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.
In this paper, we are concerned with the following Schrödinger system -Δ u_1 = λ_1 |u_1|^p(r)-2 u_1 + β |u_2|^p(r)/2 |u_1|^p(r)/2-2 u_1, x ∈ℝ^N, -Δ u_2 = λ_2 |u_2|^p(r)-2 u_2 + β |u_1|^p(r)/2 |u_2|^p(r)/2-2 u_2, x ∈ℝ^N, where N ≥ 3, λ1, λ2, β > 0, p(r)=2N N-2+f(r) with f ∈ C([0, +∞), [0, +∞)). Under some suitable assumptions on f, we prove that the system admits a positive solution if β > 0 for N ≥ 5 or β ∈ (0, β0] ∪ [β1, +∞) for N = 3,4, where 0 < β0 < β1 are some constants. In particular, the system is slightly supercritical when f ≢ 0. Delicate analysis near the origin and infinity will be involved.
In this paper, we investigate the two dimensional Schrödinger-Poisson equation (0.1) -Δ u+|x|^2u+(∫_ℝ^2ln |x-y|u^2(y)dy)u=λ u+a|u|^p-2u in ℝ^2 in the mass-supercritical case. We show that problem (0.1) admits a mountain pass solution up, which blows up as p → 4+. Then we consider the concentration and local uniqueness of up as p → 4+.
For linear self-repelling diffusions, we explicitly study deviation properties, including the deviation inequalities for some quadratic functionals and the Cramer-type moderate deviations of maximum likelihood estimators for the drift coefficients. The main methods of this paper consist of the deviation inequalities for multiple Wiener-Itô integrals and the asymptotic analysis techniques.
In this paper, a discontinuous Galerkin method is proposed to solve a singularly perturbed Volterra delay-integro-differential equation. First, the construction of a generalized Bakhvalov-type mesh and its corresponding properties are given. Furthermore, we prove an optimal parameter-uniform convergence rate s + 1 in the L2-norm, where s is the degree of the piecewise polynomial space. Finally, a numerical experiment is carried out to confirm the theoretical results of our proposed numerical method.
In this paper, we investigate the viability of solutions to some McKean-Vlasov stochastic differential equations which involve a random time change Et given by an inverse subordinator Dt. By establishing a so-called duality principle and the viability for McKean-Vlasov stochastic differential equations with the standard Brownian motion, we obtain some sufficient conditions on the viability of solutions to McKean-Vlasov stochastic differential equations driven by time-changed Brownian motion with one drift term dEt with respect to a given non-empty smooth closed set. In addition, we gain some sufficient conditions for the viability of a generalized non-empty closed convex set K by the distance function induced by K. For time-changed McKean-Vlasov stochastic differential equations with two drift terms, one driven by the random change Et and the other driven by non-random time t, by establishing some time-changed Gronwall-like inequalities, we give some sufficient conditions for the viability of the non-empty closed convex set K via the distance function induced by K.
This study examines the continuous dependence on initial data of a damped plate system that includes the p-Laplacian operator and logarithmic source terms. By applying energy decay estimates [1], we derive conclusions regarding the continuous dependence of the global solution on initial data and the coefficients associated with strong damping terms and coupled velocity terms. To estimate the continuous dependence on initial conditions, we adopt a novel approach developed by Han et al. [2], which enhances our understanding of this dependence. For the estimation of logarithmic terms, we introduce a method distinct from conventional approaches, thereby obtaining more refined results. This study contributes to the theoretical analysis of nonlinear partial differential equations and provides new insights into handling logarithmic source terms in damped plate systems.
Using the saddle point theorem and the mountain pass theorem, together with the Morse index estimates and the Morse index iteration inequalities developed by Professor Yiming Long, periodic solutions with minimal period estimates have been considered for second-order mild superquadratic Hamiltonian systems.
We establish a novel relationship connecting pressure, curvature, velocity, and gravitational acceleration in two-dimensional steady incompressible fluids with general vorticity under gravitational effects. As an application, we extend the well-posedness theory for two-dimensional stationary gravity flows with vorticity developed by Wang [J Lond Math Soc, 2024, 109(2): e12869]. Our analysis demonstrates that the minimum pressure within the fluid domain is attained precisely on the free surface, thereby establishing a new lower bound for the curvature of the free boundary.
In 1982, Alt, Caffarelli and Friedman [Comm Pure Appl Math, 1982, 35: 29–68] established the existence of the solution to the asymmetric incompressible jet flows. As a continuation of Alt-Caffarelli-Friedman’s work, we investigate the geometric shape of free boundaries of the asymmetric incompressible jet flows in this paper. More precisely, we will first show the strict monotonicity of free boundaries under the monotonicity hypotheses on the nozzle walls. Secondly, if the nozzle walls are convex to the fluid, it is proved that the free boundaries are strictly concave to the fluid. Finally, as a by-product, the optimal regularity of free boundaries at the separation points can be established.
In this paper, we focus on exploring entire solutions of finite order for a class of differential-difference equations, particularly those of the following form w_1'(z)^2+P_2^2(z)w_2(z+c)^2 = Q_1(z), w_2'(z)^2+P_1^2(z)w_1(z+c)^2 = Q_2(z), where Pi, Qi (i = 1, 2) are non-zero polynomials. We prove that the above system admits four distinct forms of solutions and derive their specific expressions. Additionally, we establish the relationship between the coefficients of the system and those of its solutions. These results extend the existing results of complex differential (difference) equations to the systems of differential-difference equations, and improve the current results related to the systems of differential-difference equations.
In this paper, we consider classical solutions to the following fractional Liouville system (-Δ)^1/2v_i=exp(∑_j∈ Iγ^ijv_j) in ℝ ∫_ℝe^vi ds<∞ for all i∈ I, where I = 1,⋯,n. Assuming that the matrix A = (γij) is nonnegative and invertible, we show that each solution vi is symmetric with respect to the point si. Furthermore, if the matrix A satisfies the irreducibility condition, the points si =1 coincide. Additionally, given that the matrix A = (γij) satisfies the condition ∑_j∈ Iγ^ij=1 , along with certain constraints on the masses vector, we employ the method of moving planes to demonstrate that all solutions conform to a standard bubble defined by a common center and scale parameters. And we show the existence of asymmetric solutions to the equations when nonnegativity is absent by applying bifurcation theory.
The aim of this paper is to study Sp remotely almost periodic (Stepanov remotely almost periodic) functions defined on 𝕋∈{ℝ,ℝ_+} with values in the Banach space B . We establish a relation between remotely almost periodic motions in the shift dynamical system (L^p_loc(𝕋, B),𝕋,σ) and Sp remotely almost periodic functions in the space of locally measurable functions L^p_loc(𝕋, B) . Using this relation, we establish some important algebraic, analytical and topological properties of Sp almost periodic functions.
In this paper, we are concerned with the large-time behavior of solution to the Cauchy problem for the 3D compressible Navier-Stokes equations for a reacting mixture in an infinite long flat nozzle domain ℝ×𝕋^2 . Under some smallness conditions on the initial perturbations, we prove that the solution to this system exists globally and tends time-asymptotically to the planar rarefaction wave, in which the background solution z̅(x,t) of the mass fraction of the reactant is nontrivial. The proof is accomplished by virtue of delicate energy method. To the best of our knowledge, this may be the first result about the nonlinear stability of the plane waves for the compressible Navier-Stokes equation for a reacting mixture.
In this paper, we study a class of Laplacian-like equation. We first deduce the Pohozaev identity of the equation and consider the minimum of the functional on the Pohozaev manifold and then study the properties of the minimum energy of the functional. Based on the properties obtained, we prove the existence of normalized solution to the equation.
A feedback control, based on delayed discrete observations, is proposed for hybrid neutral stochastic differential systems with mixed delay. Unlike conventional methods that rely on continuous mode observations, it addresses the difficulty and cost of mode identification and considers system delay. This paper successfully stabilizes unstable systems by integrating discrete state observations, discrete mode observations, and delay factors into the controller design. It also verifies the effectiveness of the theory through an example.
