We study the Dirichlet problem for a second-order linear elliptic equation in a bounded smooth domain Omega in R-n, n >= 3, with the drift b belonging to the critical weak space L-n,L-infinity(Omega). We decompose the drift b=b1+b2 in which div b(1)>= 0 and b(2) is small only in a small scale quasi-norm of L-n,L-infinity(Omega). Under this new smallness condition, we prove existence, uniqueness, and regularity estimates of weak solutions to the problem and its dual. Holder regularity and derivative estimates of weak solutions to the dual problem are also established. As a result, we prove uniqueness of very weak solutions slightly below the threshold. When b(2)=0, our results recover those by Kim and Tsai in [Existence, uniqueness, and regularity results for elliptic equations with drift terms in critical weak spaces, SIAM J. Math. Anal.52(2) (2020) 1146-1191]. Due to the new small scale quasi-norm, our results are new even when b(1)=0.

We study a class of parabolic equations in non-divergence form with measurable coefficients that are singular, degenerate, or both singular and degenerate through a weight belonging to the A_1+1/n -Muckenhoupt class of weights. Under some smallness assumption on a weighted mean oscillation of the weight, F.-H. Lin type weighted W^2,ε-estimates are proved. To prove the result, we establish a result on local quantitative lower estimates of solutions to the class of equations, which are known as the mean sojourn times of sample paths within sets. This type of estimate was proved by L. C. Evans for the class of linear elliptic equations in non-divergence form with uniformly elliptic and bounded measurable coefficients. A class of weighted parabolic cylinders intrinsically suitable for the class of equations is introduced. The parabolic ABP estimates, and a perturbation method are used to overcome the singularity and degeneracy of the coefficients. Careful analysis on regularization and truncation of the weights is performed. The paper provides foundational ingredients and estimates for the study of fully nonlinear parabolic equations with singular-degenerate coefficients.

We investigate Dirichlet boundary value problems for a class of second-order parabolic equations in divergence-form with coefficient matrices that exhibit singular and degenerate behaviors characterized by a Muckenhoupt weight class. This framework serves as the parabolic analogue to the singular-degenerate elliptic equations pioneered by Fabes, Kenig, and Seraponi. Under a smallness assumption on the partially weighted mean oscillation of the coefficients, we establish the existence, uniqueness, and local interior and boundary regularity estimates for weak solutions within appropriately defined weighted Sobolev spaces. The proofs rely on the freezing coefficient technique alongside the level-set method introduced by Caffarelli and Peral. Additionally, we develop the necessary weighted Sobolev space framework and related weighted inequalities. Finally, a compactness argument is utilized to demonstrate that solutions to these equations remain locally close, in the weighted Sobolev norm, to their frozen-coefficient counterparts.

This paper studies a class of linear parabolic equations in non-divergence form in which the leading coefficients are measurable and they can be singular or degenerate through a weight belonging to the A1+ 1 n class of Muckenhoupt weights. Krylov-Safonov Harnack inequality for solutions is proved under some smallness assumption on a weighted mean oscillation of the weight. To prove the result, we introduce a class of generic weighted parabolic cylinders and the smallness condition on the weighted mean oscillation of the weight through which several growth lemmas are established. Additionally, a perturbation method is used and the parabolic Aleksandrov-Bakelman-Pucci type maximum principle is crucially applied to suitable barrier functions to control the solutions. As corollaries, H & ouml;lder regularity estimates of solutions with respect to a quasi-distance, and a Liouville type theorem are obtained in the paper. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

We study an inviscid limit problem for a class of Navier-Stokes equations with vanishing measurable viscous coefficients in 3-dimensional spatial domains whose boundaries are oscillatory, depending on a small parameter, and become flat when the parameter converges to zero. Under some sufficient conditions on the anisotropic vanishing rates of the eigenvalues of the matrices of the viscous coefficients and the oscillatory parameter, we show that Leray-Hopf weak solutions of the Navier-Stokes equations with no slip boundary condition converge to solutions of the Euler equations in the upper half space. To prove the result, we apply a change of variables to flatten the boundaries of the spatial domains for the Navier-Stokes equations, and then construct the boundary layer terms. As the Navier-Stokes equations and the Euler equations are originally written in two different domains, additional boundary layer terms are constructed and their estimates are obtained.

This paper studies a class of linear parabolic equations with measurable coefficients in divergence form whose volumetric heat capacity coefficients are assumed to be in some Muckenhoupt class of weights. As such, the coefficients can be degenerate, singular, or both degenerate and singular. A class of weighted parabolic cylinders with a non-homogeneous quasi-distance function, and a class of weighted parabolic Sobolev spaces intrinsically suitable for the class of equations are introduced. Under some smallness assumptions on the mean oscillations of the coefficients, regularity estimates, existence, and uniqueness of weak solutions in the weighted Sobolev spaces are proved. To achieve the results, we apply the level-set method introduced by Caffarelli and Peral. Several weighted inequalities and a version of weighted Aubin-Lions compactness theorem for sequences in weighted parabolic Sobolev spaces are established.

We study a class of nondivergence form second-order degenerate linear parabolic equations in (−∞,T)×R+d with the homogeneous Dirichlet boundary condition on (−∞,T)×∂R+d, where R+d={x=(x1,x2,…,xd)∈Rd:xd>0} and T∈(−∞,∞] is given. The coefficient matrices of the equations are the product of μ(xd) and bounded positive definite matrices, where μ(xd) behaves like xdα for some given α∈(0,2), which are degenerate on the boundary {xd=0} of the domain. Under a partially weighted VMO (vanishing mean oscillation) assumption on the coefficients, we obtain the wellposedness and regularity of solutions in weighted Sobolev spaces. The results are applied to study the regularity theory of solutions to a class of degenerate viscous Hamilton-Jacobi equations.

We study a conormal boundary value problem for a class of quasilinear elliptic equations in bounded domain Ω whose coefficients can be degenerate or singular of the type dist(x, ∂Ω )^α , where ∂Ω is the boundary of Ω and α∈ (-1, ∞ ) is a given number. We establish weighted Sobolev type estimates for weak solutions under a smallness assumption on the weighted mean oscillations of the coefficients in small balls. Our approach relies on a perturbative method and several new Lipschitz estimates for weak solutions to a class of singular-degenerate quasilinear equations.

We study the incompressible stationary Navier-Stokes equations in the upper-half plane with homogeneous Dirichlet boundary condition and non-zero external forcing terms. Existence of weak solutions is proved under a suitable condition on the external forces. Weak-strong uniqueness criteria based on various growth conditions at the infinity of weak solutions are also given. This is done by employing an energy estimate and a Hardy's inequality. Several estimates of stream functions are carried out and two density lemmas with suitable weights for the homogeneous Sobolev space on 2 dimensional space are proved.

We prove trace theorems for weighted mixed norm Sobolev spaces in the upper-half space where the weight is a power function of the vertical variable. The results show the differentiability order of the trace functions depends only on the power in the weight function and the integrability power for the integration with respect to the vertical variable but not on the integrability powers for the integration with respect to the horizontal ones. They are new even in the un-weighted case and they recover classical results in the case of un-mixed norm spaces. The work is motivated by the study of regularity theory for solutions of elliptic and parabolic equations with anisotropic features and with non-homogeneous boundary conditions. The results provide an essential ingredient to the study of fractional elliptic and parabolic equations in divergence form with measurable coefficients.

In this note we establish existence and uniqueness of weak solutions of linear elliptic equation div[𝐀(x) ∇ u] = div𝐅(x), where the matrix 𝐀 is just measurable and its skew-symmetric part can be unbounded. Global reverse Hölder's regularity estimates for gradients of weak solutions are also obtained. Most importantly, we show, by providing an example, that boundedness and ellipticity of 𝐀 is not sufficient for higher integrability estimates even when the symmetric part of 𝐀 is the identity matrix. In addition, the example also shows the necessity of the dependence of α in the Hölder C^α-regularity theory on the -semi norm of the skew-symmetric part of 𝐀. The paper is an extension of classical results obtained by N. G. Meyers (1963) in which the skew-symmetric part of 𝐀 is assumed to be zero.

We study a class of non-divergence form elliptic and parabolic equations with singular first-order coefficients in an upper half space with the homogeneous Dirichlet boundary condition. In the simplest setting, the operators in the equations under consideration appear in the study of fractional heat and fractional Laplace equations. Intrinsic weighted Sobolev spaces are found in which the existence and uniqueness of strong solutions are proved under certain smallness conditions on the weighted mean oscillations of the coefficients in small parabolic cylinders. Our results are new even when the coefficients are constants and they cover the case where the weights may not be in the Ap-Muckenhoupt class.

We study the convergence of weak solutions of the Navier–Stokes equations with vanishing measurable viscous coefficients in domains with nonflat boundaries. Sufficient anisotropic conditions on the vanishing rates of the viscous coefficients are found to prove the convergence of Leray–Hopf weak solutions of the Navier–Stokes equations to solutions of the corresponding Euler equations. As the domains are not flat, we apply a change of variables to flatten the domains. We then construct explicit boundary layers for the system of Navier–Stokes equations in the upper‐half space with measurable viscous coefficients. The result is new even when the viscous coefficients are constant, and it recovers the classical results when domains are flat and with constant viscous coefficients.

We study parabolic and elliptic equations of both divergence and non-divergence form in the half space {x(d) > 0} whose coefficients are the product of x(d)(a), and uniformly nonde-generate bounded measurable matrix-valued functions, where a ? (-1, 8). As such, the coefficients are singular or degen-erate near the boundary of the half space. For equations with the conormal or Neumann boundary condition, we prove the existence, uniqueness, and regularity of solutions in weighted Sobolev spaces and mixed-norm weighted Sobolev spaces when the coefficients are only measurable in the x(d) direction and have small mean oscillation in the other directions in small cylinders. Our results are new even in the special case when the coefficients are constants.

We consider Stokes systems with measurable coefficients and Lions-type boundary conditions. We show that, in contrast to the Dirichlet boundary conditions, local boundary mixed-norm Ls,q-estimates hold for the spatial second-order derivatives of solutions, assuming the smallness of the mean oscillations of the coefficients with respect to the spatial variables in small cylinders. In the un-mixed norm case with s=q=2, the result is still new and provides local boundary Caccioppoli-type estimates. The main challenges in the work arise from the lack of regularity of the pressure and time derivatives of the solutions and from interaction of the boundary with the nonlocal structure of the system. To overcome these difficulties, our approach relies heavily on several newly developed regularity estimates for both divergence and non-divergence form parabolic equations with coefficients that are only measurable in the time variable and in one of the spatial variables.

We study parabolic equations in divergence form with coefficients which are singular or degenerate as Muckenhoupt weight functions in one spatial variable. We establish weighted reverse Hölder’s inequalities, and Lipschitz estimates for weak solutions of homogeneous equations with coefficients depending only on one spatial variable. We then use these results to prove interior, boundary, and global weighted estimates of Calderón-Zygmund type for weak solutions, assuming that the coefficients are partially vanishing mean oscillations with respect to the considered weights. The solvability in weighted Sobolev spaces is also achieved. Such results are new even for elliptic equations and our results can be readily extended to systems.

We study a class of linear parabolic equations in divergence form with degenerate coefficients on the upper half space. Specifically, the equations are considered in (-∞, T) ×ℝ^d_+, where ℝ^d_+ = {x ∈ℝ^d : x_d>0} and T∈(-∞, ∞] is given, and the diffusion matrices are the product of x_d and bounded uniformly elliptic matrices, which are degenerate at {x_d=0}. As such, our class of equations resembles well the corresponding class of degenerate viscous Hamilton-Jacobi equations. We obtain wellposedness results and regularity type estimates in some appropriate weighted Sobolev spaces for the solutions.

We prove the mixed-norm Sobolev estimates for solutions to both divergence and non-divergence form time-dependent Stokes systems with unbounded measurable coefficients having small mean oscillations with respect to the spatial variable in small cylinders. As a special case, our results imply Caccioppoli type inequalities for the Stokes systems with variable coefficients. A new ϵ-regularity criterion for Leray-Hopf weak solutions of Navier-Stokes equations is also obtained as a consequence of our regularity results, which in turn implies some borderline cases of the well-known Serrin's regularity criterion.

We revisit the well-known work of K. Masuda in 1984 on the weak–strong uniqueness of L∞L3 Leray–Hopf weak solutions of Navier–Stokes equation. We modify the argument, and extend the uniqueness result to the scaling critical anisotropic Lebesgue space with mixed-norms. As a consequence, our results cover the class of initial data and solutions which may be singular or decay with different rates along different spatial variables. The result relies on the establishment of several refined properties of solutions of the Stokes and Navier–Stokes equations in mixed-norm Lebesgue spaces which seem to be of independent interest.