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We consider an analogue of the Lieb-Thirring inequality for quantum systems with homogeneous repulsive interaction potentials, but without the antisymmetry assumption on the wave functions. We show that in the strong-coupling limit, the Lieb-Thirring constant converges to the optimal constant of the one-body Gagliardo-Nirenberg interpolation inequality without interaction., Comment: Final version to appear in Arch. Ration. Mech. Anal

The projection method to solve the incompressible Navier–Stokes equations was first studied by Chorin (Math Comput, 1969) in the framework of a finite difference method and Temam (Arch Ration Mech Anal, 1969) in the framework of a finite element method. Chorin showed convergence of approximation and its error estimates in problems with periodic boundary conditions assuming existence of a $$C^5$$ -solution, while Temam demonstrated an abstract argument to obtain a Leray–Hopf weak solution in problems on a bounded domain with the no-slip boundary condition. In the present paper, the authors extend Chorin’s result with full details to obtain convergent finite difference approximation of a Leray–Hopf weak solution to the incompressible Navier–Stokes equations on an arbitrary bounded Lipschitz domain of $${{\mathbb {R}}}^3$$ with the no-slip boundary condition and an external force. We prove unconditional solvability of our implicit scheme and strong $$L^2$$ -convergence (up to a subsequence) under the scaling condition $$ h^{3-\alpha }\le \tau $$ (no upper bound is necessary), where $$h,\tau $$ are space, time discretization parameters, respectively, and $$\alpha \in (0,2]$$ is any fixed constant. The results contain a compactness method based on a new interpolation inequality for step functions.

We show the logarithmic interpolation inequality by means of the Vishik space $${\dot{V}}^{s}_{q,\sigma ,\theta }$$ which is larger than the homogeneous Besov space $${\dot{B}}^{s}_{q,\sigma }$$ . We emphasize that $${\dot{V}}^{s}_{q,\sigma ,\theta }$$ may be the largest normed space that satisfies the logarithmic interpolation inequality. As an application of this inequality, we prove that the strong solution to the Navier–Stokes and Euler equations can be extended if the scaling invariant quantity of vorticity in the Vishik space is finite. Namely, the Beirao da Veiga- and Beale–Kato–Majda-type regularity criteria are improved in the terms of the Vishik space.

We establish global well-posedness of strong solutions to the nonhomogeneous heat conducting magnetohydrodynamic equations with non-negative density on the whole space $${\mathbb {R}}^2$$ . More precisely, under compatibility conditions for the initial data, we show the global existence and uniqueness of strong solutions. Our method relies on delicate energy estimates and a logarithmic interpolation inequality. In particular, the initial data can be arbitrarily large.

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.

We consider the minimizers for the biharmonic nonlinear Schrodinger functional $$\begin{aligned} \mathcal {E}_a(u)=\int \limits _{\mathbb {R}^d} |\Delta u(x)|^2 \mathrm{d}x + \int \limits _{\mathbb {R}^d} V(x) |u(x)|^2 \mathrm{d}x - a \int \limits _{\mathbb {R}^d} |u(x)|^{q} \mathrm{d}x \end{aligned}$$ with the mass constraint $$\int |u|^2=1$$ . We focus on the special power $$q=2(1+4/d)$$ , which makes the nonlinear term $$\int |u|^q$$ scales similarly to the biharmonic term $$\int |\Delta u|^2$$ . Our main results are the existence and blowup behavior of the minimizers when a tends to a critical value $$a^*$$ , which is the optimal constant in a Gagliardo–Nirenberg interpolation inequality.

We construct global \({\dot{H}^1\cap \dot{H}^{-1}}\) solutions to a logarithmically modified 2D Euler vorticity equation. Our main tool is a new logarithm interpolation inequality which exploits the L∞−-conservation of the vorticity.

In this paper we study a sharp Sobolev interpolation inequality on Finsler manifolds. We show that Minkowski spaces represent the optimal framework for the Sobolev interpolation inequality on a large class of Finsler manifolds: (1) Minkowski spaces support the sharp Sobolev interpolation inequality; (2) any complete Berwald space with non-negative Ricci curvature which supports the sharp Sobolev interpolation inequality is isometric to a Minkowski space. The proofs are based on properties of the Finsler–Laplace operator and on the Finslerian Bishop–Gromov volume comparison theorem.

In this paper, we prove a Gagliardo-Nirenberg type inequality on the critical Sobolev-Lorentz space with an exact growth order for the embedding constant. It turns out that the second index for the Lorentz space plays an essential role to determine the optimal growth order for the embedding constant. On the other hand, we establish a similar type interpolation inequality on the critical Besov-Lorentz space. In this case, we see that the growth order for the embedding constant can be determined by the third index for the Besov space and it is independent of the second index for the Lorentz space. Our main objective of the paper is to describe the optimal growth orders of the embedding constants by giving exact quantitative descriptions for the Gagliardo-Nirenberg type inequalities.

This paper establishes the global in time existence of classical solutions to the two-dimensional anisotropic Boussinesq equations with vertical dissipation. When only vertical dissipation is present, there is no direct control on the horizontal derivatives and the global regularity problem is very challenging. To solve this problem, we bound the derivatives in terms of the \({L^\infty}\) -norm of the vertical velocity v and prove that \({\|v\|_{L^{r}}}\) with \({2\leqq r < \infty}\) does not grow faster than \({\sqrt{r \log r}}\) at any time as r increases. A delicate interpolation inequality connecting \({\|v\|_{L^\infty}}\) and \({\|v\|_{L^r}}\) then yields the desired global regularity.

In this paper, the first boundary value problem for the second-order degenerated ellipticparabolic equations in nondivergent form is considered. Unique strong (almost everywhere) solvability of this problem in the corresponding weight Sobolev space is proved.

We discuss the existence of breathers and lower bounds on their power, in nonlinear Schr\"odinger lattices with nonlinear hopping. Our methods extend from a simple variational approach to fixed point arguments, deriving lower bounds for the power which can serve as a threshold for the existence of breather solutions. Qualitatively, the theoretical results justify non-existence of breathers below the prescribed lower bounds of the power which depend on the dimension, the parameters of the lattice as well as of the frequency of breathers. In the case of supercritical power nonlinearities we investigate the interplay of these estimates with the optimal constant of the discrete interpolation inequality. Improvements of the general estimates, taking into account the localization of the true breather solutions are derived. Numerical studies in the one dimensional lattice corroborate the theoretical bounds and illustrate that in certain parameter regimes of physical significance, the estimates can serve as accurate predictors of the breather power and its dependence on the various system parameters., Comment: To be published in Journal of Nonlinear Science

Our main purpose in this article is to establish a Gagliardo-Nirenberg type inequality in the critical Sobolev–Morrey space $H\mathcal{M}^{\frac{n}{p}}_{p,q}(\mathbb{R}^{n})$ with n∈ℕ and 1

We study a parabolic boundary-value problem with multipoint condition with respect to the time variable for a linear parabolic equation having power singularities on the coordinate planes. The result obtained is used for investigating an optimal control problem with internal and final control.

In the present work, we give the coercivity estimates for the Boltzmann collision operator Q(·, ·) without angular cut-off to clarify in which case the functional \({\langle -Q(g, f), f\rangle}\) will become the truly sub-elliptic. Based on this observation and commutator estimates in Alexandre et al. (Arch Rat Mech Anal 198:39–123, 2010), the upper bound estimates for the collision operator in Chen and He (Arch Rat Mech Anal 201(2):501–548, 2011) and the stability results in Desvillettes and Mouhot (Arch Rat Mech Anal 193(2):227–253, 2009), in the function space \({L^1_q\cap H^N}\) , we establish global well-posedness or local well-posedness for the spatially homogeneous Boltzmann equation with full-range interaction(covering most of physical collision kernels).

We prove a Galiardo–Nirenberg type pointwise interpolation inequality for special maximal functions which measure smoothness in the multidimensional case. It turns out that the classsical inequality follows from this one; it is also possible to use naturally BMO norms in the inequality. Bibliography: 6 titles.

This paper is devoted to a study of the longtime behavior of the hyperbolic equations with an arbitrary internal damping, under sharp regularity assumptions that both the principal part coefficients and the boundary of the space domain (in which the system evolves) are continuously differentiable. For this purpose, we derive a new point-wise inequality for second differential operators with symmetric coefficients. Then, based on a global Carleman estimate, we establish an estimate on the underlying resolvent operator of the equation, via which, we show the logarithmic decay rate for solutions of the hyperbolic equations.

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 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 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 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

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 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 ...

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.

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.

On estimating the eigenvalues for a class of semilinear elliptic operators, we obtain bifurcation and comparison results concerning the eigenvalues of some related linear problem.