scholarly journals Gradient weighted norm inequalities for very weak solutions of linear parabolic equations with BMO coefficients

2021 ◽  
pp. 1-15
Author(s):  
Le Trong Thanh Bui ◽  
Quoc-Hung Nguyen
Keyword(s):  
Weak Solutions ◽  
Parabolic Equations ◽  
Lipschitz Constant ◽  
Weighted Norm ◽  
Very Weak Solutions ◽  
Class A ◽  
Mean Oscillation ◽  
Muckenhoupt Class ◽  
Linear Parabolic Equations ◽  
Bmo Coefficients

In this paper, we give a short proof of the Lorentz estimates for gradients of very weak solutions to the linear parabolic equations with the Muckenhoupt class A q -weights u t − div ( A ( x , t ) ∇ u ) = div ( F ) , in a bounded domain Ω × ( 0 , T ) ⊂ R N + 1 , where A has a small mean oscillation, and Ω is a Lipchistz domain with a small Lipschitz constant.

scholarly journals Lorentz Estimates for Weak Solutions of Quasi-linear Parabolic Equations with Singular Divergence-free Drifts

2019 ◽  
Vol 71 (4) ◽  
pp. 937-982
Author(s):  
Tuoc Phan
Keyword(s):  
Weak Solutions ◽  
Parabolic Equations ◽  
Global Regularity ◽  
Perturbation Technique ◽  
Principal Term ◽  
Lorentz Spaces ◽  
Bmo Space ◽  
Local Boundary ◽  
Linear Parabolic Equations ◽  
Divergence Free

AbstractThis paper investigates regularity in Lorentz spaces for weak solutions of a class of divergence form quasi-linear parabolic equations with singular divergence-free drifts. In this class of equations, the principal terms are vector field functions that are measurable in ($x,t$)-variable, and nonlinearly dependent on both unknown solutions and their gradients. Interior, local boundary, and global regularity estimates in Lorentz spaces for gradients of weak solutions are established assuming that the solutions are in BMO space, the John–Nirenberg space. The results are even new when the drifts are identically zero, because they do not require solutions to be bounded as in the available literature. In the linear setting, the results of the paper also improve the standard Calderón–Zygmund regularity theory to the critical borderline case. When the principal term in the equation does not depend on the solution as its variable, our results recover and sharpen known available results. The approach is based on the perturbation technique introduced by Caffarelli and Peral together with a “double-scaling parameter” technique and the maximal function free approach introduced by Acerbi and Mingione.

WEIGHTED SOLVABILITY FOR PARABOLIC EQUATIONS WITH PARTIALLY BMO COEFFICIENTS AND ITS APPLICATIONS

2014 ◽  
Vol 96 (3) ◽  
pp. 396-428
Author(s):  
LIN TANG
Keyword(s):  
Bounded Domain ◽  
Parabolic Equations ◽  
Global Regularity ◽  
Morrey Spaces ◽  
Divergence Form ◽  
Bounded Mean Oscillation ◽  
Nondivergence Form ◽  
Mean Oscillation ◽  
Bmo Coefficients

AbstractWe consider the weighted $L_p$ solvability for divergence and nondivergence form parabolic equations with partially bounded mean oscillation (BMO) coefficients and certain positive potentials. As an application, global regularity in Morrey spaces for divergence form parabolic operators with partially BMO coefficients on a bounded domain is established.

Stability results for weak solutions to backward one-dimensional semi-linear parabolic equations with locally Lipschitz source

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Dinh Nho Hào ◽  
Nguyen Van Duc ◽  
Nguyen Thi Ngoc Oanh
Keyword(s):  
Inverse Problems ◽  
Weak Solutions ◽  
Parabolic Equations ◽  
Stability Estimates ◽  
One Dimensional ◽  
Nonlinear Inverse Problems ◽  
Linear Parabolic Equations ◽  
Locally Lipschitz ◽  
Stability Results

Abstract Stability estimates of Hölder type for weak solutions to backward one-dimensional semi-linear parabolic equations with locally Lipschitz source are obtained. It is noticed that stability results for weak solutions to nonlinear inverse problems are very rare in the literature.

scholarly journals A Boundary Estimate for Singular Sub-Critical Parabolic Equations

2020 ◽  
Author(s):  
Ugo Gianazza ◽  
Naian Liao
Keyword(s):  
Modulus Of Continuity ◽  
Weak Solutions ◽  
Parabolic Equations ◽  
Boundary Point ◽  
Cylindrical Domain ◽  
Boundary Estimate ◽  
Critical Range ◽  
Type Integral ◽  
Wiener Type ◽  
Singular Parabolic Equations

Abstract We prove an estimate on the modulus of continuity at a boundary point of a cylindrical domain for local weak solutions to singular parabolic equations of $p$-Laplacian type, with $p$ in the sub-critical range $\big(1,\frac{2N}{N+1}\big]$. The estimate is given in terms of a Wiener-type integral, defined by a proper elliptic $p$-capacity.

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