Parabolic Equations In Nondivergence Form Research Articles

Intrinsic Harnack’s Inequality for a General Nonlinear Parabolic Equation in Non-divergence Form

AbstractWe prove the intrinsic Harnack’s inequality for a general form of a parabolic equation that generalizes both the standard parabolic p-Laplace equation and the normalized version arising from stochastic game theory. We prove each result for the optimal range of exponents and ensure that we get stable constants.

Nonlinear Einstein paradigm of Brownian motion and localization property of solutions

We employ a generalization of Einstein's random walk paradigm for diffusion to derive a class of multidimensional degenerate nonlinear parabolic equations in nondivergence form. Specifically, in these equations, the diffusion coefficient can depend on both the dependent variable and its gradient, and it vanishes when either one of the latter does. It is known that solutions of such degenerate equations can exhibit finite speed of propagation (so‐called localization property of solutions). We give a proof of this property using a De Giorgi–Ladyzhenskaya iteration procedure for nondivergence form equations. A mapping theorem is then established to a divergence‐form version of the governing equation for the case of one spatial dimension. Numerical results via a finite‐difference scheme are used to illustrate the main mathematical results for this special case. For completeness, we also provide an explicit construction of the one‐dimensional self‐similar solution with finite speed of propagation function, in the sense of Kompaneets–Zel'dovich–Barenblatt. We thus show how the finite speed of propagation quantitatively depends on the model's parameters.

Estimates for fundamental solutions of parabolic equations in non-divergence form

We construct the fundamental solution of second order parabolic equations in non-divergence form under the assumption that the coefficients are of Dini mean oscillation in the spatial variables. We also prove that the fundamental solution satisfies a sub-Gaussian estimate. In the case when the coefficients are Dini continuous in the spatial variables and measurable in the time variable, we establish the Gaussian bounds for the fundamental solutions. We present a method that works equally for second order parabolic systems in non-divergence form.

Singular solutions to parabolic equations in nondivergence form

For any $\alpha \in (0,1)$, we construct an example of a solution to a parabolic equation with measurable coefficients in two space dimensions which has an isolated singularity and is not better that $C^\alpha$. We prove that there exists no solution to a fully nonlinear uniformly parabolic equation, in any dimension, which has an isolated singularity where it is not $C^2$ while it is analytic elsewhere, and it is homogeneous in $x$ at the time of the singularity. We build an example of a non homogeneous solution to a fully nonlinear uniformly parabolic equation with an isolated singularity, which we verify with the aid of a numerical computation.

An Intrinsic Harnack inequality for some non-homogeneous parabolic equations in non-divergence form

In this paper, we establish a scale invariant Harnack inequality for some inhomogeneous parabolic equations in a suitable intrinsic geometry dictated by the nonlinearity. The class of equations that we consider correspond to the parabolic counterpart of the equations studied by Julin in [10] where a generalized Harnack inequality was obtained which quantifies the strong maximum principle. Our version of parabolic Harnack (see Theorem 1.2) when restricted to the elliptic case is however quite different from that in [10]. The key new feature of this work is an appropriate modification of the stack of cubes covering argument which is tailored for the nonlinearity that we consider.

Growth theorems for metric spaces with applications to PDE

The paper is devoted to some extensions of the joint results by N. V. Krylov and the author on the Harnack inequalities for second order elliptic and parabolic equations in nondivergence form to metric spaces of homogeneous type. The main tools are special Landis-type growth theorems.

Lp-estimates for the Hessians of solutions to fully nonlinear parabolic equations with oblique boundary conditions

We study fully nonlinear parabolic equations in nondivergence form with oblique boundary conditions. An optimal and global Calderón-Zygmund estimate is obtained by proving that the Hessian of the viscosity solution to the oblique boundary problem is as integrable as the nonhomogeneous term in Lp spaces under minimal regularity requirement on the nonlinear operator, the boundary data and the boundary of the domain.

Weighted mixed-norm $L_p$-estimates for elliptic and parabolic equations in non-divergence form with singular coefficients

In this paper, we study non-divergence form elliptic and parabolic equations with singular coefficients. Weighted and mixed-norm L_p -estimates and solvability are established under suitable partially weighted BMO conditions on the coefficients. When the coefficients are constants, the operators are reduced to extensional operators which arise in the study of fractional heat equations and fractional Laplace equations. Our results are new even in this setting and in the unmixed norm case. For the proof, we explore and utilize the special structures of the equations to show both interior and boundary Lipschitz estimates for solutions and for higher-order derivatives of solutions to homogeneous equations. We then employ the perturbation method by using the Fefferman–Stein sharp function theorem, the Hardy–Littlewood maximal function theorem, as well as a weighted Hardy’s inequality.

An approach for weighted mixed-norm estimates for parabolic equations with local and non-local time derivatives

We give a unified approach to weighted mixed-norm estimates and solvability for both the usual and time fractional parabolic equations in nondivergence form when coefficients are merely measurable in the time variable. In the spatial variables, the leading coefficients locally have small mean oscillations. Our results extend the previous result in [5] for unmixed Lp-estimates without weights.

On the Harnack inequality for non-divergence parabolic equations

In this paper we propose an elementary proof of the Harnack inequality for linear parabolic equations in non-divergence form.

Local regularity for quasi-linear parabolic equations in non-divergence form

We consider viscosity solutions to non-homogeneous degenerate and singular parabolic equations of the p-Laplacian type and in non-divergence form. We provide local Hölder and Lipschitz estimates for the solutions. In the degenerate case, we prove the Hölder regularity of the gradient. Our study is based on a combination of the method of alternatives and the improvement of flatness estimates.

A new non-divergence diffusion equation with variable exponent for multiplicative noise removal

A new kind of model based on double degenerate parabolic equation in nondivergence form is proposed for multiplicative noise removal. We first utilize the regularization method to illustrate the existence of the weak solution for the problem and the non-expansion of support of the solution is also provided. In the numerical aspect, experiments and comparisons with other denoising models are presented in order to state the denoising capability of our model.

Gradient regularity for a singular parabolic equation in non-divergence form

In this paper we consider viscosity solutions of a class of non-homogeneous singular parabolic equations \begin{document}$ \partial_t u-|Du|^\gamma\Delta_p^N u = f, $\end{document} where $ -1<\gamma<0 $, $ 1<p<\infty $, and $ f $ is a given bounded function. We establish interior Hölder regularity of the gradient by studying two alternatives: The first alternative uses an iteration which is based on an approximation lemma. In the second alternative we use a small perturbation argument.

Null controllability for a coupled system of degenerate/singular parabolic equations in nondivergence form

Null controllability for a coupled system of degenerate/singular parabolic equations in nondivergence form

Nondivergence parabolic equations in weighted variable exponent spaces

We prove the global Calderón-Zygmund estimates for second order parabolic equations in nondivergence form in weighted variable exponent Lebesgue spaces. We assume that the associated variable exponent is log-Hölder continuous, the weight is of a certain Muckenhoupt class with respect to the variable exponent, the coefficients of the equation are the functions of small bonded mean oscillation, and the underlying domain is a C 1 , 1 C^{1,1} -domain.

Maximal and minimal solutions of second order elliptic and parabolic equations in non-divergence form with measurable coefficients

In this paper we will prove that the supremum and infimum of good solutions of the Dirichlet problem for elliptic and parabolic equations in non-divergence form with measurable coefficients, are good solutions to the same problem.

Asymptotic behavior of solutions for a doubly degenerate parabolic non-divergence form equation

This paper is concerned with the asymptotic behavior of a doubly degenerate parabolic equation in non-divergence form. The proofs are divided into three cases according to exponent values of the source, and, by using different methods, we prove the stability of the steady states. We also expand the discussion of asymptotic stability for equations with the periodic source.

Upper bounds for parabolic equations and the Landau equation

We consider a parabolic equation in nondivergence form, defined in the full space [0,∞)×Rd, with a power nonlinearity as the right-hand side. We obtain an upper bound for the solution in terms of a weighted control in Lp. This upper bound is applied to the homogeneous Landau equation with moderately soft potentials. We obtain an estimate in L∞(Rd) for the solution of the Landau equation, for positive time, which depends only on the mass, energy and entropy of the initial data.

Interior degenerate/singular parabolic equations in nondivergence form: well-posedness and Carleman estimates

We consider non-smooth general degenerate/singular parabolic equations in non-divergence form with degeneracy and singularity occurring in the interior of the spatial domain, in presence of Dirichlet or Neumann boundary conditions. In particular, we consider well posedness of the problem and then we prove Carleman estimates for the associated adjoint problem.

A Quantitative Regularity Estimate for Nonnegative Supersolutions of Uniformly Parabolic Equations

This note establishes an interior quantitative lower bound for nonnegative supersolutions of fully nonlinear uniformly parabolic equations. The result may be interpreted as a quantitative version of a growth lemma established by Krylov and Safonov for nonnegative supersolutions of linear uniformly parabolic equations in nondivergence form. Our approach is different, and follows from an application of a reverse Holder inequality. The result is the parabolic analogue of an elliptic regularity estimate established by Caffarelli, Souganidis, and Wang in the stochastic homogenization of fully nonlinear uniformly elliptic equations.