Searcharxiv⌕ Search

arXiv subjects

Hongjie Dong

Publications and source records attributed to Hongjie Dong.

At least 109 records · Page 6Linked to original sources
arXiv2020

Time analyticity for inhomogeneous parabolic equations and the Navier-Stokes equations in the half space

We prove the time analyticity for weak solutions of inhomogeneous parabolic equations with measurable coefficients in the half space with either the Dirichlet boundary condition or the conormal boundary condition under the assumption that the solution and the source term have the exponential growth of order $2$ with respect to the space variables. We also obtain the time analyticity for bounded mild solutions of the incompressible Navier-Stokes equations in the half space with the Dirichlet boundary condition. Our work is an extension of the recent work by Zhang [Proc. Amer. Math. Soc. 148 (2020)] and Dong-Zhang [J. Funct. Anal. (2020)], where the authors proved the time analyticity of solutions to the homogeneous heat equation and the Navier-Stokes equations in the whole space.

math.AP↗
arXiv2020

Existence and uniqueness of bounded stable solutions to the Peierls-Nabarro model for curved dislocations

We study the well-posedness of the vector-field Peierls-Nabarro model for curved dislocations with a double well potential and a bi-states limit at far field. Using the Dirichlet to Neumann map, the 3D Peierls-Nabarro model is reduced to a nonlocal scalar Ginzburg-Landau equation. We derive an integral formulation of the nonlocal operator, whose kernel is anisotropic and positive when Poisson's ratio $ν\in(-\frac12, \frac13)$. We then prove that any bounded stable solutions to this nonlocal scalar Ginzburg-Landau equation has a 1D profile, which corresponds to the PDE version of flatness result for minimal surfaces with anisotropic nonlocal perimeter. Based on this, we finally obtain that steady states to the nonlocal scalar equation, as well as the original Peierls-Nabarro model, can be characterized as a one-parameter family of straight dislocation solutions to a rescaled 1D Ginzburg-Landau equation with the half Laplacian.

math.AP↗
arXiv2020

The Dirichlet-conormal problem with homogeneous and inhomogeneous boundary conditions

We consider the mixed Dirichlet-conormal problem on irregular domains in $\mathbb{R}^d$. Two types of regularity results will be discussed: the $W^{1,p}$ regularity and a non-tangential maximal function estimate. The domain is assumed to be Reifenberg-flat, and the interfacial boundary is either Reifenberg-flat of co-dimension $2$ or is locally sufficiently close to a Lipschitz function of $m$ variables, where $m=1,\ldots,d-2$. For the non-tangential maximal function estimate, we also require the domain to be Lipschitz.

math.AP↗
arXiv2020

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 [6] for unmixed $L_p$-estimates without weights.

math.AP↗
arXiv2020

Time analyticity for the heat equation and Navier-Stokes equations

We prove the analyticity in time for solutions of two parabolic equations in the whole space, without any decaying or vanishing conditions. One of them involves solutions to the heat equation of exponential growth of order $2$ on $\M$. Here $\M$ is $\R^d$ or a complete noncompact manifold with Ricci curvature bounded from below by a constant. An implication is a sharp solvability condition for the Cauchy problem of the backward heat equation, which is a well known ill-posed problem. Another implication is a sharp criteria for time analyticity of solutions down to the initial time. The other pertains bounded mild solutions of the incompressible Navier-Stokes equations in the whole space. There are many long established analyticity results for the Navier-Stokes equations. See for example \cite{Ka:1} and \cite{FT:1} for spatial and time analyticity in certain integral sense, \cite{CN:1} for pointwise space-time analyticity of 3 dimensional solutions to the Cauchy problem, and also the pointwise time analyticity results of \cite{Ma:1} and \cite{Gi:1} under zero boundary condition. Our result seems to be the first general pointwise time analyticity result for the Cauchy problem for all dimensions, whose proof involves only real variable method. The proof involves a method of algebraically manipulating the integral kernel, which appears applicable to other evolution equations.

math.AP↗
arXiv2019

Nonlinear instability for the surface quasi-geostrophic equation in the supercritical regime

We consider the forced surface quasi-geostrophic equation with supercritical dissipation. We show that linear instability for steady state solutions leads to their nonlinear instability. When the dissipation is given by a fractional Laplacian, the nonlinear instability is expressed in terms of the scaling invariant norm, while we establish stronger instability claims in the setting of logarithmically supercritical dissipation. A key tool in treating the logarithmically supercritical setting is a global well-posedness result for the forced equation, which we prove by adapting and extending recent work related to nonlinear maximum principles. We believe that our proof of global well-posedness is of independent interest, to our knowledge giving the first large-data supercritical result with sharp regularity assumptions on the forcing term.

math.AP↗
arXiv2019

Second order regularity for elliptic and parabolic equations involving $p$-Laplacian via a fundamental inequality

Denote by $Δ$ the Laplacian and by $Δ_\infty $ the $\infty$-Laplacian. A fundamental inequality is proved for the algebraic structure of $ΔvΔ_\infty v$: for every $v\in C^\infty$, $$\ | { |D^2vDv|^2} - {Δv Δ_\infty v } -\frac12[|D^2v|^2-(Δv)^2]|Dv|^2\ | \le \frac{n-2}2 [|D^2v|^2{|Dv|^2}- |D^2vDv|^2 ]. $$ Based on this, we prove the following results: 1. For any $p$-harmonic functions $u$, $p\in(1,2)\cup(2,\infty)$, we have $$|Du|^{\frac{p-γ}2}Du\in W^{1,2}_{\rm loc},$$ with $γ<\min\{p+\frac{n-1}{n},3+\frac{p-1}{n-1}\}$. As a by-product, when $p\in(1,2)\cup(2,3+\frac2{n-2})$, we reprove the known $W^{2,q}_{\rm loc}$-regularity of $p$-harmonic functions for some $q>2$. 2. When $n\ge 2$ and $p\in(1,2)\cup(2,3+\frac2{n-2})$, the viscosity solutions to parabolic normalized $p $-Laplace equation have the $W_{\rm loc}^{2,q}$-regularity in the spatial variable and the $W_{\rm loc}^{1,q}$-regularity in the time variable for some $q>2$. Especially, when $n=2$ an open question in [17] is completely answered. 3. When $n\ge 1 $ and $p\in(1,2)\cup(2,3)$, the weak/viscosity solutions to parabolic $p $-Laplace equation have the $W_{\rm loc}^{2,2}$-regularity in the spatial variable and the $W_{\rm loc}^{1,2}$-regularity in the time variable. The range of $p$ (including $p=2$ from the classical result) here is sharp for the $W_{\rm loc}^{2,2}$-regularity.

math.AP↗
arXiv2019

Hessian estimates for non-divergence form elliptic equations arising from composite materials

In this paper, we prove that any $W^{2,1}$ strong solution to second-order non-divergence form elliptic equations is locally $W^{2,\infty}$ and piecewise $C^{2}$ when the leading coefficients and data are of piecewise Dini mean oscillation and the lower-order terms are bounded. Somewhat surprisingly here the interfacial boundaries are only required to be $C^{1,\text{Dini}}$. We also derive global weak-type $(1,1)$ estimates with respect to $A_{1}$ Muckenhoupt weights. The corresponding results for the adjoint operator are established. Our estimates are independent of the distance between these surfaces of discontinuity of the coefficients.

math.AP↗
arXiv2019

$C^2$ estimate for oblique derivative problem with mean Dini coefficients

We consider second-order elliptic equations in non-divergence form with oblique derivative boundary conditions. We show that any strong solutions to such problems are twice continuously differentiable up to the boundary provided that the mean oscillations of coefficients satisfy the Dini condition and the boundary is locally represented by a $C^1$ function whose first derivatives are Dini continuous. This improves a recent result in [6]. An extension to fully nonlinear elliptic equations is also presented.

math.AP↗
arXiv2019

Optimal regularity for a Dirichlet-conormal problem in Reifenberg flat domain

We study the divergence form second-order elliptic equations with mixed Dirichlet-conormal boundary conditions. The unique $W^{1,p}$ solvability is obtained with $p$ being in the optimal range $(4/3,4)$. The leading coefficients are assumed to have small mean oscillations and the boundary of domain is Reifenberg flat. We also assume that the two boundary conditions are separated by some Reifenberg flat set of co-dimension $2$ on the boundary.

math.AP↗
arXiv2019

Gradient estimates for divergence form elliptic systems arising from composite material

In this paper, we show that $W^{1,p}$ $(1\leq p<\infty)$ weak solutions to divergence form elliptic systems are Lipschitz and piecewise $C^{1}$ provided that the leading coefficients and data are of piecewise Dini mean oscillation, the lower order coefficients are bounded, and interfacial boundaries are $C^{1,\text{Dini}}$. This extends a result of Li and Nirenberg (\textit{Comm. Pure Appl. Math.} \textbf{56} (2003), 892-925). Moreover, under a stronger assumption on the piecewise $L^{1}$-mean oscillation of the leading coefficients, we derive a global weak type-(1,1) estimate with respect to $A_{1}$ Muckenhoupt weights for the elliptic systems without lower order terms.

math.AP↗
arXiv2019

$L_q$-estimates for stationary Stokes system with coefficients measurable in one direction

We study the stationary Stokes system with variable coefficients in the whole space, a half space, and on bounded Lipschitz domains. In the whole and half spaces, we obtain a priori $\dot W^1_q$-estimates for any $q\in [2,\infty)$ when the coefficients are merely measurable functions in one fixed direction. For the system on bounded Lipschitz domains with a small Lipschitz constant, we obtain a $W^1_q$-estimate and prove the solvability for any $q\in (1,\infty)$ when the coefficients are merely measurable functions in one direction and have locally small mean oscillations in the orthogonal directions in each small ball, where the direction is allowed to depend on the ball.

math.AP↗
arXiv2019

Green functions for pressure of Stokes systems

We study Green functions for the pressure of stationary Stokes systems in a (possibly unbounded) domain $Ω\subset \mathbb{R}^d$, where $d\ge 2$. We construct the Green function when coefficients are merely measurable in one direction and have Dini mean oscillation in the other directions, and $Ω$ is such that the divergence equation is solvable there. We also establish global pointwise bounds for the Green function and its derivatives when coefficients have Dini mean oscillation and $Ω$ has a $C^{1,\rm{Dini}}$ boundary. Green functions for the flow velocity of Stokes systems are also considered.

math.AP↗
arXiv2019

Dini and Schauder estimates for nonlocal fully nonlinear parabolic equations with drifts

We obtain Dini and Schauder type estimates for concave fully nonlinear nonlocal parabolic equations of order $σ\in (0,2)$ with rough and non-symmetric kernels, and drift terms. We also study such linear equations with only measurable coefficients in the time variable, and obtain Dini type estimates in the spacial variable. This is a continuation of the work [10, 11] by the first and last authors.

math.AP↗
arXiv2018

Boundary $\varepsilon$-regularity criteria for the 3D Navier-Stokes equations

We establish several boundary $\varepsilon$-regularity criteria for suitable weak solutions for the 3D incompressible Navier-Stokes equations in a half cylinder with the 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 [18], Choi-Vasseur [2], and Phuc-Guevara [6].

math.AP↗
arXiv2018

Higher-order elliptic and parabolic equations with VMO assumptions and general boundary conditions

We prove mixed $L_{p}(L_{q})$-estimates, with $p,q\in(1,\infty)$, for higher-order elliptic and parabolic equations on the half space $\R^{d+1}_{+}$ with general boundary conditions which satisfy the Lopatinskii--Shapiro condition. We assume that the elliptic operators $A$ have leading coefficients which are in the class of vanishing mean oscillations in both the time and the space variable. In the proof, we produce mean oscillation estimates for equations on the half space with general boundary conditions.

math.AP↗