De Giorgi Iteration Research Articles

Another proof of the local curvature estimate for the Ricci flow

By using the De Giorgi iteration method we will give a new simple proof of the recent result of Kotschwar et al. (J Funct Anal 271(9):2604–2630, 2016) and Sesum (Am J Math 127(6):1315–1324, 2005) on the local boundedness of the Riemannian curvature tensor of solutions of Ricci flow in terms of its inital value on a given ball and a local uniform bound on the Ricci curvature.

Stochastic De Giorgi iteration and regularity of stochastic partial differential equations

Under general conditions, we devise a stochastic version of De Giorgi iteration scheme for semilinear stochastic parabolic partial differential equation of the form \[\partial_{t}u=\operatorname{div}(A\nabla u)+f(t,x,u)+g_{i}(t,x,u)\dot{w}^{i}_{t}\] with progressively measurable diffusion coefficients. We use the scheme to show that the solution of the equation is almost surely Holder continuous in both space and time variables.

Infinitely many solutions for equations of p(x)-Laplace type with the nonlinear Neumann boundary condition

We investigate the following nonlinear Neumann boundary-value problem with associated p(x)-Laplace-type operatorwhere the function φ(x, v) is of type |v|p(x)−2v with continuous function p: → (1,∞) and both f : Ω × ℝ → ℝ and g : ∂Ω × ℝ → ℝ satisfy a Carathéodory condition. We first show the existence of infinitely many weak solutions for the Neumann problems using the Fountain theorem with the Cerami condition but without the Ambrosetti and Rabinowitz condition. Next, we give a result on the existence of a sequence of weak solutions for problem (P) converging to 0 in L∞-norm by employing De Giorgi's iteration and the localization method under suitable conditions.

Maximum principle for quasi-linear reflected backward SPDEs

This paper establishes a maximum principle for quasi-linear reflected backward stochastic partial differential equations (RBSPDEs for short). We prove the existence and uniqueness of the weak solution to RBSPDEs allowing for non-zero Dirichlet boundary conditions and, using a stochastic version of De Giorgi's iteration, establish the maximum principle for RBSPDEs on a general domain. The maximum principle for RBSPDEs on a bounded domain and the maximum principle for backward stochastic partial differential equations (BSPDEs for short) on a general domain can be obtained as byproducts. Finally, the local behavior of the weak solutions is considered.

Local L∞-estimates, weak Harnack inequality, and stochastic continuity of solutions of SPDEs

We consider stochastic partial differential equations under minimal assumptions: the coefficients are merely bounded and measurable and satisfy the stochastic parabolicity condition. In particular, the diffusion term is allowed to be scaling-critical. We derive local supremum estimates with a stochastic adaptation of De Giorgi's iteration and establish a weak Harnack inequality for the solutions. The latter is then used to obtain pointwise almost sure continuity.

Partial Regularity of the Suitable Weak Solutions to the Multi-dimensional Incompressible Boussinesq Equations

This work is concerned with the partial regularity of the suitable weak solutions to the Boussinesq equations in \(\mathbb {R}^{n}\) where \(n=3,\,4\). By means of the De Giorgi iteration method developed in Vasseur (Nonlinear Differ Equ Appl 14(5–6):753–785, 2007), Wang, Wu (J Differ Equ 256(3):1224–1249, 2014), we obtain that \(n-2\) dimensional parabolic Hausdorff measure of the possible singular points set of the suitable weak solutions to this system is zero. Particularly, we obtain some interior regularity criteria only in terms of the scaled mixed norm of velocity for the suitable weak solutions to the Boussinesq equations, which implies that the potential singular points may only stem from the velocity field.

A-priori bounds and existence for solutions of weighted elliptic equations with a convection term

Abstract We investigate weighted elliptic equations containing a convection term with variable exponents that are subject to Dirichlet or Neumann boundary condition. By employing the De Giorgi iteration and a localization method, we give a-priori bounds for solutions to these problems. The existence of solutions is also established using Brezis’ theorem for pseudomonotone operators.

Uniformly attracting limit sets for the critically dissipative SQG equation

We consider the global attractor of the critical surface quasi-geostrophic (SQG) semigroup S(t) on the scale-invariant space . It was shown in [] that this attractor is finite dimensional, and that it attracts uniformly bounded sets in for any , leaving open the question of uniform attraction in . In this paper we prove the uniform attraction in , by combining ideas from the De Giorgi iteration and nonlinear maximum principles.

Extinction of solutions to a class of fast diffusion systems with nonlinear sources

Y. Wang In this paper, the finite time extinction of solutions to the fast diffusion system ut=div(|∇u|p − 2∇u) + vm, vt=div(|∇v|q − 2∇v) + un is investigated, where 1 < p,q < 2, m,n > 0 and is a bounded smooth domain. After establishing the local existence of weak solutions, the authors show that if mn > (p − 1)(q − 1), then any solution vanishes in finite time provided that the initial data are ‘comparable’; if mn = (p − 1)(q − 1) and Ω is suitably small, then the existence of extinction solutions for small initial data is proved by using the De Giorgi iteration process and comparison method. On the other hand, for 1 < p = q < 2 and mn < (p − 1)2, the existence of at least one non-extinction solution for any positive smooth initial data is proved. Copyright © 2015 John Wiley & Sons, Ltd.

A unified proof on the partial regularity for suitable weak solutions of non-stationary and stationary Navier–Stokes equations

The partial regularity of the suitable weak solutions to the Navier–Stokes equations in Rn with n=2,3,4 and the stationary Navier–Stokes equations in Rn for n=2,3,4,5,6 are investigated in this paper. Using some elementary observation of these equations together with De Giorgi iteration method, we present a unified proof on the results of Caffarelli, Kohn and Nirenberg [1], Struwe [17], Dong and Du [5], and Dong and Strain [7]. Particularly, we obtain the partial regularity of the suitable weak solutions to the 4d non-stationary Navier–Stokes equations, which improves the previous result of [5], where Dong and Du studied the partial regularity of smooth solutions of the 4d Navier–Stokes equations at the first blow-up time.

Remarks on partial regularity for suitable weak solutions of the incompressible magnetohydrodynamic equations

In this paper, we are concerned with the partial regularity for suitable weak solutions of the tri-dimensional magnetohydrodynamic equations. With the help of the De Giorgi iteration method, we obtain the results proved by He and Xin (C. He, Z. Xin, Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations, J. Funct. Anal. 227 (2005) 113–152), namely, the one dimensional parabolic Hausdorff measure of the possible singular points of the velocity field and the magnetic field is zero.

Maximum principle for quasi-linear backward stochastic partial differential equations

In this paper we are concerned with the maximum principle for quasi-linear backward stochastic partial differential equations (BSPDEs for short) of parabolic type. We first prove the existence and uniqueness of the weak solution to quasi-linear BSPDEs with the null Dirichlet condition on the lateral boundary. Then using the De Giorgi iteration scheme, we establish the maximum estimates and the global maximum principle for quasi-linear BSPDEs. To study the local regularity of weak solutions, we also prove a local maximum principle for the backward stochastic parabolic De Giorgi class.

A priori bounds for weak solutions to elliptic equations with nonstandard growth

In this paper we study elliptic equations with a nonlinear conormal derivative boundary condition involving nonstandard growth terms. By means of the localization method and De Giorgi's iteration technique we derive global a priori bounds for weak solutions of such problems.

Periodic doubly degenerate parabolic equation with nonlocal terms

This paper is a remark on Gao and Wang (2007) [4] . Under some weaker conditions than [4] , we establish the existence of nontrivial nonnegative periodic solutions for a doubly degenerate parabolic equation with nonlocal terms. The key is the use of De Giorgi iteration technique and the theory of Leray–Schauder degree.

Boundedness of weak solutions to evolutionary partial integro-differential equations with discontinuous coefficients

We investigate linear and quasilinear evolutionary partial integro-differential equations of second order which include time fractional evolution equations of time order less than one. By means of suitable energy estimates and De Giorgi's iteration technique we establish results asserting the global boundedness of appropriately defined weak solutions of these problems. We also show that a maximum principle holds for such equations.

On the [formula omitted]-estimate in Calabi–Yau theorem

We use the technique of De Giorgi iteration to show that the C 0 -norm of the normalized solution of the complex Monge–Ampère equation could be controlled by the L p -norm of the coefficient.