous and bounded on $\mathbb{R}^N$ and that the global existence assertion also holds true if we assume that ${\bf g}$ is locally Lipschitz and satisfies the growth condition $|{\bf g}(\xi) |\leq c|\xi|^\alpha$ for some $c>0, \alpha\in (2, 3)$, $\sup_{\mathbb{R}^N}|\nabla u_0|<\infty$, and the norm of $u_0$ in the space $L^{\frac{(\alpha-1)N}{3-\alpha}}(\mathbb{R}^N) $ is sufficiently small. This is done by exploring various properties of the biharmonic heat kernel. In the initial boundary value problem, we assume that ${\bf g}$ is continuous and satisfies the growth condition $|{\bf g}(\xi) |\leq c|\xi|^\alpha+c$ for some $c, \alpha\in (0,\infty)$. Our investigations reveal that if $\alpha\leq 1$ we have global existence of a weak solution, while if $1<\alpha<\frac{N^2+2N+4}{N^2}$ only a local existence theorem can be established. Our method here is based upon a new interpolation inequality, which may be of interest in its own right.

Interpolation inequalities on the sphere: linear vs. nonlinear flows

This paper is devoted to sharp interpolation inequalities on the sphere and their proof using flows. The method explains some rigidity results and proves uniqueness in related semilinear elliptic equations. Nonlinear flows allow to cover the interval of exponents ranging from Poincare to Sobolev inequality, while an intriguing limitation (an upper bound on the exponent) appears in the carre du champ method based on the heat flow. We investigate this limitation, describe a counter-example for exponents which are above the bound, and obtain improvements below.

Observability Inequalities from Measurable Sets for Some Abstract Evolution Equations

This paper studies the observability from measurable sets in time for an evolution equation (in a Hilbert space): $u'=Au$ ($t\geq0)$, with an observation operator $B$. We obtain such an observability inequality in two different settings on $(A,B)$. In the first setting, $A$ generates an analytic semigroup, $B$ is an admissible observation operator for this semigroup, and $(A,B)$ satisfies an observability inequality from time intervals. By the propagation estimate of analytic functions and a telescoping series method (provided in the current paper), we build up the desired inequality for this setting. In the second setting, $A$ generates a $C_0$ semigroup, $B$ is a linear and bounded operator, and $(A, B)$ satisfies a spectral-like condition. By methods developed by Phung and Wang [J. Eur. Math. Soc. (JEMS), 15 (2013), pp. 681--703] and Apraiz et al. [J. Eur. Math. Soc. (JEMS), 16 (2014), pp. 2433--2475], we first obtain an interpolation inequality at one time point, and then derive the desired observability inequality for the second setting. This observability inequality is applied to get the bang-bang property of time optimal control problems for several kinds of differential equations.

Regularity criteria for Navier-Stokes and related system

We show some regularity criteria for Navier-Stokes equations, the harmonic heat flow, two liquid crystals models, and a model for magneto-elastic materials. The method of proof depends on a systematic use of interpolation inequalities in Besov spaces and is independent on logarithmic inequalities.

The Cauchy problem and blow-up phenomena for a new integrable two-component peakon system with cubic nonlinearities

This paper studies the local well-posedness and blow-up phenomena for a new integrable two-component peakon system in the Besov space. Firstly, by utilizing the Littlewood-Paley theory, the logarithmic interpolation inequality and the Osgood's Lemma, we investigate the existence and uniqueness of the solution to the system in the critical Besov space $B_{2, 1}^{\frac{1}{2}}(\mathbb{R})× B_{2, 1}^{\frac{1}{2}}(\mathbb{R})$, and show that the data-to-solution mapping is Holder continuous. Secondly, we derive a blow-up criteria for the Cauchy problem in the critical Besov space. Finally, with suitable conditions on the initial data, a new blow-up criteria for the system is obtained by virtue of the global conservative property of the potential densities $m$ and $n$ along the characteristics and the blow-up criteria at hand.

Groundstates and radial solutions to nonlinear Schrödinger–Poisson–Slater equations at the critical frequency

We study the nonlocal Schr\"odinger-Poisson-Slater type equation $$ - \Delta u + (I_\alpha \ast |u|^p)|u|^{p - 2} u= |u|^{q-2}u\quad\text{in \(\mathbb{R}^N\),} $$ where $N\in\mathbb{N}$, $p>1$, $q>1$ and $I_\alpha$ is the Riesz potential of order $\alpha\in(0,N).$ We introduce and study the Coulomb-Sobolev function space which is natural for the energy functional of the problem and we establish a family of associated optimal interpolation inequalities. We prove existence of optimizers for the inequalities, which implies the existence of solutions to the equation for a certain range of the parameters. We also study regularity and some qualitative properties of solutions. Finally, we derive radial Strauss type estimates and use them to prove the existence of radial solutions to the equation in a range of parameters which is in general wider than the range of existence parameters obtained via interpolation inequalities.

Weighted fast diffusion equations (Part Ⅰ): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities

In this paper we consider a family of Caffarelli-Kohn-Nirenberg interpolation inequalities (CKN), with two radial power law weights and exponents in a subcritical range. We address the question of symmetry breaking: are the optimal functions radially symmetric, or not ? Our intuition comes from a weighted fast diffusion (WFD) flow: if symmetry holds, then an explicit entropy - entropy production inequality which governs the intermediate asymptotics is indeed equivalent to (CKN), and the self-similar profiles are optimal for (CKN). We establish an explicit symmetry breaking condition by proving the linear instability of the radial optimal functions for (CKN). Symmetry breaking in (CKN) also has consequences on entropy - entropy production inequalities and on the intermediate asymptotics for (WFD). Even when no symmetry holds in (CKN), asymptotic rates of convergence of the solutions to (WFD) are determined by a weighted Hardy-Poincar{e} inequality which is interpreted as a linearized entropy - entropy production inequality. All our results rely on the study of the bottom of the spectrum of the linearized diffusion operator around the self-similar profiles, which is equivalent to the linearization of (CKN) around the radial optimal functions, and on variational methods. Consequences for the (WFD) flow will be studied in Part II of this work.

Weighted interpolation inequalities: a perturbation approach

We study optimal functions in a family of Caffarelli–Kohn–Nirenberg inequalities with a power-law weight, in a regime for which standard symmetrization techniques fail. We establish the existence of optimal functions, study their properties and prove that they are radial when the power in the weight is small enough. Radial symmetry up to translations is true for the limiting case where the weight vanishes, a case which corresponds to a well-known subfamily of Gagliardo–Nirenberg inequalities. Our approach is based on a concentration-compactness analysis and on a perturbation method which uses a spectral gap inequality. As a consequence, we prove that optimal functions are explicit and given by Barenblatt-type profiles in the perturbative regime.

Green functions for weighted subelliptic p-Laplace operator constructed by Hörmander's vector fields

This paper deals with the weighted subelliptic p-Laplace operator constructed by Hörmander's vector fields:Lpu=divX(〈A(x)Xu(x),Xu(x)〉p−22A(x)Xu(x)), where u∈W1,p(U,w),1<p<Q,andA(x) is a bounded measurable and m×m symmetric matrix satisfyingλ−1w(x)2/p|ξ|2⩽〈A(x)ξ,ξ〉⩽λw(x)2/p|ξ|2,ξ∈Rm,w(x)∈Ap. We first prove existence of the modified Green function for Lp by virtue of Minty–Browder theorem and then existence of the Green function for Lp by checking the convergence of sequence of modified Green functions. Next, we derive upper bounds of the modified Green function for Lp by establishing the interpolation inequality in the weighted weak Lp spaces. Finally, the bounds of the Green function for Lp are also obtained by virtue of results for the modified Green function and the compact weighted embedding theorem.

Geodesic interpolation inequalities on Heisenberg groups

In this Note, we present geodesic versions of the Borell–Brascamp–Lieb, Brunn–Minkowski and entropy inequalities on the Heisenberg group Hn. Our arguments use the Riemannian approximation of Hn combined with optimal mass-transportation techniques.

Global existence and uniqueness theorem to 2-D incompressible non-resistive MHD system with non-equilibrium background magnetic field

In this article, we consider the global existence and uniqueness of the solution to the 2D incompressible non-resistive MHD system with non-equilibrium background magnetic field. Our result implies that a strong enough non-equilibrium background magnetic field will guarantee the stability of the nonlinear MHD system. Beside the classical energy method, the interpolation inequalities and the algebraic structure of the equations coming from the incompressibility of the fluid are crucial in our arguments.

Interpolation inequalities in pattern formation

We prove some interpolation inequalities which arise in the analysis of pattern formation in physics. They are the strong version of some already known estimates in weak form that are used to give a lower bound of the energy in many contexts (coarsening and branching in micromagnetics and superconductors). The main ingredient in the proof of our inequalities is a geometric construction which was first used by Choksi, Conti, Kohn, and one of the authors in [4] in the study of branching in superconductors.

Interpolation Inequalities, Nonlinear Flows, Boundary Terms, Optimality and Linearization

This paper is devoted to the computation of the asymptotic boundary terms in entropy methods applied to a fast diffusion equation with weights associated with Caffarelli-Kohn-Nirenberg interpolation inequalities. So far, only elliptic equations have been considered and our goal is to justify, at least partially, an extension of the carre du champ / Bakry-Emery / Renyi entropy methods to parabolic equations. This makes sense because evolution equations are at the core of the heuristics of the method even when only elliptic equations are considered, but this also raises difficult questions on the regularity and on the growth of the solutions in presence of weights. We also investigate the relations between the optimal constant in the entropy–en- tropy production inequality, the optimal constant in the information–information production inequality, the asymptotic growth rate of generalized Renyi entropy powers under the action of the evolution equation and the optimal range of parameters for symmetry breaking issues in Caffarelli-Kohn-Nirenberg inequalities, under the assumption that the weights do not introduce singular boundary terms at x = 0. These considerations are new even in the case without weights. For instance, we establish the equivalence of carre du champ and Renyi entropy methods and explain why entropy methods produce optimal constants in entropy–entropy production and Gagliardo-Nirenberg inequalities in absence of weights, or optimal symmetry ranges when weights are present.