Acta Mathematica Scientia最新文献

This paper is concerned with the Cauchy problem for a 3D fluid-particle interaction model in the so-called flowing regime in ℝ³. Under the smallness assumption on both the external potential and the initial perturbation of the stationary solution in some Sobolev spaces, the existence and uniqueness of global smooth solutions in H³ of the system are established by using the careful energy method.

For a general normed vector space, a special optimal value function called a maximal time function is considered. This covers the farthest distance function as a special case, and has a close relationship with the smallest enclosing ball problem. Some properties of the maximal time function are proven, including the convexity, the lower semicontinuity, and the exact characterizations of its subdifferential formulas.

Evolution and interaction of plane waves of the multidimensional zero-pressure gas dynamics system leads to the study of the corresponding one dimensional system. In this paper, we study the initial value problem for one dimensional zero-pressure gas dynamics system. Here the first equation is the Burgers equation and the second one is the continuity equation. We consider the solution with initial data in the space of bounded Borel measures. First we prove a general existence result in the algebra of generalized functions of Colombeau. Then we study in detail special solutions with δ-measures as initial data. We study interaction of waves originating from initial data concentrated on two point sources and interaction with classical shock/rarefaction waves. This gives an understanding of plane-wave interactions in the multidimensional case. We use the vanishing viscosity method in our analysis as this gives the physical solution.

We study the global unique solutions to the 2-D inhomogeneous incompressible MHD equations, with the initial data (u₀, B₀) being located in the critical Besov space (dot{B}_{p,1}^{{-1}+{{2}over{p}}}(mathbb{R}^{2})) (1 < p < 2) and the initial density ρ₀ being close to a positive constant. By using weighted global estimates, maximal regularity estimates in the Lorentz space for the Stokes system, and the Lagrangian approach, we show that the 2-D MHD equations have a unique global solution.

In the present paper, we prove the existence, non-existence and multiplicity of positive normalized solutions (λ_c, u_c) ∈ ℝ × H¹ (ℝ^N) to the general Kirchhoff problem

satisfying the normalization constraint (int_{mathbb{R}^N}u^2{rm d}x=c), where M ∈ C([0, ∞)) is a given function satisfying some suitable assumptions. Our argument is not by the classical variational method, but by a global branch approach developed by Jeanjean et al. [J Math Pures Appl, 2024, 183: 44–75] and a direct correspondence, so we can handle in a unified way the nonlinearities g(s), which are either mass subcritical, mass critical or mass supercritical.

Abstract

In this article, we investigate the (big) Hankel operator H_f on the Hardy spaces of bounded strongly pseudoconvex domains Ω in ℂⁿ. We observe that H_f is bounded on H^p (Ω) (1 < p < ∞) if f belongs to BMO and we obtain some characterizations for H_f on H² (Ω) of other pseudoconvex domains. In these arguments, Amar’s L^p-estimations and Berndtsson’s L²-estimations for solutions of the ({{bar partial }_b}) -equation play a crucial role. In addition, we solve Gleason’s problem for Hardy spaces H^p(Ω) (1 ≤ p ≤ ∞) of bounded strongly pseudoconvex domains.

Abstract

We investigate the global existence of strong solutions to a non-isothermal ideal gas model derived from an energy variational approach. We first show the global well-posedness in the Sobolev space H² (ℝ³) for solutions near equilibrium through iterated energy-type bounds and a continuity argument. We then prove the global well-posedness in the critical Besov space (dot{boldsymbol{B}}_{boldsymbol{2,1}}^{boldsymbol{3/2}}) by showing that the linearized operator is a contraction mapping under the right circumstances.

Abstract

By introducing the Carathéodory metric, we establish the Schwarz lemma at the boundary for holomorphic self-mappings on the unit p-ball B _p ⁿ of ℂⁿ. Furthermore, the boundary rigidity theorem for holomorphic self-mappings defined on B _p ⁿ is obtained. These results cover the boundary Schwarz lemma and rigidity result for holomorphic self-mappings on the unit ball for p = 2, and the unit polydisk for p = ∞, respectively.

Let qλ (z) = 1 + λsinh(ς), 0 < λ < 1/sinh(1) be a non-vanishing analytic function in the open unit disk. We introduce a subclass ({{cal S}^ * }({q_lambda })) (qλ) of starlike functions which contains the functions (mathfrak{f}) such that (z{mathfrak{f}^prime }/mathfrak{f}) is subordinated by qλ. We establish inclusion and radii results for the class ({{cal S}^ * }) (qλ) for several known classes of starlike functions. Furthermore, we obtain sharp coefficient bounds and sharp Hankel determinants of order two for the class ({{cal S}^ * }) (qλ). We also find a sharp bound for the third Hankel determinant for the case λ = 1/2.