Quoc-Hung Nguyen, Yannick Sire, Le Xuan Truong
We provide several regularity results for non-homogeneous drift-diffusion equations with applications to general dissipative SQG. Our results unify in a rather simple way several previously known results. We build the estimates on an algebraic identity (for the refined energy argument) which relates any integral operator with pure powers of the laplacian.
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We prove that every quasiconformal mapping from the harmonic $β$-Bloch space between the unit ball and a spatial domain with $C^1$ boundary is globally $α$-Hölder continuous for $α<1-β$, with the Hölder coefficient that does not depend neither on the mapping nor on $β$. An analogous result also holds for Lipschitz continuous, quasiconformal harmonic mappings for $α<1$. This extends some results from the complex plane obtained by Warschawski in \cite{Warschawski} for conformal mappings and Kalaj in \cite{Kalaj6} for quasiconformal harmonic mappings.
We consider non degenerate Brownian SDEs with H{ö}lder continuous in space diffusion coefficient and unbounded drift with linear growth. We derive two sided bounds for the associated density and pointwise controls of its derivatives up to order two under some additional spatial H{ö}lder continuity assumptions on the drift. Importantly, the estimates reflect the transport of the initial condition by the unbounded drift through an auxiliary, possibly regularized, flow.
In this paper we shall prove the existence, uniqueness and global H$\ddot{o}$lder continuity for the Dirichlet problem of a class of Monge-Ampère type equations which may be degenerate and singular on the boundary of convex domains. We will establish a relation of the H$\ddot{o}$lder exponent for the solutions with the convexity for the domains.
We consider parabolic PDEs associated with fractional type operators drifted by non-linear singular first order terms. When the drift enjoys some boundedness properties in appropriate Lebesgue and Besov spaces, we establish by exploiting a priori Besov-type estimates, the H{ö}lder continuity of the solutions. In particular, we handle the almost critical case in whole generality.
We establish strong uniqueness for a class of degenerate SDEs of weak H{ö}rmander type under suitable H{ö}lder regularity conditions for the associated drift term. Our approach relies on the Zvonkin transform which requires to exhibit good smoothing properties of the underlying parabolic PDE with rough, here H{ö}lder, drift coefficients and source term. Such regularizing effects are established through a perturbation technique (forward parametrix approach) which also heavily relies on appropriate duality properties on Besov spaces. For the method employed, we exhibit some sharp thresholds on t…
We investigate some smoothness properties for a transport-diffusion equation involving a class of non-degerate L{é}vy type operators with singular drift. Our main argument is based on a duality method using the molecular decomposition of Hardy spaces through which we derive some H{ö}lder continuity for the associated parabolic PDE. This property will be fulfilled as far as the singular drift belongs to a suitable Morrey-Campanato space for which the regularizing properties of the L{é}vy operator suffice to obtain global H{ö}lder continuity.
Improved Regularity for Nonlocal Hamilton--Jacobi Equations with Superlinear Hamiltonians
研究一类由x依赖积分微分算子和强制超线性哈密顿量驱动的非局部Hamilton-Jacobi方程的粘性解正则性，在一般结构假设下建立Hölder正则性，并在弱椭圆性条件下证明全局Lipschitz连续性，通过反例说明假设的尖锐性。
We show that there exist solutions of drift-diffusion equations in two dimensions with divergence-free super-critical drifts, that become discontinuous in finite time. We consider classical as well as fractional diffusion. However, in the case of classical diffusion and time-independent drifts we prove that solutions satisfy a modulus of continuity depending only on the local $L^1$ norm of the drift, which is a super-critical quantity.
In this paper, we study local uniform continuity of nonnegative weak solutions to degenerate diffusion-drift equations in the form \[ u_{t} = Δu^{m} + \nabla\cdot \left( B (x,t) \, u\right), \quad \text{for } m \geq 1 \] assuming a vector field $B \in L^{p}_t L^{q}_{x}$. Regarding local Hölder continuity, we provide a sharp condition on $p$ and $q$, which is referred to as the subcritical region. In the critical region, the divergence-free condition is essential to providing uniform continuity which depends on the modulus continuity of $B$.
The aim of this note is to prove sharp regularity estimates for solutions of the continuity equation associated to vector fields of class $W^{1,p}$ with $p>1$. The regularity is of "logarithmic order" and is measured by means of suitable versions of Gargliardo's seminorms.
Within the global setting of singular drifts in Morrey-Campanato spaces presented in [6], we study now the H{ö}lder regularity properties of the solutions of a transport-diffusion equation with nonlinear singular drifts that satisfy a Besov stability property. We will see how this Besov information is relevant and how it allows to improve previous results. Moreover, in some particular cases we show that as the nonlinear drift becomes more regular, in the sense of Morrey-Campanato spaces, the additional Besov stability property will be less useful.
This paper is devoted to existence and uniqueness results for classes of nonlinear diffusion equations (or systems) which may be viewed as regular perturbations of Wasserstein gradient flows. First, in the case. where the drift is a gradient (in the physical space), we obtain existence by a semi-implicit Jordan-Kinderlehrer-Otto scheme. Then, in the nonpotential case, we derive existence from a regularization procedure and parabolic energy estimates. We also address the uniqueness issue by a displacement convexity argument.
We address the persistence of Hölder continuity for weak solutions of the linear drift-diffusion equation with nonlocal pressure \[ u_t + b \cdot \grad u - \lap u = \grad p,\qquad \grad\cdot u =0 \] on $[0,\infty) \times \R^{n}$, with $n \geq 2$. The drift velocity $b$ is assumed to be at the critical regularity level, with respect to the natural scaling of the equations. The proof draws on Campanato's characterization of Hölder spaces, and uses a maximum-principle-type argument by which we control the growth in time of certain local averages of $u$. We provide an estimate that does not depend…
Investigating for interior regularity of viscosity solutions to the fully nonlinear elliptic equation $$F(x,u,\triangledown u,\triangledown ^2 u)=0,$$ we establish the interior $C^{1+1}$ continuity under the assumptions that $F$ is uniformly elliptic, H$\ddot o$lder continuous and satisfies the natural structure conditions of fractional order, but without the concavity assumption of $F$. These assumptions are weaker and the result is stronger than that of Caffarelli and Wang[1], Chen[2].
We prove uniqueness for continuity equations in Hilbert spaces $H$. The corresponding drift $F$ is assumed to be in a first order Sobolev space with respect to some Gaussian measure. As in previous work on the subject, the proof is based on commutator estimates which are infinite dimensional analogues to the classical ones due to DiPerna-Lions. Our general approach is, however, quite different since, instead of considering renormalized solutions, we prove a dense range condition implying uniqueness. In addition, compared to known results by Ambrosio-Figalli and Fang-Luo, we use a different app…
A Classical Elliptic Regularity Approach to Almost Harmonic Maps and Related Systems
提出基于Campanato型离散迭代和Caccioppoli估计的统一抽象框架，证明在稳定于缩放且满足离散振荡衰减公理的容许对中，映射自动局部Hölder连续，并达到最优指数，避免了现有理论中的对偶、不等式和移动标架等技术。
We prove the well-posedness of some non-linear stochastic differential equations in the sense of McKean-Vlasov driven by non-degenerate symmetric $α$-stable Lévy processes with values in $R^d$ under some mild H{ö}lder regularity assumptions on the drift and diffusion coefficients with respect to both space and measure variables. The methodology developed here allows to consider unbounded drift terms even in the so-called super-critical case, i.e. when the stability index $α\in (0, 1)$. New strong well-posedness results are also derived from the previous analysis.
We study the propagation of regularity of solutions to a three dimensional system of linear parabolic PDE known as the kinematic dynamo equations. The divergence free drift velocity is assumed to be at the critical regularity level with respect to the natural scaling of the equations.
We consider an equation with drift and either critical or supercritical fractional diffusion. Under a regularity assumption for the vector field that is marginally stronger than what is required for Holder continuity of the solutions, we prove that the solution becomes immediately differentiable with Holder continuous derivatives. Therefore, the solutions to the equation are classical.
In this paper we provide regularity results for active scalars that are weak solutions of almost critical drift-diffusion equations in general surfaces. This includes models of anisotropic non-homogeneous media and the physically motivated case of the two-dimensional sphere. Our finest result deals with the critical surface quasigeostrophic equation on the round sphere.
This paper concerns the boundary behavior of solutions of certain fully nonlinear equations with a general drift term. We elaborate on the non-homogeneous generalized Harnack inequality proved by the second author in (Julin, ARMA -15), to prove a generalized Carleson estimate. We also prove boundary Hölder continuity and a boundary Harnack type inequality.
We consider nonlinear Kolmogorov-Fokker-Planck type equations of the form \begin{equation}\label{abeqn} (\partial_t+X\cdot\nabla_Y)u=\nabla_X\cdot(A(\nabla_X u,X,Y,t)). \end{equation} The function $A=A(ξ,X,Y,t):\R^m\times\R^m\times\R^m\times\R\to\R^m$ is assumed to be continuous with respect to $ξ$, and measurable with respect to $X,Y$ and $t$. $A=A(ξ,X,Y,t)$ is allowed to be nonlinear but with linear growth. We establish higher integrability and local boundedness of weak sub-solutions, weak Harnack and Harnack inequalities, and H{ö}lder continuity with quantitative estimates. In addition we e…
Stochastic flows for Hölder drifts and transport/continuity equations with noise
该研究通过Zvonkin型变换证明了带Hölder漂移SDE的微分同胚随机流存在性，建立相关抛物PDE的适定性与最优正则性，进而得到带输运噪声的随机输运与连续性方程的适定性，弥补确定性情形下解可能不唯一的问题。
We consider a non-autonomous Cauchy problem involving linear operators associated with time-dependent forms $a(t;.,.):V\times V\to {\mathbb{C}}$ where $V$ and $H$ are Hilbert spaces such that $V$ is continuously embedded in $H$. We prove $H$-maximal regularity under a new regularity condition on the form $a$ with respect to time; namely H{ö}lder continuity with values in an interpolation space. This result is best suited to treat Robin boundary conditions. The maximal regularity allows one to use fixed point arguments to some non linear parabolic problems with Robin boundary conditions.
For second-order elliptic or parabolic equations with subcritical or critical drifts, it is well-known that the Harnack inequality holds and their bounded weak solutions are Hölder continuous. We construct time-independent supercritical drifts in $L^{n-λ}(\mathbb{R}^n)$ with arbitrarily small $λ>0$ such that the Harnack inequality and the Hölder continuity fail in both the elliptic and the parabolic cases, thus confirming a conjecture by Seregin, Silvestre, Sverak and Zlatos. These results are sharp, and they also apply to a toy model of the axi-symmetric Navier-Stokes equations in space dimen…
We consider the stochastic continuity equation associated to an Itô diffusion with irregular drift and diffusion coefficients. We give regularity conditions under which weak solutions are renormalized in the sense of DiPerna/Lions, and prove well-posedness in $L^p$. As an application, we give a new proof of renormalizability (hence uniqueness) of weak solutions to the stochastic continuity equation when the diffusion matrix is constant and the drift only belongs to $L^q_tL^p$, where $\frac{2}{q} + \frac{n}{p} <1$, without resorting to the regularity of the stochastic flow or a duality method.
The continuity equation in the Heisenberg-periodic case: a representation formula and an application to Mean Field Games
本文给出了Heisenberg群上连续性方程弱解的表示公式，并据此证明一阶平均场博弈弱解的温和性。
We analyze the propagation of Lipschitz continuity of solutions to various linear and nonlinear drift-diffusion systems, with and without incompressibility constraints. Diffusion is assumed to be either fractional or classical. Such equations model the incompressible Navier-Stokes systems, generalized viscous Burgers-Hilbert equation and various active scalars. We derive conditions that guarantee the propagation of Lipschitz regularity by the incompressible NSE in the form of a non-local, one dimensional viscous Burgers-type inequality. We show the analogous inequality is always satisfied for …
On higher integrability for $p(x)$-Laplacian equations with drift
本文在变指数p(x)连续模的较弱条件下，通过广义Gehring引理和修正的Sobolev-Poincaré不等式，建立了带漂移项p(x)-Laplace方程弱解梯度的高可积性，推广了无漂移项情形下的已有结果。
For $n\geq 3$, we study the existence and asymptotic properties of the fundamental solution for elliptic operators in nondivergence form, ${\mathcal L}(x,\partial_x)=a_{ij}(x)\partial_i\partial_j+b_k(x)\partial_k$, where the $a_{ij}$ have modulus of continuity $ω(r)$ satisfying the square-Dini condition and the $b_k$ are allowed mild singularities of order $r^{-1}ω(r)$. A singular integral is introduced that controls the existence of the fundamental solution. We give examples that show the singular drift $b_k\partial_k$ may act as a perturbation that does not dramatically change the fundamenta…
We prove well-posedness for a class of second-order SPDEs with multiplicative Wiener noise and doubly nonlinear drift in divergence form involving maximal monotone graphs defined everywhere, on which neither growth nor continuity assumptions are imposed.
Uniqueness and regularity of weak solutions of a drift-diffusion system for perovskite solar cells
针对钙钛矿太阳能电池的非定常漂移扩散模型，通过建立载流子密度梯度的改进可积性，结合正则化连续性方程与标量拟线性椭圆方程的正则性结果，证明了该模型弱解的唯一性，补充了先前工作中弱解存在性与有界性的结论。
In this paper linear stochastic transport and continuity equations with drift in critical $L^{p}$ spaces are considered. In this situation noise prevents shocks for the transport equation and singularities in the density for the continuity equation, starting from smooth initial conditions. Specifically, we first prove a result of Sobolev regularity of solutions, which is false for the corresponding deterministic equation. The technique needed to reach the critical case is new and based on parabolic equations satisfied by moments of first derivatives of the solution, opposite to previous works …
This article is concerned with a porous medium equation whose pressure law is both nonlinear and nonlocal, namely $\partial_t u = { \nabla \cdot} \left(u \nabla(-Δ)^{\fracα{2}-1}u^{m-1} \right)$ where $u:\mathbb{R}_+\times \mathbb{R}^N \to \mathbb{R}_+$, for $0<α<2$ and $m\geq2$. We prove that the $L^1\cap L^\infty$ weak solutions constructed by Biler, Imbert and Karch (2015) are locally H{ö}lder-continuous in time and space. In this article, the classical parabolic De Giorgi techniques for the regularity of PDEs are tailored to fit this particular variant of the PME equation. In the spirit of…
We provide here global Schauder-type estimates for a chain of integro-partial differential equations (IPDE) driven by a degenerate stable Ornstein-Uhlenbeck operator possibly perturbed by a deterministic drift, when the coefficients lie in some suitable anisotropic H{ö}lder spaces. Our approach mainly relies on a perturbative method based on forward parametrix expansions and, due to the low regularizing properties on the degenerate variables and to some integrability constraints linked to the stability index, it also exploits duality results between appropriate Besov Spaces. In particular, our…
Analysis and Numerics of a Stationary Drift-Diffusion Model for Electrical Discharge in MEMS
本文分析并数值模拟了MEMS中电放电的稳态漂移-扩散模型，证明了弱解的存在性，并开发了有限元求解器。
On the existence and uniqueness of weak solutions to elliptic equations with a singular drift
本文针对三维有界Lipschitz域中带奇异漂移项的椭圆方程Dirichlet问题，在参数α<0且漂移散度为正的不利条件下，证明了在轴上取零值的唯一弱解的存在性与Hölder连续性。
We use De Giorgi techniques to prove Hölder continuity of weak solutions to a class of drift-diffusion equations, with $L^2$ initial data and divergence free drift velocity that lies in $L_{t}^{\infty}BMO_{x}^{-1}$. We apply this result to prove global regularity for a family of active scalar equations which includes the advection-diffusion equation that has been proposed by Moffatt in the context of magnetostrophic turbulence in the Earth's fluid core.
We consider degenerate porous medium equations with a divergence type of drift terms. We establish the existence of $L^{q}$-weak solutions (satisfying energy estimates or even further with moment and speed estimates in Wasserstein spaces), in case the drift term belongs to a sub-scaling (including scaling invariant) class depending on $q$ and $m$ caused by the nonlinear structure of diffusion, which is a major difference compared to that of a linear case. It is noticeable that the classes of drift terms become wider if the drift term is divergence-free. Similar conditions of gradients of drift…
Divergence-free drifts decrease concentration
本文研究了无散漂移场对对流-扩散方程解集中度的影响，发现其能降低绝对连续性模量，且在特定条件下导致解的方差、熵和$L^p$范数变化。
We study evolution equations of drift-diffusion type when various parameters are random. Motivated by applications in pedestrian dynamics, we focus on the case when the total mass is, due to boundary or reaction terms, not conserved. After providing existence and stability for the deterministic problem, we consider uncertainty in the data. Instead of a sensitivity analysis we propose to measure functionals of the solution, so-called quantities of interest (QoI), by involving scalarizing statistics. For these summarizing statistics we provide probabilistic continuity results.
Recently, using DiGiorgi-type techniques, Caffarelli and Vasseur showed that a certain class of weak solutions to the drift diffusion equation with initial data in $L^2$ gain Hölder continuity provided that the BMO norm of the drift velocity is bounded uniformly in time. We show a related result: a uniform bound on BMO norm of a smooth velocity implies uniform bound on the $C^β$ norm of the solution for some $β>0.$ We use elementary tools involving control of Hölder norms using test functions. In particular, our approach offers a third proof of the global regularity for the critical surface qu…
We prove absolute continuity of the law of the solution, evaluated at fixed points in time and space, to a parabolic dissipative stochastic PDE on $L^2(G)$, where $G$ is an open bounded domain in $\mathbb{R}^d$ with smooth boundary. The equation is driven by a multiplicative Wiener noise and the nonlinear drift term is the superposition operator associated to a real function which is assumed to be monotone, locally Lipschitz continuous, and growing not faster than a polynomial. The proof, which uses arguments of the Malliavin calculus, crucially relies on the well-posedness theory in the mild …
We prove global well-posedness for a class of dissipative semilinear stochastic evolution equations with singular drift and multiplicative Wiener noise. In particular, the nonlinear term in the drift is the superposition operator associated to a maximal monotone graph everywhere defined on the real line, on which no continuity nor growth assumptions are imposed. The hypotheses on the diffusion coefficient are also very general, in the sense that the noise does not need to take values in spaces of continuous, or bounded, functions in space and time. Our approach combines variational techniques …
On the well-posedness of (nonlinear) rough continuity equations
本文研究非 Lipschitz 漂移下的 RDE 与 RPDE，建立线性粗糙连续性/输运方程适定性及流表示，并推广到非线性方程与二维粗糙 Euler 方程。
Stochastic Itô Equations and Parabolic Second-Order Equations with singular Drift
本书研究带奇异漂移的随机伊藤方程理论，利用Morrey空间刻画奇异性，推导相关PDE结果，给出该类伊藤方程弱解与强解存在唯一的最优条件，成果具创新性。
We prove existence and uniqueness of strong solutions, as well as continuous dependence on the initial datum, for a class of fully nonlinear second-order stochastic PDEs with drift in divergence form. Due to rather general assumptions on the growth of the nonlinearity in the drift, which, in particular, is allowed to grow faster than polynomially, existing techniques are not applicable. A well-posedness result is obtained through a combination of a priori estimates on regularized equations, interpreted both as stochastic equations as well as deterministic equations with random coefficients, an…
We consider a $N$-particle interacting particle system with the vision geometrical constraints and reflected noises, proposed as a model for collective behavior of individuals. We rigorously derive a continuity-type of mean-field equation with discontinuous kernels and the normal reflecting boundary conditions from that stochastic particle system as the number of particles $N$ goes to infinity. More precisely, we provide a quantitative estimate of the convergence in law of the empirical measure associated to the particle system to a probability measure which possesses a density which is a weak…
Infinite-dimensional nonlinear stationary Fokker-Planck-Kolmogorov equations
本文证明在无穷维空间上，具有中心高斯测度γ、单位扩散算子和形如-x+v(p,x)的漂移的非线性平稳Fokker-Planck-Kolmogorov方程存在概率解，其中v是取值于γ的Cameron-Martin空间H的有界映射，且v定义在由概率密度构成的L^2(γ)的子集E与X的乘积空间上。
Reflected diffusion, no-flux continuity equations and confined Lagrangian flows in bounded domains
该研究针对有界域中反射扩散的边际分布流，给出密度/通量对存在正则拉格朗日流的充分条件，构造反例说明边界假设不可放松，还建立无通量Fokker-Planck方程的唯一性结果，为反射扩散模型的ODE采样提供数学依据。
Regularity preservation in Kolmogorov equations for non-Lipschitz coefficients under Lyapunov conditions
本文针对非Lipschitz系数的Kolmogorov方程，在Lyapunov条件下证明漂移和扩散系数的Lipschitz常数分别为$o(\log V)$和$o(\sqrt{\log V})$时，可替代全局Lipschitz连续性以保持正则性。
We prove existence and uniqueness of solutions to a class of stochastic semilinear evolution equations with a monotone nonlinear drift term and multiplicative noise, considerably extending corresponding results obtained in previous work of ours. In particular, we assume the initial datum to be only measurable and we allow the diffusion coefficient to be locally Lipschitz-continuous. Moreover, we show, in a quantitative fashion, how the finiteness of the $p$-th moment of solutions depends on the integrability of the initial datum, in the whole range $p \in ]0,\infty[$. Lipschitz continuity of t…
One-dimensional Feynman--Kac regularity under the Engelbert--Schmidt conditions
该研究针对系数满足Engelbert–Schmidt条件的时齐一维扩散过程，证明了Feynman–Kac半群的正则性，指出连续性足以保证内部$C^{1,2}$正则性，并通过实例说明结论在二维情形不成立。
Drift-diffusion equations with saturation
针对具有紧支集非线性迁移率（不一定凹）的饱和型漂移扩散方程，证明了$L^1$压缩$C_0$半群的存在性，研究了$ω$-极限性质、自由边界出现及自由能极小值，并分析了结构保持隐式有限体积格式的收敛性。
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