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We prove the first ever pointwise estimates of the (unrestricted) Green tensor and the associated pressure tensor of the nonstationary Stokes system in the half-space, for every space dimension greater than one. The force field is not necessarily assumed to be solenoidal. The key is to find a suitable Green tensor formula which maximizes the tangential decay, showing in particular the integrability of Green tensor derivatives. With its pointwise estimates, we show the symmetry of the Green tensor, which in turn improves pointwise estimates. We also study how the solutions converge to the initial data, and the (infinitely many) restricted Green tensors acting on solenoidal vector fields. As applications, we give new proofs of existence of mild solutions of the Navier–Stokes equations in L q , pointwise decay, and uniformly local L q spaces in the half-space. [ABSTRACT FROM AUTHOR]

For the Stokes system in the half space, Kang [Math. Ann. 331 (2005), pp. 87-109] showed that a solution generated by a compactly supported, Hölder continuous boundary flux may have unbounded normal derivatives near the boundary. In this paper we first prove explicit global pointwise estimates of the above solution, showing in particular that it has finite global energy and its derivatives blow up everywhere on the boundary away from the flux. We then use the above solution as a profile to construct solutions of the Navier-Stokes equations which also have finite global energy and unbounded normal derivatives due to the flux. Our main tool is the pointwise estimates of the Green tensor of the Stokes system proved by us in an earlier paper. We also examine the Stokes flows generated by dipole bumps' boundary flux, and identify the regions where the normal derivatives of the solutions tend to positive or negative infinity near the boundary. [ABSTRACT FROM AUTHOR]

In this paper, we study the regularity and pointwise estimates of forward self-similar solutions of fractional Navier-Stokes system under the critical case. By employing a Caffarelli,Kohn and Nirenberg-type iteration, L ∞ estimates of the self-similar solution's profile are established, which is a key ingredient to ensure that the global weighted energy estimate procedure used in [28] is performed under the critical case. As a product, its natural pointwise bounds are recovered. Moreover, to obtain the optimal spatial decay estimate of self-similar solution's profile, a new technique is required due to lack of the related regularity theory. [ABSTRACT FROM AUTHOR]

We investigate a regularity for weak solutions of the following generalized Leray equations (−Δ)α V − 2α−1/2α V + V ⋅ ∇ V − 1/2α x ⋅ ∇V + ∇P = 0, which arises from the study of self-similar solutions to the generalized Navier-Stokes equations in R3. Firstly, by making use of the vanishing viscosity and developing non-local effects of the fractional diffusion operator, we prove uniform estimates for weak solutions V in the weighted Hilbert space Hαω(R3). Via the differences characterization of Besov spaces and the bootstrap argument, we improve the regularity for weak solution from Hαω(R3) to Hω1+α(R3). This regularity result, together with linear theory for the non-local Stokes system, leads to pointwise estimates of V which allow us to obtain a natural pointwise property of the self-similar solution constructed by Lai, Miao, and Zheng [Adv. Math. 352 (2019), pp. 981–1043]. In particular, we obtain an optimal decay estimate of the self-similar solution to the classical Navier-Stokes equations by means of the special structure of Oseen tensor. This answers the question proposed by Tsai Comm. Math. Phys., 328 (2014), pp. 29–44. [ABSTRACT FROM AUTHOR]

In this paper, we provide some new regularity criteria for suitable weak solutions to the 3D nematic liquid crystal flow in terms of the velocity field. More precisely, we prove that the suitable weak solution ( u , d ) is regular provided that either the scaled L p , q norm of u , or the scaled L p , q norm of ∇ × u is sufficiently small. [ABSTRACT FROM AUTHOR]

In this paper, we consider the radial singular solution of Δ 2 u = − 1 u p , in R N with u > 0 , p > 0 . By using some elementary ordinary differential equation arguments, we prove the above equation admits no radial singular solution for N = 3 , p ≥ 3 . In addition, the exact asymptotic behaviors of the radial singular solution as r → 0 is established for p = 1 , N ≥ 4 , which have a significant difference with the explicit singular solution. As a product, the singularity of any radial singular solution is of type II if p = 1 , N ≥ 4 . [ABSTRACT FROM AUTHOR]

Let B ⊂ ℝ be the unit ball centered at the origin. The authors consider the following biharmonic equation: where $$p > \frac{{n + 4}}{{n - 4}}$$ and v is the outward unit normal vector. It is well-known that there exists a λ* > 0 such that the biharmonic equation has a solution for λ ∈ (0, λ*) and has a unique weak solution u* with parameter λ = λ*, called the extremal solution. It is proved that u* is singular when n ≥ 13 for p large enough and satisfies $$u* \leqslant {r^{ - \frac{4}{{p - 1}}}} - 1$$ on the unit ball, which actually solve a part of the open problem left in [Dàvila, J., Flores, I., Guerra, I., Multiplicity of solutions for a fourth order equation with power-type nonlinearity, Math. Ann., 348(1), 2009, 143-193]. [ABSTRACT FROM AUTHOR]

Here we give a self-contained new proof of the existence of singular solutions to a class Dirichlet problem with a singular nonlinearity. These results were obtained by Dávila–Flores–Guerra (Dávila et al., 2009). Our proof is much more direct and simpler. [ABSTRACT FROM AUTHOR]

We consider the entire solution of the semilinear biharmonic equation Δ 2 u = e u , in R N , N ≥ 1 . For N = 3 , we obtain the asymptotic of the entire nonradial solution which extends the results of Lai and Ye (in press). Besides, a new singular solution of the above equation is constructed by derivative estimates for N ≥ 13 . [ABSTRACT FROM AUTHOR]

We establish the global well-posedness of the Landau-Lifshitz-Gilbert equation in $$\mathbb R^n$$ for any initial data $$\mathbf{m}_0\in H^1_*(\mathbb R^n,\mathbb S^2)$$ whose gradient belongs to the scaling critical Morrey space $$M^{2,2}(\mathbb R^n)$$ with small norm $$\displaystyle \Vert \nabla \mathbf{m}_0\Vert _{M^{2,2}(\mathbb R^n)}$$ . The method is based on priori estimates of a dissipative Schrödinger equation of Ginzburg-Landau types obtained from the Landau-Lifshitz-Gilbert equation by the moving frame technique. [ABSTRACT FROM AUTHOR]

We analyze pull-in instability of electrostatically actuated microelectromechanical systems, and we find that as the device size is reduced, the effect of the Casimir force becomes more important. In the miniaturization process there is a minimum size for the device below which the system spontaneously collapses with zero applied voltage. According to the mathematical analysis, we obtain a set U in the plane, such that elements of U correspond to minimal stable solutions of a two-parameter mathematical model. For points on the boundary $${\Upsilon}$$ of U, there exists weak solutions to this model, which are called extremal solutions. More refined properties of stable solutions-such as regularity, stability, uniqueness-are also established. [ABSTRACT FROM AUTHOR]

Abstract: Here we give a self-contained new proof of the asymptotic behavior of the radially symmetric entire solution of These results were obtained by I. Guerra in [4]. Our proof is much more direct and simpler. [Copyright &y& Elsevier]

Abstract: We consider the following nonlinear eigenvalue problems where is smooth bounded domain and is a nonnegative function. Our study is motivated by a simple Micro-Electromechanical Systems (MEMS) device model. We find some new phenomena about the regularity results by the classification of entire solutions. Besides, we also consider the uniqueness of solutions by Rellich–Pohozaev type estimates. [Copyright &y& Elsevier]

In this paper, we study the existence of periodic solutions to a class of functional differential system. By using Schauder,s fixed point theorem, we show that the system has aperiodic solution under given conditions. Finally, four examples are given to demonstrate the validity of our main results. [ABSTRACT FROM AUTHOR]

In this paper, we study asymptotic behavior of solution near 0 for a class of elliptic problem. The uniqueness of singular solution is established. [ABSTRACT FROM AUTHOR]

This paper deals with the generalized Swift-Hohenberg equation, proposed by Stoop et al. (2015), on the symmetry-breaking model that describes wrinkling morphology and pattern selection in curved elastic bilayer materials. More precisely, with help of the Mountain Pass theorem, we establish the existence of nontrivial periodic solutions to this equation. [ABSTRACT FROM AUTHOR]