Generalization of the Lieb–Thirring–Araki Inequality and Its Applications

The matrix eigenvalue is very important in matrix analysis, and it has been applied to matrix trace inequalities, such as the Lieb–Thirring–Araki theorem and Thompson–Golden theorem. In this manuscript, we obtain a matrix eigenvalue inequality by using the Stein–Hirschman operator interpolation inequality; then, according to the properties of exterior algebra and the Schur-convex function, we provide a new proof for the generalization of the Lieb–Thirring–Araki theorem and Furuta theorem.

The Lieb–Thirring Inequality for Interacting Systems in Strong-Coupling Limit

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.

Optimality of Serrin type extension criteria to the Navier-Stokes equations

We prove that a strong solution u to the Navier-Stokes equations on (0, T) can be extended if either u ∈ L θ (0, T; U ˙ ∞ , 1 / θ , ∞ − α $\begin{array}{} \displaystyle \dot{U}^{-\alpha}_{\infty,1/\theta,\infty} \end{array}$ ) for 2/θ + α = 1, 0 < α < 1 or u ∈ L 2(0, T; V ˙ ∞ , ∞ , 2 0 $\begin{array}{} \displaystyle \dot{V}^{0}_{\infty,\infty,2} \end{array}$ ), where U ˙ p , β , σ s $\begin{array}{} \displaystyle \dot{U}^{s}_{p,\beta,\sigma} \end{array}$ and V ˙ p , q , θ s $\begin{array}{} \displaystyle \dot{V}^{s}_{p,q,\theta} \end{array}$ are Banach spaces that may be larger than the homogeneous Besov space B ˙ p , q s $\begin{array}{} \displaystyle \dot{B}^{s}_{p,q} \end{array}$ . Our method is based on a bilinear estimate and a logarithmic interpolation inequality.

Global in Time Results for a Parabolic Equation Solution in Non-rectangular Domains

This article deals with the parabolic equation ∂tw − c(t)∂2x w = f in D, D = { (t, x) ∈ R2 : t > 0, φ1 (t) < x < φ2(t) } with φi : [0,+∞[→ R, i = 1, 2 and c : [0,+∞[→ R satisfying some conditions and the problem is supplemented with boundary conditions of Dirichlet-Robin type. We study the global regularity problem in a suitable parabolic Sobolev space. We prove in particular that for f ∈ L2(D) there exists a unique solution w such that w, ∂tw, ∂jw ∈ L2(D), j = 1, 2. Notice that the case of bounded non-rectangular domains is studied in [9]. The proof is based on energy estimates after transforming the problem in a strip region combined with some interpolation inequality. This work complements the results obtained in [19] in the case of Cauchy-Dirichlet boundary conditions

THE HARDY AND HEISENBERG INEQUALITIES IN MORREY SPACES

We use the Morrey norm estimate for the imaginary power of the Laplacian to prove an interpolation inequality for the fractional power of the Laplacian on Morrey spaces. We then prove a Hardy-type inequality and use it together with the interpolation inequality to obtain a Heisenberg-type inequality in Morrey spaces.