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Register a new userGagliardo-Nirenberg inequalities in Lorentz type spaces and energy equality for the Navier-Stokes system
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In this paper, we derive some new Gagliardo-Nirenberg type inequalities in Lorentz type spaces without restrictions on the second index of Lorentz norms, which generalize almost all known corresponding results. Our proof mainly relies on the Bernstein inequalities in Lorentz spaces, the embedding relation among various Lorentz type spaces, and Littlewood-Paley decomposition techniques. In addition, we establish several novel criteria in terms of the velocity or the gradient of the velocity in Lorentz spaces for energy conservation of the 3D Navier-Stokes equations. Particularly, we improve the classical Shinbrots condition for energy balance to allow both the space-time directions of the velocity to be in Lorentz spaces.
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We prove a Lieb-Thirring type inequality for potentials such that the associated Schr{o}dinger operator has a pure discrete spectrum made of an unbounded sequence of eigenvalues. This inequality is equivalent to a generalized Gagliardo-Nirenberg inequality for systems. As a special case, we prove a logarithmic Sobolev inequality for infinite systems of mixed states. Optimal constants are determined and free energy estimates in connection with mixed states representations are also investigated.
With a view towards Riemannian or sub-Riemannian manifolds, RCD metric spaces and specially fractals, this paper makes a step further in the development of a theory of heat semigroup based $(1,p)$ Sobolev spaces in the general framework of Dirichlet spaces. Under suitable assumptions that are verified in a variety of settings, the tools developed by D. Bakry, T. Coulhon, M. Ledoux and L. Saloff-Coste in the paper Sobolev inequalities in disguise allow us to obtain the whole family of Gagliardo-Nirenberg and Trudinger-Moser inequalities with optimal exponents. The latter depend not only on the Hausdorff and walk dimensions of the space but also on other invariants. In addition, we prove Morrey type inequalities and apply them to study the infimum of the exponents that ensure continuity of Sobolev functions. The results are illustrated for fractals using the Vicsek set, whereas several conjectures are made for nested fractals and the Sierpinski carpet.
It is well-known that a Leray-Hopf weak solution in $L^4 (0,T; L^4(Omega))$ for the incompressible Navier-Stokes system is persistence of energy due to Lions [19]. In this paper, it is shown that Lionss condition for energy balance is also valid for the weak solutions of the isentropic compressible Navier-Stokes equations allowing vacuum under suitable integrability conditions on the density and its derivative. This allows us to establish various sufficient conditions implying energy equality for the compressible flow as well as the non-homogenous incompressible Navier-Stokes equations. This is an improvement of corresponding results obtained by Yu in [32, Arch. Ration. Mech. Anal., 225 (2017)], and our criterion via the gradient of the velocity partially answers a question posed by Liang in [18, Proc. Roy. Soc. Edinburgh Sect. A (2020)].
In this paper, we derive several new sufficient conditions of non-breakdown of strong solutions for for both the 3D heat-conducting compressible Navier-Stokes system and nonhomogeneous incompressible Navier-Stokes equations. First, it is shown that there exists a positive constant $varepsilon$ such that the solution $(rho,u,theta)$ to full compressible Navier-Stokes equations can be extended beyond $t=T$ provided that one of the following two conditions holds (1) $rho in L^{infty}(0,T;L^{infty}(mathbb{R}^{3}))$, $uin L^{p,infty}(0,T;L^{q,infty}(mathbb{R}^{3}))$ and $$| u|_{L^{p,infty}(0,T;L^{q,infty}(mathbb{R}^{3}))}leq varepsilon, ~~text{with}~~ {2/p}+ {3/q}=1, q>3;$$ (2) $lambda<3mu,$ $rho in L^{infty}(0,T;L^{infty}(mathbb{R}^{3}))$, $thetain L^{p,infty}(0,T;L^{q,infty}(mathbb{R}^{3}))$ and $$|theta|_{L^{p,infty}(0,T; L^{q,infty}(mathbb{R}^{3}))}leq varepsilon, ~~text{with}~~ {2/p}+ {3/q}=2, q>3/2.$$ To the best of our knowledge, this is the first continuation theorem allowing the time direction to be in Lorentz spaces for the compressible fluid. Second, we establish some blow-up criteria in anisotropic Lebesgue spaces to the full Navier-Stokes system. Third, without the condition on $rho$ in (0.1) and (0.3), the results also hold for the 3D nonhomogeneous incompressible Navier-Stokes equations. The appearance of vacuum in these systems could be allowed.
We prove that the energy equality holds for weak solutions of the 3D Navier-Stokes equations in the functional class $L^3([0,T);V^{5/6})$, where $V^{5/6}$ is the domain of the fractional power of the Stokes operator $A^{5/12}$.
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