Calculus of Variations and Partial Differential Equations最新文献

Whether the global existence and uniqueness of strong solutions to n-dimensional incompressible magnetohydrodynamic (MHD for short) equations with only kinematic viscosity or magnetic diffusion holds true or not remains an outstanding open problem. In recent years, stared from the pioneer work by Lin and Zhang (Commun Pure Appl Math 67(4):531–580, 2014), much more attention has been paid to the case when the magnetic field close to an equilibrium state (the background magnetic field for short). Specifically, when the background magnetic field satisfies the Diophantine condition (see (1.2) for details), Chen et al. (Sci China Math 41:1–10, 2022) first studied the perturbation system and established the decay estimates and asymptotic stability of its solutions in 3D periodic domain (mathbb {T}^3), which was then improved to (H^{(3+2beta )r+5+(alpha +2beta )}(mathbb {T}^2)) for 2D periodic domain (mathbb {T}^2) and any (alpha >0), (beta >0) by Zhai (J Differ Equ 374:267–278, 2023). In this paper, we seek to find the optimal decay estimates and improve the space where the global stability is taking place. Through deeply exploring and effectively utilizing the structure of perturbation system, we discover a new dissipative mechanism, which enables us to establish the decay estimates in the Sobolev spaces with much lower regularity. Based on the above discovery, we greatly reduce the initial regularity requirement of aforesaid two works from (H^{4r+7}(mathbb {T}^3)) and (H^{(3+2beta )r+5+(alpha +2beta )}(mathbb {T}^2)) to (H^{(3r+3)^+}(mathbb {T}^n)) for (r>n-1) when (n=3) and (n=2) respectively. Additionally, we first present the linear stability result via the method of spectral analysis in this paper. From which, the decay estimates obtained for the nonlinear system can be seen as sharp in the sense that they are in line with those for the linearized system.

Variational models for cohesive fracture are based on the idea that the fracture energy is released gradually as the crack opening grows. Recently, [21] proposed a variational approximation via (Gamma )-convergence of a class of cohesive fracture energies by phase-field energies of Ambrosio-Tortorelli type, which may be also used as regularization for numerical simulations. In this paper we address the question of the asymptotic behaviour of critical points of the phase-field energies in the one-dimensional setting: we show that they converge to a selected class of critical points of the limit functional. Conversely, each critical point in this class can be approximated by a family of critical points of the phase-field functionals.

Let (nge 2) and (Omega ) be a bounded Lipschitz domain of (mathbb {R}^n). Assume that (L_R) is a second-order divergence form elliptic operator having real-valued, bounded, symmetric, and measurable coefficients on (L^2(Omega )) with the Robin boundary condition. In this article, via first obtaining the Hölder estimate of the heat kernels of (L_R), the authors establish a new atomic characterization of the Hardy space (H^p_{L_R}(Omega )) associated with (L_R). Using this, the authors further show that, for any given (pin (frac{n}{n+delta _0},1]),

where (H^p_{L_D}(Omega )) and (H^p_{L_N}(Omega )) denote the Hardy spaces on (Omega ) associated with the corresponding elliptic operators respectively having the Dirichlet and the Neumann boundary conditions, (H^p_z(Omega )) and (H^p_r(Omega )) respectively denote the “supported type” and the “restricted type” Hardy spaces on (Omega ), and (delta _0in (0,1]) is the critical index depending on the operators (L_D), (L_N), and (L_R). The authors then obtain the boundedness of the Riesz transform (nabla L_R^{-1/2}) on the Lebesgue space (L^{p}(Omega )) when (pin (1,infty )) [if (p>2), some extra assumptions are needed] and its boundedness from (H_{L_R}^{p}(Omega )) to (L^{p}(Omega )) when (pin (0,1]) or to (H^{p}_r(Omega )) when (pin (frac{n}{n+1},1]). As applications, the authors further obtain the global regularity estimates, in (L^{p}(Omega )) when (pin (0,p_0)) and in (H^{p}_r(Omega )) when (pin (frac{n}{n+1},1]), for the inhomogeneous Robin problem of (L_R) on (Omega ), where (p_0in (2,infty )) is a constant depending only on n, (Omega ), and the operator (L_R). The main novelties of these results are that the range ((0,p_0)) of p for the global regularity estimates in the scale of (L^p(Omega )) is sharp and that, in some sense, the space (X{:}{=}H^1_{L_R}(Omega )) is also optimal to guarantee both the boundedness of (nabla L^{-1/2}_R) from X to (L^1(Omega )) or to (H^1_r(Omega )) and the global regularity estimate (Vert nabla uVert _{L^{frac{n}{n-1}} (Omega ;,mathbb {R}^n)}le CVert fVert _{X}) for inhomogeneous Robin problems with C being a positive constant independent of both u and f.

In this paper we study global nonlinear stability for the Dirac–Klein–Gordon system in two and three space dimensions for small and regular initial data. In the case of two space dimensions, we consider the Dirac–Klein–Gordon system with a massless Dirac field and a massive scalar field, and prove global existence, sharp time decay estimates and linear scattering for the solutions. In the case of three space dimensions, we consider the Dirac–Klein–Gordon system with a mass parameter in the Dirac equation, and prove uniform (in the mass parameter) global existence, unified time decay estimates and linear scattering in the top order energy space.

We obtain a priori (C^{1,1}) estimates for some Hessian quotient equations with positive Lipschitz right hand sides, through studying a twisted special Lagrangian equation. The results imply the interior (C^{2,alpha }) regularity for (C^0) viscosity solutions to (sigma _2=f^2(x)) in dimension 3, with positive Lipschitz f(x).

This is the first in a series of papers devoted to the blow up analysis for the quenching phenomena in a parabolic MEMS equation. In this paper, we first give an optimal Hölder estimate for solutions to this equation by using the blow up method and some Liouville theorems on stationary two-valued caloric functions, and then establish a convergence theory for sequences of uniformly Hölder continuous solutions. These results are also used to prove a stratification theorem on the rupture set ({u=0}).

We study complete minimal surfaces in (mathbb {R}^n) with finite total curvature and embedded planar ends. After conformal compactification via inversion, these yield examples of surfaces stationary for the Willmore bending energy (mathcal {W}: =frac{1}{4} int |vec H|^2). In codimension one, we prove that the (mathcal {W})-Morse index for any inverted minimal sphere or real projective plane with m such ends is exactly (m-3=frac{mathcal {W}}{4pi }-3). We also consider several geometric properties—for example, the property that all m asymptotic planes meet at a single point—of these minimal surfaces and explore their relation to the (mathcal {W})-Morse index of their inverted surfaces.

This study examines an initial-boundary value problem involving the system

in a smoothly bounded domain (Omega subset mathbb {R}^n) with no-flux boundary conditions, where (m, nge 1). The motility function (phi in C^0([0,infty )) cap C^3((0,infty ))) is positive on ((0,infty )) and satisfies

for some (alpha >0). Through distinct approaches, we establish that, for sufficiently regular initial data, in two- and higher-dimensional contexts, if (alpha in [1,2m)), then ((star )) possesses global weak solutions, while in one-dimensional settings, the same conclusion holds for (alpha >0), and notably, the solution remains uniformly bounded when (alpha ge 1). Furthermore, for the one-dimensional case where (alpha ge 1), the bounded solution additionally possesses the convergence property that

with (u_{infty }in L^{infty }(Omega )). Further conditions on the initial data enable the identification of admissible initial data for which (u_{infty }) exhibits spatial heterogeneity.

In this paper, we consider the existence and asymptotic behavior of infinitely many nodal solutions of Kirchhoff-type equations with an asymptotically cubic nonlinear term without oddness assumptions. Combining variational methods and convex analysis techniques, we show, for any positive integer k, the existence of a radial nodal solution that changes sign exactly k times. Meanwhile, we prove that the energy of such solution is an increasing function of k. Moreover, the asymptotic behavior of these solutions are also studied upon varying the parameters. By using different analytical approaches, the question of the existence of infinite solutions to some elliptic nonlinear equations is addressed without invoking oddness assumptions. At the same time, we propose a method to overcome the difficulties caused by the complicated competition between the nonlocal term and the asymptotically cubic nonlinearity.

We compute the indicial roots of the Lichnerowicz Laplacian on Ricci-flat cones and give a detailed description of the corresponding radially homogeneous tensor fields in its kernel. For a Ricci-flat conifold (M, g) which may have asymptotically conical as well as conically singular ends, we compute at each end a lower bound for the order with which the metric converges to the tangent cone. As a special subcase of our result, we show that any Ricci-flat ALE manifold ((M^n,g)) is of order n and thereby close a small gap in a paper by Cheeger and Tian.