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We are concerned with interior and global gradient estimates for solutions to a class of singular quasilinear elliptic equations with measure data, whose prototype is given by the p-Laplace equation -- Δp u = μ with p ∈ (1,2). The cases when p ∈ (2 - 1/n, 2) and p ∈ (3n-2/2n-1, 2 - 1/n] were studied in Duzaar and Mingione [J. Funct. Anal. 259 (2010), 379-418] and Nguyen and Phuc [J. Funct. Anal. 278 (2020), art. 108391] respectively. In this paper, we improve the results of Nguyen and Phuc and address the open case when p ∈ (1, 3n-2/2n-1]. Interior and global modulus of continuity estimates of the gradients of solutions are also established. [ABSTRACT FROM AUTHOR]

We study nonlocal elliptic and parabolic equations on C1,τ open sets in weighted Sobolev spaces, where τ ∈ (0, 1). The operators we consider are infinitesimal generators of symmetric stable Lévy processes, whose Lévy measures are allowed to be very singular. Additionally, for parabolic equations, the measures are assumed to be merely measurable in the time variable. [ABSTRACT FROM AUTHOR]

We are concerned with interior and global gradient estimates for solutions to a class of singular quasilinear elliptic equations with measure data, whose prototype is given by the p-Laplace equation --Δpu = μ with p ∈ (1, 2). The cases when p ∈ (2 -- 1/n, 2) and p ∈ (3n--2/2n--1, 2 -- 1/n] were studied in Duzaar and Mingione [J. Funct. Anal. 259 (2010), 379-418] and Nguyen and Phuc [J. Funct. Anal. 278 (2020), art. 108391] respectively. In this paper, we improve the results of Nguyen and Phuc and address the open case when p ∈ (1, 3n--2/2n--1)]. Interior and global modulus of continuity estimates of the gradients of solutions are also established. [ABSTRACT FROM AUTHOR]

We present a priori estimates and unique solvability results in the mixed-norm Lebesgue spaces for the kinetic Kolmogorov-Fokker-Planck (KFP) equation in divergence form. The leading coefficients are bounded uniformly nondegenerate with respect to the velocity variable v and satisfy a vanishing mean oscillation (VMO) type condition. We consider the L2 case separately and treat more general equations, which include the relativistic KFP equation. This paper is a continuation of our previous work on Lp estimates for KFP equations in nondivergence form. [ABSTRACT FROM AUTHOR]

In this paper, we investigate pointwise time analyticity of solutions to nonlocal parabolic equations in the settings of ℝd and a complete Riemannian manifold M. On the one hand, in ℝd, we prove that any solution u=u(t,x) to ..., where ... is a nonlocal operator of order α, is time analytic in (0,1] if u satisfies the growth condition ... for any ... and ... . We also obtain pointwise estimates for ..., where ... is the fractional heat kernel. Furthermore, under the same growth condition, we show that the mild solution is the unique solution. On the other hand, in a manifold M, we also prove the time analyticity of the mild solution under the same growth condition and the time analyticity of the fractional heat kernel when M satisfies the Poincaré inequality and the volume doubling condition. Moreover, we also study the time and space derivatives of the fractional heat kernel in Rd using the method of Fourier transform and contour integrals. We find that when ..., the fractional heat kernel is time analytic at t=0 when x≠0, which differs from the standard heat kernel. As corollaries, we obtain a sharp solvability condition for the backward nonlocal parabolic equations and time analyticity of some nonlinear nonlocal parabolic equations with power nonlinearity of order p . These results are related to those in [9] and [22], which deal with local equations. [ABSTRACT FROM AUTHOR]

Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is a serious threat to global development. Rapid and accurate diagnosis is critical for containing the pandemic and treating patients in time. As the gold standard for SARS-CoV-2 diagnosis, the qualitative reverse transcription-PCR (RT-qPCR) test has long been criticized for its long detection time. In this study, we optimized the primers and probes targeting SARS-CoV-2 ORF1ab and N gene designed by the Chinese Center for Disease Control and Preventions (CDC) to increase their Tm values to meet the optimal elongation temperature of Taq DNA polymerase, thus greatly shortened the elongation time. The higher elongation temperature in turn narrowed the temperature range of the reaction and saved more time. In addition, by shortening the distance between the fluorophore at the 5' end and the quencher in the middle we got a probe with higher signal-to-noise ratio. Finally, by using all these measures and optimized RT-qPCR program we successfully reduced the time (nucleic acid extraction step is not included) for nucleic acid test from 74 min to 26 min. [ABSTRACT FROM AUTHOR]

We study stationary Stokes systems in divergence form with piecewise Dini mean oscillation (DMO) coefficients and data in a bounded domain containing a finite number of subdomains with C1,Dini boundaries. We prove that if (u, p) is a weak solution of the system, then (Du, p) is bounded and piecewise continuous. The corresponding results for stationary Navier—Stokes systems are also established, from which the Lipschitz regularity of the stationary H¹-weak solution in dimensions d = 2, 3, 4 is obtained. Our results can be applied to stationary Stokes systems and Navier—Stokes systems with the second-order term div(τSu), where Su = ½ (Du + (Du)T) is the strain tensor and τ is a positive piecewise DMO scalar function. [ABSTRACT FROM AUTHOR]

We obtain the regularity of solutions in Sobolev spaces for the mixed Dirichletconormal problem for parabolic operators in cylindrical domains with time-dependent separations, which is the first of its kind. Assuming the boundary of the domain to be Reifenberg-flat and the separation to be locally sufficiently close to a Lipschitz function of m variables, where m = 0, . . ., d-2, with respect to the Hausdorff distance, we prove the unique solvability for p ∈ (2(m+2/(m+3), 2(m+2)/(m+1))). In the case when m = 0, the range p ∈ (4/3, 4) is optimal in view of the known results for Laplace equations. [ABSTRACT FROM AUTHOR]

We construct the Green function for second-order elliptic equations in nondivergence form when the mean oscillations of the coefficients satisfy the Dini condition. We show that Green's function is a bounded mean oscillation in the domain and establish logarithmic pointwise bounds. We also obtain pointwise bounds for first and second derivatives of Green's function. [ABSTRACT FROM AUTHOR]

We establish several boundary ε-regularity criteria for suitable weak solutions to the three-dimensional incompressible Navier--Stokes equations in a half cylinder with the zero Dirichlet boundary condition on the flat boundary. Our proofs are based on delicate iteration arguments and interpolation techniques. These results extend and provide alternative proofs for the earlier interior results by Vasseur [NODEA Nonlinear Differential Equations Appl., 14 (2007), pp. 753--785], Choi-Vasseur [Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), pp. 899--945], and Phuc-Guevara [Calc. Var. Partial Differential Equations, 56 (2017), pp. 1--16]. [ABSTRACT FROM AUTHOR]

In this paper, we show that W1,p (1 ≤ p < ∞ ) weak solutions to divergence form elliptic systems are Lipschitz and piecewise C¹ provided that the leading coefficients and data are of piecewise Dini mean oscillation, the lower-order coefficients are bounded, and interfacial boundaries are C1,Dini. This extends a result of Li and Nirenberg [Comm. Pure Appl. Math., 56 (2003), pp. 892-925]. Moreover, under a stronger assumption on the piecewise L¹-mean oscillation of the leading coefficients, we derive a global weak-type (1,1) estimate with respect to A1 Muckenhoupt weights for the elliptic systems without lower-order terms. [ABSTRACT FROM AUTHOR]

We will present in this paper a linear theory on a flow under the effect of a selfinduced electric field past an obstacle governed by the Vlasov–Poisson–Boltzmann equation. The bulk velocity at infinity acts like a driving force on the flow so that some nontrivial stationary solution profile exists. Similar to Ukai and Asano’s work on the exterior problem for the Boltzmann equation, the spectrum of the linear operator is used through an approach in the spirit of the limiting absorption in scattering theory. However, to take care of the electric potential that satisfies the coupled Poisson equation, a new decomposition of the solution operator is needed to show the compactness. [ABSTRACT FROM AUTHOR]

We construct fundamental solutions of second-order parabolic systems of divergence form with bounded and measurable leading coefficients and divergence free first-order coefficients in the class of BMOx−1 , under the assumption that weak solutions of the system satisfy a certain local boundedness estimate. We also establish Gaussian upper bounds for such fundamental solutions under the same conditions. [ABSTRACT FROM AUTHOR]

We prove generalized Fefferman-Stein type theorems on sharp functions with Ap weights in spaces of homogeneous type with either finite or infinite underlying measure. We then apply these results to establish mixednorm weighted Lp-estimates for elliptic and parabolic equations/systems with (partially) BMO coefficients in regular or irregular domains. [ABSTRACT FROM AUTHOR]

The paper is a comprehensive study of Lp and Schauder estimates for higher-order divergence type parabolic systems with discontinuous coefficients on a half space and cylindrical domains with the conormal derivative boundary conditions. For the Lp estimates, we assume that the leading coefficients are only bounded and measurable in the t variable and have vanishing mean oscillations (VMOx) with respect to x. We also prove the Schauder estimates in two situations: the coefficients are Holder continuous only in the x variable; the coefficients are Hölder continuous in the t variable as well on the lateral boundary. [ABSTRACT FROM AUTHOR]

We studyan elliptic equation with measurable coefficients arising from photo-acoustic imaging in inhomogeneous media. We establish Hölder continuity of weak solutions and obtain pointwise bounds for Green's functions subject to Dirichlet or Neumann condition. [ABSTRACT FROM AUTHOR]

We study boundary gradient estimates for second-order divergence form parabolic and elliptic systems in C1,σ -domains. The coefficients and data are assumed to be Hölder in the time variable and all but one spatial variables. This type of systems arises from the problems of linearly elastic laminates and composite materials. [ABSTRACT FROM AUTHOR]

We prove the Wp1,2-estimate and solvability for the Dirichlet problem of second-order parabolic equations in simple convex polytopes with time irregular coefficients, when p ∈ [1, 2). We also consider the corresponding Neumann problem in a half space when p ∈ (2, ∞). Similar results are obtained for equations in a half space with coefficients which are measurable in a tangential direction and have small mean oscillations in the other directions. Equations with discontinuous coefficients in nonsmooth domains emerge from problems in mechanics, engineering, and biology, to name a few fields. [ABSTRACT FROM AUTHOR]

We consider a one-dimensional nonlocal nonlinear equation of the form ?tu = (?-au)?xu-??ßu, where? = (-?xx) 1/2 is the fractional Laplacian and ? = 0 is the viscosity coefficient. We primarily consider the regime 0 < a < 1 and 0 = ß = 2 for which the model has nonlocal drift, fractional dissipation, and captures essential features of the 2D a-patch models. In the critical and subcritical range 1 - a = ß = 2, we prove global wellposedness for arbitrarily large initial data in Sobolev spaces. In the full supercritical range 0 = ß < 1 - a, we prove formation of singularities in finite time for a class of smooth initial data. Our proof is based on a novel nonlocal weighted inequality which can be of independent interest. [ABSTRACT FROM AUTHOR]

We study similarity solutions to the multidimensional aggregation equation ut + div(uv) = 0, v = - ▽K * u with general power-law kernels K such that ▽K(x) = x|x α - 2, α ∈ (2 - d, 2). We analyze the equation in different regimes of the parameter α. In the case when α ∈ [4 - d, 2), we give a characterization of all the "first-kind" radially symmetric similarity solutions. We prove that any such solution is a linear combination of a delta ring and a delta mass at the origin. On the other hand, when α ∈ (2 - d, 4 - d), we show that there exist multi delta-ring similarity solutions in ℝd. In particular, our results imply that multi delta-ring similarity solutions exist in three dimensions if α is just a little bit below 1. [ABSTRACT FROM AUTHOR]

We establish the solvability of second order divergence-type parabolic systems in Sobolev spaces. The leading coefficients are assumed to be merely measurable in one spatial direction on each small parabolic cylinder with the spatial direction allowed to depend on the cylinder. In the other orthogonal directions and the time variable, the coefficients have locally small mean oscillations. We also obtain the corresponding Wp1-solvability of second order elliptic systems in divergence form. This type of system arises from the problems of linearly elastic laminates and composite materials. Our results are new even for scalar equations, and the proofs differ from and simplify the methods used previously in [H. Dong and D. Kim, Arch. Ration. Mech. Anal., 196 (2010), pp. 25-70]. As an application, we improve a result by Chipot, Kinderlehrer, and Vergara-Caffarelli [Arch. Ration. Mech. Anal., 96 (1986), pp. 81-96] on gradient estimates for elasticity system Dα(Aαβ(x1)Dβu) = f, which typically arises in homogenization of layered materials. We relax the condition on f from Hk,k ≥ d/2, to Lp with p > d. [ABSTRACT FROM AUTHOR]

The solvability in Sobolev spaces is proved for divergence form second order elliptic equations in the whole space, a half space, and a bounded Lipschitz domain. For equations in the whole space or a half space, the leading coefficients a ij are assumed to be only measurable in one direction and have locally small BMO semi-norms in the other directions. For equations in a bounded domain, additionally we assume that a ij have small BMO semi-norms in a neighborhood of the boundary of the domain. We give a unified approach of both the Dirichlet boundary problem and the conormal derivative problem. We also investigate elliptic equations in Sobolev spaces with mixed norms under the same assumptions on the coefficients. [ABSTRACT FROM AUTHOR]

We study regularity criteria for the d-dimensional incompressible Navier-Stokes equations. We prove in this paper that if $${u \in L_\infty^tL_{d}^x((0,T)\times \mathbb{R}^d)}$$ is a Leray-Hopf weak solution, then u is smooth and unique in $${(0, T)\times \mathbb{R}^d}$$ . This generalizes a result by Escauriaza, Seregin and Šverák [5]. Additionally, we show that if T = ∞ then u goes to zero as t goes to infinity. [ABSTRACT FROM AUTHOR]

We study Green's matrices for divergence form, second order strongly elliptic systems with bounded measurable coefficients in two dimensional domains. We establish existence, uniqueness, and pointwise estimates of Green's matrices. [ABSTRACT FROM AUTHOR]

We study boundary blow-up solutions of semilinear elliptic equations Lu = u+p with p > 1, or Lu = eau with a > 0, where L is a second order elliptic operator with measurable coefficients. Several uniqueness theorems and an existence theorem are obtained. [ABSTRACT FROM AUTHOR]

We consider the Dirichlet problem for two types of degenerate elliptic Hessian equations. New results about solvability of the equations in the C 1,1 space are provided. [ABSTRACT FROM AUTHOR]

We consider degenerate parabolic and elliptic equations of second order with C¹ and C² coefficients. Error bounds for certain types of finite-difference schemes are obtained. [ABSTRACT FROM AUTHOR]