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1.
We generalize the concept of entropy solutions for parabolic equations with L1-data and consider a class of nonlinear history-dependent degenerated elliptic-parabolic equations including problems with a fractional time derivative such as with Dirichlet boundary conditions and initial condition, where 0<γ?1. We show uniqueness of entropy solutions for general L1-data by Kruzhkov's method of doubling variables. Moreover, existence in the nondegenerated case, i.e. b≡id, is shown by using the concept of generalized solutions of a corresponding abstract Volterra equation.  相似文献   

2.
This paper presents the study of maximal regularity properties for anisotropic differential-operator equations with VMO (vanishing mean oscillation) coefficients. We prove that the corresponding differential operator is separable and is a generator of analytic semigroup in vector-valued Lp spaces. Moreover, discreetness of spectrum and completeness of root elements of this operator is obtained.  相似文献   

3.
In this paper, we prove existence and uniqueness of an entropy solution to the A-obstacle problem, for L 1-data. We also extend the Lewy?CStampacchia inequalities to the general framework of L 1-data and show convergence and stability results. We then prove that the free boundary has finite (N ? 1)-Hausdorff measure, which completes previous works on this subject by Caffarelli for the Laplace operator and by Lee and Shahgholian for the p-Laplace operator when p?>?2.  相似文献   

4.
We investigate a limiting uniqueness criterion to the Navier-Stokes equations. We prove that the mild solution is unique under the class , where bmo-1 is the “critical” space including Ln. As an application of uniqueness theorem, we also consider the local well-posedness of Navier-Stokes equations in bmo-1.  相似文献   

5.
The equation ut=Δp(u1/(p−1)) for p>1 is a nonlinear generalization of the heat equation which is also homogeneous, of degree 1. For large time asymptotics, its links with the optimal Lp-Euclidean logarithmic Sobolev inequality have recently been investigated. Here we focus on the existence and the uniqueness of the solutions to the Cauchy problem and on the regularization properties (hypercontractivity and ultracontractivity) of the equation using the Lp-Euclidean logarithmic Sobolev inequality. A large deviation result based on a Hamilton-Jacobi equation and also related to the Lp-Euclidean logarithmic Sobolev inequality is then stated.  相似文献   

6.
Area integral functions are introduced for sectorial operators on Lp-spaces. We establish the equivalence between the square and area integral functions for sectorial operators on Lp spaces. This follows that the results of Cowling, Doust, McIntosh, Yagi, and Le Merdy on Hinfin functional calculus of sectorial operators on Lp-spaces hold true when the square functions are replaced by the area integral functions.  相似文献   

7.
We prove uniqueness and existence of certain nonlinear stochastic partial differential equations (SPDEs) of divergence type defined on C 1-domains. Some L p and Hölder estimates of the solution and its gradient are also obtained.  相似文献   

8.
We study the comparison principle for degenerate parabolic-hyperbolic equations with initial and nonhomogeneous boundary conditions. We prove a comparison theorem for any entropy sub- and supersolution. The L1 contractivity and, therefore, uniqueness of entropy solutions has been obtained so far by some authors, but it seems that any comparison theorem is not proven. The method used there is the doubling variable technique due to Kru?kov. Our method is based upon the kinetic formulation and the kinetic techniques. By developing the kinetic techniques for degenerate parabolic-hyperbolic equations with boundary conditions, we can obtain a comparison property which obviously extends the L1 contractive property.  相似文献   

9.
We obtain a strong solution in charge critical space L2(R) of the Thirring system and Federbusch equations in one space dimension by using solution representation of the models. The uniqueness is obtained for the solution ΨL([0,T];L2(R)∩L4(R)). A decay of local charge and asymptotic behavior of the field can be shown directly.  相似文献   

10.
We consider the strongly elliptic operator A of order 2m in the divergence form with bounded measurable coefficients and assume that the coefficients of top order are uniformly continuous. For 1<p<∞, A is a bounded linear operator from the Lp Sobolev space Hm,p into Hm,p. We will prove that (Aλ)−1 exists in Hm,p for some λ and estimate its operator norm.  相似文献   

11.
We first prove a local weighted integral inequality for conjugate A-harmonic tensors. Then, as an application of our local result, we prove a global weighted integral inequality for conjugate A-harmonic tensors in Ls(μ)-averaging domains, which can be considered as a generalization of the classical result. Finally, we give applications of the above results to quasiregular mappings.  相似文献   

12.
Given a Calderón-Zygmund (C-Z for short) operatorT, which satisfies Hörmander condition, we prove that: ifT maps all the characteristic atoms toWL 1, thenT is continuous fromL p toL p (1 <p < ∞). So the study of strong continuity on arbitrary function inL p has been changed into the study of weak continuity on characteristic functions.  相似文献   

13.
本文研究下面这两个函数,(1-|x|^ρ1)+^α和(1-|x|^ρ2)+^α,其中ρ1,ρ2和α均为正实数,文献上称这类函数为广义Bochner-Riesz乘子.本文将证明,当α给定时,对任意的ρ1〉0和ρ2〉0,这两个函数作为乘子,其乘子算子的L^p有界性和Hp有界性是等价的.  相似文献   

14.
15.
The aim of this paper is twofold: to prove, for L 1-data, the existence and uniqueness of an entropy solution to the obstacle problem for nonlinear elliptic equations with variable growth, and to show some convergence and stability properties of the corresponding coincidence set. The latter follow from extending the Lewy-Stampacchia inequalities to the general framework of L 1.  相似文献   

16.
We prove uniqueness for extended real-valued lower semicontinuous viscosity solutions of the Bellman equation forL -control problems. This result is then used to prove uniqueness for lsc solutions of Hamilton-Jacobi equations of the form –u t +H(t, x, u, –Du)=0, whereH(t, x, r, p) is convex inp. The remaining assumptions onH in the variablesr andp extend the currently known results.Supported in part by Grant DMS-9300805 from the National Science Foundation.  相似文献   

17.
In this paper, we construct local solution with highly oscillating initial velocity and then get the global strong solution in the LpLp based Besov space which improves the result of J. Qian, Z. Zhang (2010) [25] and X. Hu, D. Wang (2011) [14]. The local existence and uniqueness lies on the Lagrange coordinate transform and the contraction mapping theorem. The global result lies on a decomposition of the system and some commutator estimates. In the last part, we prove a time-decay in the critical Besov space framework which seems to have little investigation. The proof is based on the properties of the Green's matrix and various interpolations between Besov type spaces.  相似文献   

18.
Sufficient conditions for the complete convergence for the partial sums and the random selected partial sums of Lp-mixingales are given. Necessary conditions are also discussed.  相似文献   

19.
In this paper we present some results on the global existence of weak solutions to a nonlinear variational wave equation and some related problems. We first introduce the main tools, the L p Young measure theory and related compactness results, in the first section. Then we use the L p Young measure theory to prove the global existence of dissipative weak solutions to the asymptotic equation of the nonlinear wave equation, and comment on its relation to Camassa-Holm equations in the second section. In the third section, we prove the global existence of weak solutions to the original nonlinear wave equation under some restrictions on the wave speed. In the last section, we present global existence of renormalized solutions to two-dimensional model equations of the asymptotic equation, which is also the so-called vortex density equation arising from sup-conductivity.  相似文献   

20.
We first obtain the Lp-Lq estimates of solutions to the Cauchy problem for one-dimensional damped wave equation
  相似文献   

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