Partial regularity of the gradients of the solutions of nonlinear parabolic systems of nondiagonal form

One establishes the partial C^1, -regularity for the generalized solutions of nondiagonal quasilinear second-order parabolic systems of the form

Partial regularity and singular set of solutions to second order parabolic systems

In this paper we deal with the study of regularity properties of weak solutions to nonlinear, second-order parabolic systems of the type

     

Partial regularity for quasilinear nonuniformly parabolic systems

The partial C^1,λ-regularity of weak solutions to nonuniform parabolic systems is established. Bibliography: 22 titles. Translated fromProblemy Matematicheskogo Analiza, No. 16. 1997, pp. 88–111.     

Partial regularity for parabolic systems with non-standard growth

We study non-linear parabolic systems with non-standard p(z)-growth conditions and establish that the gradient of weak solutions is locally H?lder continuous with H?lder exponent b ? (0,1){\beta \in (0,1)} with respect to the parabolic metric on an open set of full Lebesgue measure, provided the exponent function p(z) itself is H?lder continuous with exponent β with respect to the parabolic metric.     

Partial regularity for weak solutions of nonlinear elliptic systems: the subquadratic case

We consider weak solutions of second order nonlinear elliptic systems of divergence type under subquadratic growth conditions. Via the method of -harmonic approximation we give a characterization of regular points up to the boundary which extends known results from the quadratic and superquadratic case. The proof yields directly the optimal higher regularity on the regular set.     

Partial regularity and singular sets of solutions of higher order parabolic systems

In the present paper we provide a broad survey of the regularity theory for non-differentiable higher order parabolic systems of the type

Partial regularity for subquadratic parabolic systems with continuous coefficients

We establish the partial regularity of solutions to quasilinear parabolic systems with elliptic part that grows subquadratically. More precisely, it is shown that there is an open subset with full measure, of the solution’s domain, on which the solution is H?lder continuous. A key feature in this article is that we only require the coefficients of the system to be continuous with respect to the first two arguments. To prove the result, we use the A-caloric approximation method and an intrinsic scaling. To accommodate the subquadratic growth an adaptation of the A-caloric approximation lemma is also provided.     

Partial regularity of traces of solutions of higher-order nonlinear elliptic systems

For traces of generalized solutions of elliptic systems on smooth manifolds, we study the dependence of the Hausdorff dimension of the set of points at which a solution is not smooth on the modulus of ellipticity of a system.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 11, pp. 1495–1502, November, 1993.     

Explicit decay estimates for solutions to nonlinear parabolic systems

The aim of this paper is to investigate a class of nonlinear parabolic systems with initial and boundary values of Dirichlet type, when the nonlinearities depend on the gradient of the solution. Sufficient conditions on data are established in order to preclude blow up and to deduce that the solution decays exponentially in time. Moreover, an upper bound of its gradient is derived.     

Explicit decay estimates for solutions to nonlinear parabolic systems

The aim of this paper is to investigate a class of nonlinear parabolic systems with initial and boundary values of Dirichlet type, when the nonlinearities depend on the gradient of the solution. Sufficient conditions on data are established in order to preclude blow up and to deduce that the solution decays exponentially in time. Moreover, an upper bound of its gradient is derived.     

Partial regularity for weak solutions of a nonlinear elliptic equation

For scalar non-linear elliptic equations, stationary solutions are defined to be critical points of a functional with respect to the variations of the domain. We consideru a weak positive solution of −Δu=u ^α in -Δu=u ^α in Ω ⊂ ℝ ⁿ , which is stationary. We prove that the Hausdorff dimension of the singular set ofu is less thann−2α+1/α−1, if α≥n+2/n−2.     

Partial regularity for weak solutions of stationary navier-stokes systems

This article is concerned with the partial regularity for the weak solutions of stationary Navier-Stokes system under the controllable growth condition.By A-harmonic approximation technique,the optimal regularity is obtained.     

A priori estimates for solutions to systems of nonlinear parabolic equations

A class of nondiagonal systems of nonlinear parabolic equations that can be reduced to a scalar parabolic equation in the phase space of a larger dimension is described. In view of such a reduction, it is possible to state the maximum principle for solutions to systems of nonlinear parabolic equations and derive a priori C^2+α-estimates for a solution to the Cauchy problem. Bibliography: 19 titles. Translated fromProblemy Matematicheskogo Analiza, No. 16. 1997, pp. 41–67.     

Partial regularity of solutions to a class of degenerate systems

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 .

     

Partial regularity for nonlinear elliptic systems with continuous growth exponent

For weak solutions of nonlinear elliptic systems of the type ${- {\rm div}a(x, u(x), Du(x)) = 0,}$ with nonstandard p(x) growth, we show interior partial Hölder continuity for any Hölder exponent ${\alpha \in (0,1)}$ , provided that the exponent function is ‘logarithmic Hölder continuous’. The result also covers the up to now open partial regularity for systems with constant growth with exponent p less than two in the case of merely continuous dependence on the spacial variable x.     

Viscosity solutions of fully nonlinear parabolic systems

In this paper, we discuss the viscosity solutions of the weakly coupled systems of fully nonlinear second-order degenerate parabolic equations and their Cauchy-Dirichlet problem. We prove the existence, uniqueness and continuity of viscosity solution by combining Perron's method with the technique of coupled solutions. The results here generalize those in Proc. London Math. Soc. 63 (1991) 212-240 and Comm. Partial Differential Equations 16 (1991) 1095-1128.     

Upper-lower solutions for nonlinear parabolic systems and their applications

The method of upper-lower solutions for nonlinear parabolic systems without the assumption of quasi-monotonicity is obtained. An application is provided, by using the method developed in this paper, involving the existence of positive solutions to certain time-dependent interaction systems arising in biological and medical sciences. Furthermore, the existence of the ω-limit of given system is studied.