In the first part we introduce the A-caloric approximation lemma, a parabolic analogue of the harmonic approximation lemma of De Giorgi [Sem. Scuola Normale Superiore Pisa (1960–1961); Lectures in Math., ETH Zürich, Birkhäuser, Basel, 1996] in the version of Simon. This allows to prove optimal partial regularity results for solutions in an elementary way, under minimal and natural assumptions. In the second part we provide estimates for the parabolic Hausdorff dimension of the singular sets of solutions; the proof makes use of parabolic fractional Sobolev spaces.  相似文献   

11.
Optimal time and space regularity for solutions of degenerate differential equations     
Alberto Favaron 《Central European Journal of Mathematics》2009,7(2):249-271
We derive optimal regularity, in both time and space, for solutions of the Cauchy problem related to a degenerate differential equation in a Banach space X. Our results exhibit a sort of prevalence for space regularity, in the sense that the higher is the order of regularity with respect to space, the lower is the corresponding order of regularity with respect to time.   相似文献   

12.
Partial regularity and singular sets of solutions of higher order parabolic systems     
Verena Bögelein 《Annali di Matematica Pura ed Applicata》2009,188(1):61-122
In the present paper we provide a broad survey of the regularity theory for non-differentiable higher order parabolic systems of the type
  首页 | 本学科首页   官方微博 | 高级检索  
相似文献
 共查询到20条相似文献,搜索用时 0 毫秒
1.
We consider the system , in coupled with suitable initial-boundary conditions, where is a bounded domain in with smooth boundary and is a continuous and positive function of . Our main result is that under some conditions on there exists a relatively open subset of such that is locally Hölder continuous on , the interior of is empty, and is essentially bounded on .

  相似文献   


2.
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.  相似文献   

3.
In this paper, we study the local behavior of the solutions to the three-dimensional magnetohydrodynamic equations. we are interested in both the uniform gradient estimates for smooth solutions and regularity of weak solutions. It is shown that, in some neighborhood of (x0,t0), the gradients of the velocity field u and the magnetic field B are locally uniformly bounded in L norm as long as that either the scaled local L2-norm of the gradient or the scaled local total energy of the velocity field is small, and the scaled local total energy of the magnetic field is uniformly bounded. These estimates indicate that the velocity field plays a more dominant role than that of the magnetic field in the regularity theory. As an immediately corollary we can derive an estimates of Hausdorff dimension on the possible singular set of a suitable weak solution as in the case of pure fluid. Various partial regularity results are obtained as consequences of our blow-up estimates.  相似文献   

4.
5.
In this paper, we establish Hölder regularity of weak solutions to the diagonal divergence form degenerate quasilinear parabolic system related to Hörmander type vector fields by deriving a parabolic Poincaré inequality and the higher integrability of gradients of weak solutions.  相似文献   

6.
7.
In this paper we deal with the study of regularity properties of weak solutions to nonlinear, second-order parabolic systems of the type
  相似文献   

8.
Sufficient conditions are given for the solutions to the (fully nonlinear, degenerate) elliptic equation F(x,u,Du,D2u)=0 in Ω to satisfy |u(x)−u(y)|?Cα|xy| for some α∈(0,1) when xΩ and y∈∂Ω.  相似文献   

9.
10.
We present a new, complete approach to the partial regularity of solutions to non-linear, second order parabolic systems of the form
ut−divA(x,t,u,Du)=0.
Initially, we prove a partial regularity result with the method of A-polycaloric approximation, which is a parabolic analogue of the harmonic approximation lemma of De Giorgi. Moreover, we prove better estimates for the maximal parabolic Hausdorff-dimension of the singular set of weak solutions, using fractional parabolic Sobolev spaces. Thereby, we also consider different situations, which yield a better dimension reduction result, including the low dimensional case and coefficients A(z, D m u), independent of the lower order derivatives of u.   相似文献   

13.
Partial regularity of weak solutions to a PDE system with cubic nonlinearity     
Jian-Guo Liu  Xiangsheng Xu 《Journal of Differential Equations》2018,264(8):5489-5526
In this paper we investigate regularity properties of weak solutions to a PDE system that arises in the study of biological transport networks. The system consists of a possibly singular elliptic equation for the scalar pressure of the underlying biological network coupled to a diffusion equation for the conductance vector of the network. There are several different types of nonlinearities in the system. Of particular mathematical interest is a term that is a polynomial function of solutions and their partial derivatives and this polynomial function has degree three. That is, the system contains a cubic nonlinearity. Only weak solutions to the system have been shown to exist. The regularity theory for the system remains fundamentally incomplete. In particular, it is not known whether or not weak solutions develop singularities. In this paper we obtain a partial regularity theorem, which gives an estimate for the parabolic Hausdorff dimension of the set of possible singular points.  相似文献   

14.
On improved regularity of weak solutions of some degenerate,Anisotropic elliptic systems     
Tilak Bhattacharya  Francesco Leonetti 《Annali di Matematica Pura ed Applicata》1996,170(1):241-255
Summary We consider a (possibly) vector-valued function u: RN, Rn, minimizing the integral , 2-2/(n*1)<p<2, whereD i u=u/x i or some more general functional retaining the same behaviour, we prove higher integrability for Du: D1 u,..., Dn–1 u Lp/(p-1) and Dnu L2; this result allows us to get existence of second weak derivatives: D(D1 u),...,D(Dn–1u)L2 and D(Dn u) L p.This work has been supported by MURST and GNAFA-CNR.  相似文献   

15.
Partial regularity of weak solutions for Landau-Lifshitz system with potential     
YuMing Chu  XianGao Liu 《中国科学A辑(英文版)》2009,52(5):981-990
In this note, we prove the partial regularity of stationary weak solutions for the Landau-Lifshitz system with general potential in four or three dimensional space. As well known, in general, since the constraint of the methods, in order to get the partial regularity of stationary weak solution of the Landau-Lifshitz system with potential, we need to add some very strongly conditions on the potential. The main difficulty caused by potential is how to find the equation satisfied by the scaling function, which breaks down the blow-up processing. We estimate directly Morrey’s energy to avoid the difficulties by blowing up. This work was supported by National Natural Science Foundation of China (Grant Nos. 10631020, 60850005) and the Natural Science Foundation of Zhejiang Province (Grant No. D7080080)  相似文献   

16.
On the partial regularity of weak solutions of nonlinear parabolic systems     
Mariano Giaquinta  Michael Struwe 《Mathematische Zeitschrift》1982,179(4):437-451
  相似文献   

17.
Local existence and regularity for a class of degenerate parabolic equations     
Sigurd Angenent 《Mathematische Annalen》1988,280(3):465-482
  相似文献   

18.
Local regularity for solutions of anisotropic obstacle problems     
Hongya Gao  Qiuhua Huang 《Nonlinear Analysis: Theory, Methods & Applications》2012
  相似文献   

19.
Optimal partial regularity for weak solutions of nonlinear sub-elliptic systems in Carnot groups   总被引:1,自引:0,他引:1  
Jialin Wang  Pengcheng Niu 《Nonlinear Analysis: Theory, Methods & Applications》2010,72(11):4162-180
This paper is concerned with partial regularity for weak solutions to nonlinear sub-elliptic systems in divergence form in Carnot groups. The technique of A-harmonic approximation introduced by Simon and developed by Duzaar and Grotowski is adapted to our context. We establish Caccioppoli type inequalities and partial regularity with optimal local Hölder exponents for horizontal gradients of weak solutions to systems under super-quadratic natural structure conditions and super-quadratic controllable structure conditions, respectively.  相似文献   

20.
Global and nonglobal weak solutions to a degenerate parabolic system     
Peidong Lei  Sining Zheng 《Journal of Mathematical Analysis and Applications》2006,324(1):177-198
This paper deals with a degenerate parabolic system coupled via general reaction terms of power type. Global weak solutions are obtained by means of energy estimates and the De Giorgi's technique. In particular, the criterion for global nonexistence of weak solutions is proved by introducing suitable weak sub-solutions together with a weak comparison principle. In summary, the critical exponent for weak solutions of the degenerate parabolic system is determined.  相似文献   

设为首页 | 免责声明 | 关于勤云 | 加入收藏

Copyright©北京勤云科技发展有限公司  京ICP备09084417号