Solutions of Navier-Stokes Equations with Coriolis Force in the Rotational Framework

In this article, for 0 ≤m<∞ and the index vectors q=(q_1,q_2 ,q_3 ),r=(r_1,r_2,r_3) where 1≤q_i≤∞,1<r_i<∞ and 1≤i≤3, we study new results of Navier-Stokes equations with Coriolis force in the rotational framework in mixed-norm Sobolev-Lorentz spaces H ̇^(m,r,q) (R^3), which are more general than the classical Sobolev spaces. We prove the existence and uniqueness of solutions to the Navier-Stokes equations (NSE) under Coriolis force in the spaces L^∞([0, T]; H ̇^(m,r,q) ) by using topological arguments, the fixed point argument and interpolation inequalities. We have achieved new results compared to previous research in the Navier-Stokes problems.

Global regularity for the micropolar Rayleigh-Bénard problem with only velocity dissipation

This paper is concerned with the global regularity problem on the micropolar Rayleigh-Bénard problem with only velocity dissipation in $\mathbb {R}^{d}$ with $d=2\ or\ 3$ . By fully exploiting the special structure of the system, introducing two combined quantities and using the technique of Littlewood-Paley decomposition, we establish the global regularity of solutions to this system in $\mathbb {R}^{2}$ . Moreover, we obtain the global regularity for fractional hyperviscosity case in $\mathbb {R}^{3}$ by employing various techniques including energy methods, the regularization of generalized heat operators on the Fourier frequency localized functions and logarithmic Sobolev interpolation inequalities.

Maximal $$L^q$$-Regularity for Parabolic Hamilton–Jacobi Equations and Applications to Mean Field Games

In this paper we investigate maximal $$L^q$$ L q -regularity for time-dependent viscous Hamilton–Jacobi equations with unbounded right-hand side and superlinear growth in the gradient. Our approach is based on the interplay between new integral and Hölder estimates, interpolation inequalities, and parabolic regularity for linear equations. These estimates are obtained via a duality method à la Evans. This sheds new light on the parabolic counterpart of a conjecture by P.-L. Lions on maximal regularity for Hamilton–Jacobi equations, recently addressed in the stationary framework by the authors. Finally, applications to the existence problem of classical solutions to Mean Field Games systems with unbounded local couplings are provided.

On Some Remarkable Points and Line Segments in Triangles

Let ra, rb, rc be the radii, and OA, OB, OC the centers of tangent circles at the vertices to the circumcircle of a triangle ABC and to the opposite sides. In the paper [Andrica D., Marinescu D.S. New interpolation inequalities to Euler's R≥2 // Forum Geometricorum. 2017. Vol. 17], the authors proved that 4/R £ 1/ra + 1/rb +1/rc £2/r. In the paper [Isaev I., Maltsev Yu., Monastyreva A. On some relations in geometry of a triangle // Journal of Classical Geometry. 2018. Vol. 4], it is given the following generalization of these inequalities: 1/ra + 1/rb +1/rc=2/R+1/r. In that paper, we find the area of the triangle OAOBOC (see Theorem 1). We prove some relations for the numbers R-ra, R-rb, R-rc, where R is the circumradius of a triangle ABC. Namely, we find the expressions 1/R-ra+1/R-rb + 1/R-rc и a/R-ra+b/R-rb + c/R-rc by means by the parameters p, R and r (see Theorem 2). We estimate these values (see Theorem 3). Finally, using the results of paper [Maltsev Yu., Monastyreva A. On some relations for a triangle // International Journal of Geometry. 2019. Vol. 8 (1)] and representing the expression of (1-cos(αβ))(1-cos(β-γ))(1-cos(α-γ)) by means of p, R, r, we prove new proof of the fundamental triangle inequality (see Corollary 2).

Sobolev–Kantorovich inequalities under CD(0,∞) condition

We refine and generalize several interpolation inequalities bounding the [Formula: see text] norm of a probability density with respect to the reference measure [Formula: see text] by its Sobolev norm and the Kantorovich distance to [Formula: see text] on a smooth weighted Riemannian manifold satisfying [Formula: see text] condition.

Energy decay of some boundary coupled systems involving wave\ Euler-Bernoulli beam with one locally singular fractional Kelvin-Voigt damping

<p style='text-indent:20px;'>In this paper, we investigate the energy decay of hyperbolic systems of wave-wave, wave-Euler-Bernoulli beam and beam-beam types. The two equations are coupled through boundary connection with only one localized non-smooth fractional Kelvin-Voigt damping. First, we reformulate each system into an augmented model and using a general criteria of Arendt-Batty, we prove that our models are strongly stable. Next, by using frequency domain approach, combined with multiplier technique and some interpolation inequalities, we establish different types of polynomial energy decay rate which depends on the order of the fractional derivative and the type of the damped equation in the system.</p>

Inequalities involving Aharonov–Bohm magnetic potentials in dimensions 2 and 3

This paper is devoted to a collection of results on nonlinear interpolation inequalities associated with Schrödinger operators involving Aharonov–Bohm magnetic potentials, and to some consequences. As symmetry plays an important role for establishing optimality results, we shall consider various cases corresponding to a circle, a two-dimensional sphere or a two-dimensional torus, and also the Euclidean spaces of dimensions 2 and 3. Most of the results are new and we put the emphasis on the methods, as very little is known on symmetry, rigidity and optimality in the presence of a magnetic field. The most spectacular applications are new magnetic Hardy inequalities in dimensions [Formula: see text] and [Formula: see text].

Improved Interpolation Inequalities and Stability

For exponents in the subcritical range, we revisit some optimal interpolation inequalities on the sphere with carré du champ methods and use the remainder terms to produce improved inequalities. The method provides us with lower estimates of the optimal constants in the symmetry breaking range and stability estimates for the optimal functions. Some of these results can be reformulated in the Euclidean space using the stereographic projection.