It is well-established in harmonic analysis that Calderón-Zygmund convolution operators adapted to non-isotropic dilations are bounded on non-isotropic Hardy spaces. Similarly, product Calderón-Zygmund operators are known to be bounded on classical product Hardy spaces. The principal contribution of this paper is to extend these results by demonstrating that classical Calderón-Zygmund convolution operators with non-isotropic homogeneity remain bounded on product Hardy spaces.
Let Ω ⊂ ℂn be a bounded domain covered by the polydisc 𝔻^n through a proper holomorphic mapping Φ. In this paper, we study the Lp regularity and give the Lp-norm estimate for the Bergman projection of Ω. As applications, we obtain the upper bounds of Lp-norm for the Bergman projections on the symmetrized polydisc and monomial polyhedra.
In this paper, we study a two-phase flow system consisting of the compressible pressureless Euler equations and the incompressible Navier-Stokes equations coupled through the drag force. Owing to the lack of a pressure term in the Euler equations, the classical existence theory for hyperbolic-parabolic systems cannot be applied to this particular system. To solve this problem, we employ the spectral analysis method. At the beginning, we formulate the energy estimates for the velocities. Subsequently, in order to acquire the uniform estimates of the density, we make use of the spectral analysis method to derive a more rapid time-decay rate for the velocity of the Euler flow, specifically ∇ u_H^3≤ C(1+t)^-5/4 . After obtaining the above decay estimate, we manage to prove the uniform boundedness of the density. Eventually, leveraging the uniform estimates of the velocities and the continuity argument, we establish the global well-posedness of this two-phase flow system. Meanwhile, it is also proved that the velocities (u, v) decay to the motionless state at the optimal algebraic time-decay rates in L2-norm. Our results suggest that the smoothing effect of Navier-Stokes equations can be propagated to Euler equations through the drag force.
In this paper, we investigate the following elliptic system with Sobolev critical growth -Δ u_1 = G_1(|y|) u_1^2^*-1 + 1/2 u_1^2^*/2-1 u_2^2^*/2, y ∈ℝ^N, -Δ u_2 = G_2(|y|) u_2^2^*-1 + 1/2 u_2^2^*/2-1 u_1^2^*/2, y ∈ℝ^N, u_1, u_2 > 0, u_1, u_2 ∈ D^1,2(ℝ^N), where N ≥ 5, G1 (r) and G2(r) are positive radial potentials, 2^*=2N/N-2 . We construct an unbounded sequence of non-radial positive vector solutions of synchronized type via Lyapunov-Schmidt reduction argument. Different from the single equation, it is worth noting that when N ≥ 5, the coupling exponent 2^*/2-1=2/N-2<1 , which poses a serious obstacle to applying the perturbation argument directly. This constitutes the main difficulty of this paper and reflects the significant differences between the coupled system and a single equation. As a matter of fact, it is necessary to give an accurate point-wise estimate of the error term so that it can be absolutely controlled by less than one multiple of the approximate solution. To this end, we improve the decaying order of the error term iteratively, and it should be an effective way to deal with the ill coupled terms.
This paper focuses on the two-dimensional hydrostatic Navier-Stokes equations in the strip ℝ×𝕋 . We first prove the long-time well-posedness for the hydrostatic Navier-Stokes equations in Sobolev space Hm when the initial data is a small perturbation of some convex function. Then we justify the limit from the anisotropic Navier-Stokes equations to the hydrostatic Navier-Stokes equations and get the optimal convergence rate in Hm.
