Green's functions for parabolic systems of second order in time-varying domains
Hongjie Dong Seick Kim
Communications on Pure & Applied Analysis 2014, 13(4): 1407-1433 doi: 10.3934/cpaa.2014.13.1407
We construct Green's functions for divergence form, second order parabolic systems in non-smooth time-varying domains whose boundaries are locally represented as graph of functions that are Lipschitz continuous in the spatial variables and $1/2$-Hölder continuous in the time variable, under the assumption that weak solutions of the system satisfy an interior Hölder continuity estimate. We also derive global pointwise estimates for Green's function in such time-varying domains under the assumption that weak solutions of the system vanishing on a portion of the boundary satisfy a certain local boundedness estimate and a local Hölder continuity estimate. In particular, our results apply to complex perturbations of a single real equation.
keywords: Green's function time-varying domain. parabolic system Green's matrix
Schauder estimates for a class of non-local elliptic equations
Hongjie Dong Doyoon Kim
Discrete & Continuous Dynamical Systems - A 2013, 33(6): 2319-2347 doi: 10.3934/dcds.2013.33.2319
We prove Schauder estimates for a class of non-local elliptic operators with kernel $K(y)=a(y)/|y|^{d+\sigma}$ and either Dini or Hölder continuous data. Here $0 < \sigma < 2$ is a constant and $a$ is a bounded measurable function, which is not necessarily to be homogeneous, regular, or symmetric. As an application, we prove that the operators give isomorphisms between the Lipschitz--Zygmund spaces $\Lambda^{\alpha+\sigma}$ and $\Lambda^\alpha$ for any $\alpha>0$. Several local estimates and an extension to operators with kernels $K(x,y)$ are also discussed.
keywords: Lévy processes. Non-local elliptic equations Schauder estimates
Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space
Hongjie Dong Dapeng Du
Discrete & Continuous Dynamical Systems - A 2008, 21(4): 1095-1101 doi: 10.3934/dcds.2008.21.1095
By adapting a method in [11] with a suitable modification, we show that the critical dissipative quasi-geostrophic equations in $R^2$ has global well-posedness with arbitrary $H^1$ initial data. A decay in time estimate for homogeneous Sobolev norms of solutions is also discussed.
keywords: Higher regularity Quasi-geostrophic equations. Global well-posedness
On a generalized maximum principle for a transport-diffusion model with $\log$-modulated fractional dissipation
Hongjie Dong Dong Li
Discrete & Continuous Dynamical Systems - A 2014, 34(9): 3437-3454 doi: 10.3934/dcds.2014.34.3437
We consider a transport-diffusion equation of the form $\partial_t \theta + v \cdot \nabla \theta + \nu \mathcal{A} \theta = 0$, where $v$ is a given time-dependent vector field on $\mathbb R^d$. The operator $\mathcal{A}$ represents log-modulated fractional dissipation: $\mathcal{A}=\frac {|\nabla|^{\gamma}}{\log^{\beta}(\lambda+|\nabla|)}$ and the parameters $\nu\ge 0$, $\beta\ge 0$, $0\le \gamma \le 2$, $\lambda>1$. We introduce a novel nonlocal decomposition of the operator $\mathcal{A}$ in terms of a weighted integral of the usual fractional operators $|\nabla|^{s}$, $0\le s \le \gamma$ plus a smooth remainder term which corresponds to an $L^1$ kernel. For a general vector field $v$ (possibly non-divergence-free) we prove a generalized $L^\infty$ maximum principle of the form $ \| \theta(t)\|_\infty \le e^{Ct} \| \theta_0 \|_{\infty}$ where the constant $C=C(\nu,\beta,\gamma)>0$. In the case $\text{div}(v)=0$ the same inequality holds for $\|\theta(t)\|_p$ with $1\le p \le \infty$. Under the additional assumption that $\theta_0\in L^2$, we show that $\|\theta(t)\|_p$ is uniformly bounded for $2\le p\le \infty$. At the cost of a possible exponential factor, this extends a recent result of Hmidi [7] to the full regime $d\ge 1$, $0\le \gamma \le 2$ and removes the incompressibility assumption in the $L^\infty$ case.
keywords: nonlocal operators fractional dissipation Generalized maximum principle transport-diffusion equations nonlocal decomposition.
Dissipative quasi-geostrophic equations in critical Sobolev spaces: Smoothing effect and global well-posedness
Hongjie Dong
Discrete & Continuous Dynamical Systems - A 2010, 26(4): 1197-1211 doi: 10.3934/dcds.2010.26.1197
We study the critical and super-critical dissipative quasi-geostrophic equations in $\R^2$ or $\T^2$. An optimal local smoothing effect of solutions with arbitrary initial data in $H^{2-\gamma}$ is proved. As a main application, we establish the global well-posedness for the critical 2D quasi-geostrophic equations with periodic $H^1$ data. Some decay in time estimates are also provided.
keywords: quasi-geostrophic equations. global well-posedness critical and super-critical higher regularity
Neumann problem for non-divergence elliptic and parabolic equations with BMO$_x$ coefficients in weighted Sobolev spaces
Doyoon Kim Hongjie Dong Hong Zhang
Discrete & Continuous Dynamical Systems - A 2016, 36(9): 4895-4914 doi: 10.3934/dcds.2016011
We prove the unique solvability in weighted Sobolev spaces of non-divergence form elliptic and parabolic equations on a half space with the homogeneous Neumann boundary condition. All the leading coefficients are assumed to be only measurable in the time variable and have small mean oscillations in the spatial variables. Our results can be applied to Neumann boundary value problems for stochastic partial differential equations with BMO$_x$ coefficients.
keywords: $L_p$ estimates weighted Sobolev spaces parabolic equations.
Conormal derivative problems for stationary Stokes system in Sobolev spaces
Jongkeun Choi Hongjie Dong Doyoon Kim
Discrete & Continuous Dynamical Systems - A 2018, 38(5): 2349-2374 doi: 10.3934/dcds.2018097

We prove the solvability in Sobolev spaces of the conormal derivative problem for the stationary Stokes system with irregular coefficients on bounded Reifenberg flat domains. The coefficients are assumed to be merely measurable in one direction, which may differ depending on the local coordinate systems, and have small mean oscillations in the other directions. In the course of the proof, we use a local version of the Poincaré inequality on Reifenberg flat domains, the proof of which is of independent interest.

keywords: Stokes system Reifenberg flat domains measurable coefficients conormal derivative boundary condition

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