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In the paper long-time behavior of solutions derived from the dissipative MKdV equation in the unbounded domain is studied. The Sobolev interpolation inequality and prior estimate on time t are applied to show the existence of solution in unbounded domain. Moreover, operator decomposition techniques and Kuratowskii α -noncompacted measures are applied to study the smooth property of the solution. Existence of global attractor for dissipative MKdV equation in H 2 ( R 1 ) is proved. Then, the computational stability of universal double-time differential format is proved by using nature analysis approach and dissipative conservative scheme theory. Finally, accordant conclusion with theoretical proof is drawn by numerical simulation.

We use an interpolation inequality on Besov spaces to show some logarithmically improved regularity criteria for Navier–Stokes equations, the harmonic heat flow, the Landau–Lifshitz equations, and the Landau–Lifshitz–Maxwell system. Copyright © 2009 John Wiley & Sons, Ltd.

In this paper, we study the inhomogeneous Schrodinger equation $$i\varphi_{t}=-\triangle\varphi -|x|^{b}|\varphi|^{p-1}\varphi,\quad x\in \mathbb{R}^{N}.$$ By using variational methods and a refined interpolation inequality, we establish some simple but sharp conditions on the solutions which exist globally or blow up in a finite time. An interesting result is that we obtain the existence of global solution for arbitrarily large data.

In this paper, we describe the range of the Lp-norm of a function under fixed Lp-norms with two other different exponents p and under a natural multiplicative restriction of the type of the Muckenhoupt condition. Particular cases of such results are simple inequalities as the interpolation inequality between two Lp-norms as well as such nontrivial inequalities as the Gehring inequality or the reverse Holder inequality for Mackenhoupt weights. The basic method of our paper is the search for the exact Bellman function of the corresponding extremal problem. Bibliography: 5

In this work we develop fully discrete (in time and space) numerical schemes for two-dimensional incompressible fluids with mass diffusion, also so-called Kazhikhov-Smagulov models. We propose at most H 1 -conformed finite elements (only globally continuous functions) to approximate all unknowns (velocity, pressure and density), although the limit density (solution of continuous problem) will have H 2 regularity. A backward Euler in time scheme is considered decoupling the computation of the density from the velocity and pressure. Unconditional stability of the schemes and convergence towards the (unique) global in time weak solution of the models is proved. Since a discrete maximum principle cannot be ensured, we must use a different interpolation inequality to obtain the strong estimates for the discrete density, from the used one in the continuous case. This inequality is a discrete version of the Gagliardo-Nirenberg interpolation inequality in 2D domains. Moreover, the discrete density is truncated in some adequate terms of the velocity-pressure problem.

In this paper, we study the initial-boundary value problem of one class of nonlinear Schrodinger equations described in molecular crystals. Furthermore, the existence of the global solution is obtained by means of interpolation inequality and a priori estimation.

In this paper, we first give an important interpolation inequality. Secondly, we use this inequality to prove the existence of local and global solutions of an inhomogeneous Schrodinger equation. Thirdly, we construct several invariant sets and prove the existence of blowing up solutions. Finally, we prove that for any $\omega>0$ the standing wave $e^{i \omega t} \phi (x)$ related to the ground state solution $\phi$ is strongly unstable.

On a compact Riemannian manifold with boundary, we prove a spectral inequality for the bi-Laplace operator in the case of so-called "clamped" boundary conditions , that is, homogeneous Dirichlet and Neumann conditions simultaneously. We also prove a resolvent estimate for the generator of the damped plate semigroup associated with these boundary conditions. The spectral inequality allows one to observe finite sums of eigenfunctions for this fourth-order elliptic operator, from an arbitrary open subset of the manifold. Moreover, the constant that appears in the inequality grows as exp(C$\mu$ 1/4) where $\mu$ is the largest eigenvalue associated with the eigenfunctions appearing in the sum. This type of inequality is known for the Laplace operator. As an application, we obtain a null-controllability result for a higher-order parabolic equation. The resolvent estimate provides the spectral behavior of the plate semigroup generator on the imaginary axis. This type of estimate is known in the case of the damped wave semigroup. As an application , we deduce a stabilization result for the damped plate equation, with a log-type decay. The proofs of both the spectral inequality and the resolvent estimate are based on the derivation of different types of Carleman estimates for an elliptic operator related to the bi-Laplace operator: in the interior and at some boundaries. One of these estimates exhibits a loss of one full derivative. Its proof requires the introduction of an appropriate semi-classical calculus and a delicate microlocal argument.

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.

In this paper we prove two results about the inviscid limit of the Navier-Stokes system. The first one concerns the convergence in H s of a sequence of solutions to the Navier-Stokes system when the viscosity goes to zero and the initial data is in H s . The second result deals with the best rate of convergence for vortex patch initial data in 2 and 3 dimensions. We present here a simple proof which also works in the 3D case. The 3D case is new.

We give a simple direct proof of the interpolation inequality $ \|\nabla f\|^2_{L^{2p}} \le C \|f\|_{\rm BMO} \|f\|_{W^{2,p}}$ , where $ 1 . For $p=2$ this inequality was obtained by Meyer and Riviere via a different method, and it was applied to prove a regularity theorem for a class of Yang–Mills fields. We also extend the result to higher derivatives, sharpening all those cases of classical Gagliardo–Nirenberg inequalities where the norm of the function is taken in $L^\infty$ and other norms are in $L^q$ for appropriate $q>1$ .

We study a specific example of energy-driven coarsening in two space dimensions. The energy is ∫|∇∇u|2 + (1 - | ∇u|2)2; the evolution is the fourth-order PDE representing steepest descent. This equation has been proposed as a model of epitaxial growth for systems with slope selection. Numerical simulations and heuristic arguments indicate that the standard deviation of u grows like t1/3, and the energy per unit area decays like t-1/3. We prove a weak, one-sided version of the latter statement: The time-averaged energy per unit area decays no faster than t-1/3. Our argument follows a strategy introduced by Kohn and Otto in the context of phase separation, combining (i) a dissipation relation, (ii) an isoperimetric inequality, and (iii) an ODE lemma. The interpolation inequality is new and rather subtle; our proof is by contradiction, relying on recent compactness results for the Aviles-Giga energy. © 2003 Wiley Periodicals, Inc.

In this paper we give two existence theorems for a class of elliptic problems in an Orlicz-Sobolev space setting concerning both the sublinear and the superlinear case with Neumann boundary conditions. We use the classical critical point theory with the Cerami (PS)-condition.

We show a sharp version of the Sobolev inequality of the Beale--Kato--Majda and the Kozono--Taniuchi type in Lizorkin--Triebel space. As an application of this inequality, the regularity problem under the critical condition to the gradient flow of the harmonic map into a sphere is considered in the class $L^2(0,T;BMO({\mathbb R}^n;{\mathbb S}^{m}))$, where BMO is the class of functions of bounded mean oscillations.

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, ∞).

This work is concerned with smooth periodic solutions for the compressible Navier–Stokes equations coupled with the Maxwell equations through the Lorentz force. The existence and uniqueness of the global smooth solution is established by using energy method.

In this paper we obtain well-posedness for a class of semilinear weakly degenerate reaction-diffusion systems with Robin boundary conditions. This result is obtained through a Gagliardo-Nirenberg interpolation inequality and some embedding results for weighted Sobolev spaces.

The Boussinesq equations describe the three-dimensional incompressible fluid moving under the gravity and the earth rotation which come from atmospheric or oceanographic turbulence where rotation and stratification play an important role. In this paper, we investigate the Cauchy problem of the three-dimensional incompressible Boussinesq equations. By commutator estimate, some interpolation inequality, and embedding theorem, we establish a blow-up criterion of weak solutions in terms of the pressurepin the homogeneous Besov spaceḂ∞,∞0.

We present a new approach to the Marcinkiewicz interpolation inequality for the distribution function of the Hilbert transform, and prove an "abstract" version of this inequality. The approach uses "logarithmic determinants" and new estimates of canonical products of genus one., 32 pages

Interpolation inequalities are important to estimate solutions of nonlinear partial dierential equations. Those associated with interpolation methods are almost trivial. The Gagliardo-Nirenberg inequalities are now standard. Escobedo and Vega [1] proposed new interpolation inequalities of Gagliardo-Nirenberg type in the study of nonlinear Dirac equations. Their proof breaks down in the endpoint case, while the corresponding limiting inequality reduces Miyakawa’s inequalitiy [3], which requires a totally dierent proof. In this talk I present new interpolation inequalitites in the framework of homogeneous Besov spaces, which reduces those two types as special cases.

We prove the Gagliardo–Nirenberg type inequality ‖u‖ W θs,p/θ ⩽c(n) p p−1 θ 1−s 1−θ θ/p ‖u‖ θ W s,p ‖u‖ 1−θ L ∞ , where 0 W s,p ( R n ) . The dependence of the constant factor in the right-hand side on each of the parameters s, θ, and p is precise in a sense.

We consider two standard models of surface-energy-driven coarsening: a constant-mobility Cahn-Hilliard equation, whose large-time behavior corresponds to Mullins-Sekerka dynamics; and a degenerate-mobility Cahn-Hilliard equation, whose large-time behavior corresponds to motion by surface diffusion. Arguments based on scaling suggest that the typical length scale should behave as \(\) in the first case and \(\) in the second. We prove a weak, one-sided version of this assertion – showing, roughly speaking, that no solution can coarsen faster than the expected rate. Our result constrains the behavior in a time-averaged sense rather than pointwise in time, and it constrains not the physical length scale but rather the perimeter per unit volume. The argument is simple and robust, combining the basic dissipation relations with an interpolation inequality and an ODE argument.

We consider positive solutions of the Cauchy problem in for the equation $$u_t=u^p\,\Delta u+u^q,\quad p\geq1,\; q\geq 1$$\nopagenumbers\end and show that concerning global solvability, the number q = p + 1 appears as a critical growth exponent. Copyright © 2002 John Wiley & Sons, Ltd.

We prove new point-wise inequalities involving the gradient of a function u is an element of C-1(R-n), the modulus of continuity w of the gradient delu, and a certain maximal function M(lozenge)u a ...

We prove the nonlinear stability in Lp, with $1\le p\le 2$, of particular steady solutions of the Vlasov--Poisson system for charged particles in the whole space R6 . Our main tool is a functional associated to the relative entropy or Casimir-energy functional.

Brezis and Mironescu have announced several years ago that for a compact manifold $N^n \subset \mathbb{R}^��$ and for real numbers $0 < s < 1$ and $1 \le p < \infty$ the class $C^\infty(\overline{Q}^m; N^n)$ of smooth maps on the cube with values into $N^n$ is dense with respect to the strong topology in the Sobolev space $W^{s, p}(Q^m; N^n)$ when the homotopy group $��_{\lfloor sp \rfloor}(N^n)$ of order $\lfloor sp \rfloor$ is trivial. The proof of this beautiful result is long and rather involved. Under the additional assumption that $N^n$ is $\lfloor sp \rfloor$ simply connected, we give a shorter proof of their result. Our proof for $sp \ge 1$ is based on the existence of a retraction of $\mathbb{R}^��$ onto $N^n$ except for a small subset in the complement of $N^n$ and on the Gagliardo-Nirenberg interpolation inequality for maps in $W^{1, q} \cap L^\infty$. In contrast, the case $sp < 1$ relies on the density of step functions on cubes in $W^{s, p}$.

In this note, we study a nonlocal variational problem modeling microphase separation of diblock copolymers ([22], [3], [21]). We apply certain new tools developed in [5] to determine the principal part of the asymptotic expansion of the minimum free energy. That is, we prove a scaling law for the minimum energy and confirm that it is attained by a simple periodic lamellar structure. A previous result of Ohnishi et al. [23] was for one space dimension. Here, we obtain a similar result for the full three-dimensional problem.

We show that transport inequalities, similar to the one derived by M. Talagrand (1996, Geom. Funct. Anal. 6 , 587–600) for the Gaussian measure, are implied by logarithmic Sobolev inequalities. Conversely, Talagrand's inequality implies a logarithmic Sobolev inequality if the density of the measure is approximately log-concave, in a precise sense. All constants are independent of the dimension and optimal in certain cases. The proofs are based on partial differential equations and an interpolation inequality involving the Wasserstein distance, the entropy functional, and the Fisher information.

We obtain sharp pointwise error estimates for relaxation approximation to scalar conservation laws with piecewise smooth solutions. We first prove that the first-order partial derivatives for the perturbation solutions are uniformly upper bounded (the so-called Lip+ stability). A one-sided interpolation inequality between classical L1 error estimates and Lip+ stability bounds enables us to convert a global L1 result into a (nonoptimal) local estimate. Optimal error bounds on the weighted error then follow from the maximum principle for weakly coupled hyperbolic systems. The main difficulties in obtaining the Lip+ stability and the optimal pointwise errors are how to construct appropriate "difference functions" so that the maximum principle can be applied.

Assuming that the n th iterate of the Laplacian Δ n f belongs to L r ( R d ), we investigate for 0 ≤ k n the validity of the following interpolation inequality:[formula]where D ξ i is the derivative in the direction ξ i .

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, 51pages

International audience; 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 carré 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.; Cet article est consacré à la conjecture de Lin-Ni pour une équation semi-linéaire elliptique avec non-linéarité super-linéaire,sous-critique et des conditions de Neumann homogènes. Nous établissons un résultat de rigidité, c’est-à-dire nous prouvons que la seule solution positive est constante si le param`etre du problème est en-dessous d’une borne explicite, reliée à la constante optimale d’une inégalité d’interpolation de Gagliardo-Nirenberg-Sobolev et aussi à une inégalité de Keller-Lieb-Thirring optimale. Nos résultats sont également valides dans un régime sous-linéaire. La borne de rigidité est obtenue par des méthodes de flots non-linéaires inspirées de résultats récents sur les variétés compactes, qui unifient des techniques d’équations elliptiques non-linéaires et la méthode du carré du champ en théorie des semi-groupes. Notre méthode requiert la convexité du domaine. Elle repose sur des quantités intégrales, prend en compte des estimations spectrales et fournit des inégalités améliorées.

We prove interpolation estimates between Morrey-Campanato spaces and Sobolev spaces. These estimates give in particular concentration-compactness inequalities in the translation-invariant and in the translation- and dilation-invariant case. They also give in particular interpolation estimates between Sobolev spaces and functions of bounded mean oscillation. The proofs rely on Sobolev integral representation formulae and maximal function theory. Fractional Sobolev spaces are also covered., 12 pages

We study the global existence of weak solutions to a multi-dimensional simplified Ericksen-Leslie system for compressible flows of nematic liquid crystals with large initial energy in a bounded domain $\Omega\subset \mathbb{R}^N$, where N=2 or 3. By exploiting a maximum principle, Nirenberg's interpolation inequality and a smallness condition imposed on the $N$-th component of initial direction field $\mf{d}_0$ to overcome the difficulties induced by the supercritical nonlinearity $|\nabla{\mathbf d}|^2{\mathbf d}$ in the equations of angular momentum, and then adapting a modified three-dimensional approximation scheme and the weak convergence arguments for the compressible Navier-Stokes equations, we establish the global existence of weak solutions to the initial-boundary problem with large initial energy and without any smallness condition on the initial density and velocity., Comment: arXiv admin note: substantial text overlap with arXiv:1210.3565

In this paper we introduce three types of inequalities related to the Lorentz spaces on a measure space (M,m).

In this paper, we establish a priori estimates for the three-dimensional compressible Euler equations with moving physical vacuum boundary, the $\gamma$-gas law equation of state for $\gamma=2$ and the general initial density $\ri \in H^5$. 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 play a vital role in our energy estimates and obtain some extra estimates for the space-time derivatives of the velocity in $L^3$., Comment: 51 pages. arXiv admin note: text overlap with arXiv:1003.4721, arXiv:0906.0289 by other authors

In 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 Poincar\'e 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., Comment: 35+14 pages, comments are welcome, added remark on relation between weak and strong curvature condition

Variable Hilbert scales are constructed using the spectral theory of self-adjoint operators in Hilbert spaces. An embedding and an interpolation theorem (based on Jenssen's inequality) are proved. They generalize known results about “ordinary” Hilbert scales derived by Natterer [Applic. Anal., 18 (1984), pp.29-37]. Bounds on best possible and actual errors for regularization methods are obtained by applying the interpolation inequality. These bounds extend the standard ones, and, in particular, include exponential and logarithmic error laws. Similar results were established earlier by Hegland [SIAM J. Numer. Anal., 29 (1992), pp. 1446-14611 for compact operators only. Here, they are generalized to include unbounded operators. A detailed discussion of Tikhonov regularization by Nair et al. [Tech. Rep. MR8-94, CMA, Aust. Nat. Uni., 19941 indicates that parameter choice strategies, which were thought to be suboptimal, can give substantially higher convergence rates than the so-called optimal choices! This im...

An interpolation inequality of Nirenberg, involving Lebesgue-space norms of functions and their derivatives, is modified, replacing one of the norms by a Holder norm.

In this work we are concerned with the minimization problem associated to the best constant Cn,α in the following interpolation inequality ∫∫|u(x)|α|u(y)|α|x−y|n−2dxdy≤Cn,α‖∇ψ‖L2nα−(n+2)‖ψ‖L2(n+2)−(n−2)α, u∈H1(Rn) for 2≤α

An easy proof of the John-Nirenberg inequality is provided by merely using the Calderón-Zygmund decomposition. Moreover, an interpolation inequality is presented with the help of the John-Nirenberg inequality.

An interpolation inequality for the total variation of the gradient of a composite function is derived by applying the coarea formula. A bound for the pressure integral is studied by establishing ana priori estimate for a solution of the Grad-Shafranov equation of plasma equilibrium. A weak formulation of the Grad-Shafranov equation is given to include singular current profiles.

A parabolic problem is considered with a nonlocal constraint in place of one of the standard boundary conditions. Well-posedness of the problem is proved in a weighted, fractional Sobolev space with the problem data proposed in related weighted spaces. This comes as an intrinsic requirement of the problem. The proof uses an interpolation inequality for norms of fractional Sobolev spaces and is based on an interesting choice of the test function.