Guy Metivier, Kevin Zumbrun
In this paper we prove the continuity of stable subspaces associated to parabolic-hyperbolic boundary value problems, for limiting values of parameters. The analysis is based on the construction performed in a previous paper of Kreiss' type symmetrizers.
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We prove the boundedness of the time derivative in the parabolic Signorini problem, as well as establish its Hölder continuity at regular free boundary points.
Continuity estimates for degenerate parabolic double-phase equations via nonlinear potentials
本文研究了退化抛物双相方程的正则性，利用方程右端的椭圆Riesz位势为有界弱解建立了连续性估计。
We establish the pointwise continuity of bounded weak solutions to of a class of scalar parabolic equations and strongly coupled parabolic systems. Our approach to the regularity theory of parabolic scalar equations is quite elementary and its applications to strongly coupled systems does not require higher $L^p$ integrability of derivatives.
We prove uniqueness for backward parabolic equations whose coefficients are Osgood continuous in time for $t>0$ but not at $t=0$.
We study local boundedness and Hölder continuity of a parabolic equation involving the fractional $p$-Laplacian of order $s$, with $0<s<1$, $2\leq p < \infty$, with a general right hand side. We focus on obtaining precise Hölder continuity estimates. The proof is based on a perturbative argument using the already known Hölder continuity estimate for solutions to the equation with zero right hand side.
We present a local almost everywhere regularity result for a general nonlinear non-diagonal parabolic system, which main part depends on symmetric part of the gradient.
For a general class of divergence type quasi-linear degenerate parabolic equations with measurable coeffcients and lower order terms from non-linear Kato-type classes, we prove local boundedness and continuity of solutions, and the intrinsic Harnack inequality for positive solutions.
On the modulus of continuity of solutions to nonlocal parabolic equations
本文在极小及稍强尾部条件下，分别量化并推导了非局部抛物型方程局部有界弱解的一般连续性模与Hölder连续性模，并在具有可测核的非局部$p$-Laplacian框架下证明了这些正则性估计。
We give a proof of the Hölder continuity of weak solutions of certain degenerate doubly nonlinear parabolic equations in measure spaces. We only assume the measure to be a doubling non-trivial Borel measure which supports a Poincaré inequality. The proof discriminates between large scales, for which a Harnack inequality is used, and small scales, that require intrinsic scaling methods.
We establish the local Hölder continuity for the nonnegative weak solutions of certain doubly nonlinear parabolic equations possessing a singularity in the time derivative part and a degeneracy in the principal part. The proof involves the method of intrinsic scaling and the setting is a measure space equipped with a doubling non-trivial Borel measure supporting a Poincaré inequality.
This paper deals with two separate but related results. First we consider weak solutions to a parabolic operator with Hörmander vector fields. Adapting the iteration scheme of Jürgen Moser for elliptic and parabolic equations in $\mathbb{R}^n$ we show a parabolic Harnack inequality. Then, after proving the Harnack inequality for weak solutions to equations of the form $u_t = \sum X_i (a_{ij} X_j u)$ we use this to show Hölder continuity. We assume the coefficients are bounded and elliptic. The iteration scheme is a tool that may be adapted to many settings and we extend this to nonlinear parab…
This paper considers a certain doubly singular parabolic equations with one singularity occurs in the time derivative, whose model is \begin{equation*} \partial_tβ(u)-\operatorname{div}|Du|^{p-2}Du\ni0,\qquad \text{in}\quad Ω\times(0,T)\end{equation*} where $Ω\subset\mathbb{R}^N$ and $N\geq3$. We show that the bounded weak solutions are locally continuous in the range $$2-ε_0\leq p<2,$$ provided $ε_0>0$ is small enough, and the continuity is stable as $p\to2$.
For proper minimizers of parabolic variational integrals with linear growth with respect to $|Du|$, we establish a necessary and sufficient condition for $u$ to be continuous at a point $(x_o,t_o)$, in terms of a sufficient fast decay of the total variation of $u$ about $(x_o,t_o)$ (see (1.4) below). These minimizers arise also as {proper} solutions to the parabolic $1$-laplacian equation. Hence, the continuity condition continues to hold for such solutions (§ 3).
Continuity estimates for doubly degenerate parabolic equations with lower order terms via nonlinear potentials
本文针对非齐次双非线性抛物方程，建立了有界非负弱解的连续性估计，该估计通过椭圆Riesz势刻画右端项的影响。
In this paper we study the regularity of weak solutions to an elliptic-parabolic system modeling natural network formation. The system is singular and involves cubic nonlinearity. Our investigation reveals that weak solutions are Hölder continuous when the space dimension $N$ is $2$. This is achieved via an inequality associated with the Stummel-Kato class of functions and refinement of a lemma originally due to S. Campanato and C. B. Morrey (\cite{G}, p. 86).
Hölder Continuity for Fully Fractional Parabolic Equations with Space-time Nonlocal Operators
本文研究含空间分数阶扩散与Marchaud型分数阶时间导数的全分数阶抛物型方程，在不假设解和非齐次项有界的条件下，引入时间依赖非局部尾项，利用非局部De Giorgi迭代技术证明弱解的局部有界性与局部Hölder连续性。
Hölder Continuity and Harnack estimate for non-homogeneous parabolic equations
本文针对非散度型非齐次抛物型方程，建立了时间向前的内在Harnack不等式，由此导出解的Hölder连续性，并给出量化强最小值原理的全局Harnack型估计，为广义Harnack不等式提供了新证明。
In this paper we investigate the zero-relaxation limit of the following multi-D semilinear hyperbolic system in pseudodifferential form: W_{t}(x,t) + (1/epsilon) A(x,D) W(x,t) = (1/epsilon^2) B(x,W(x,t)) + (1/epsilon) D(W(x,t)) + E(W(x,t)). We analyse the singular convergence, as epsilon tends to 0, in the case which leads to a limit system of parabolic type. The analysis is carried out by using the following steps: (i) We single out algebraic ``structure conditions'' on the full system, motivated by formal asymptotics, by some examples of discrete velocity models in kinetic theories. (ii) We …
Attractor Continuity for Semilinear Parabolic Equations on Thin Domains with Degenerating Outward Peaks
研究半线性抛物方程在退化向外尖峰薄域上吸引子的连续性及收敛速度。
We prove the continuity of bounded solutions for a wide class of parabolic equations with $(p,q)$-growth $$ u_{t}-{\rm div}\left(g(x,t,|\nabla u|)\,\frac{\nabla u}{|\nabla u|}\right)=0, $$ under the generalized non-logarithmic Zhikov's condition $$ g(x,t,{\rm v}/r)\leqslant c(K)\,g(y,τ,{\rm v}/r), \quad (x,t), (y,τ)\in Q_{r,r}(x_{0},t_{0}), \quad 0<{\rm v}\leqslant Kλ(r), $$ $$ \quad \lim\limits_{r\rightarrow0}λ(r)=0, \quad \lim\limits_{r\rightarrow0} \frac{λ(r)}{r}=+\infty, \quad \int_{0} λ(r)\,\frac{dr}{r}=+\infty. $$ In particular, our results cover new cases of double-phase parabolic equat…
In this paper, we prove gradient continuity estimates for viscosity solutions to $Δ_{p}^N u- u_t= f$ in terms of the scaling critical $L(n+2,1 )$ norm of $f$, where $Δ_{p}^N$ is the game theoretic normalized $p-$Laplacian operator defined in (1.2) below. Our main result, Theorem 2.5 constitutes borderline gradient continuity estimate for $u$ in terms of the modified parabolic Riesz potential $\mathbf{P}^{f}_{n+1}$ as defined in (2.8) below. Moreover, for $f \in L^{m}$ with $m>n+2$, we also obtain Hölder continuity of the spatial gradient of the solution $u$, see Theorem 2.6 below. This improve…
We analyze the dynamics of the flow generated by a nonlinear parabolic problem when some reaction and potential terms are concentrated in a neighborhood of the boundary. We assume that this neighborhood shrinks to the boundary as a parameter εgoes to zero. Also, we suppose that the "inner boundary" of this neighborhood presents a highly oscillatory behavior. Our main goal here is to show the continuity of the family of attractors with respect to ε. Indeed, we prove upper semicontinuity under the usual properties of regularity and dissipativeness and, assuming hyperbolicity of the equilibria, w…
Gradient continuity for the parabolic $(1,\,p)$-Laplace system
本文针对抛物型$(1,p)$-拉普拉斯方程组，克服算子在梯度退化区一致椭圆性失效的难点，通过近似方程组与经典正则性方法证明截断梯度的局部赫尔德连续性，最终得到空间梯度的时空连续性，还拓展到带Uhlenbeck结构的广义方程组。
Regularity for obstacle problems to anisotropic parabolic equations
本文沿用Dibenedetto的内在标度法，在障碍仅满足Hölder连续性的条件下，证明了部分各向异性抛物方程障碍问题弱解的局部Hölder连续性。
We prove that the moduli of continuity of viscosity solutions to fully nonlinear parabolic partial differential equations are viscosity subsolutions of suitable parabolic equations of one space variable. As applications, we obtain sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations with bounded initial data, via comparison with one-dimensional solutions. This work extends multiple results of Andrews and Clutterbuck for quasilinear equations to fully nonlinear equations.
Here we generalize quasilinear parabolic $p-$Laplacian type equations to obtain the prototype equation as \[ u_t - \text{div} (g(|Du|)/ |Du| \cdot Du) = 0, \] where a nonnegative, increasing, and continuous function $g$ trapped in between two power functions $|Du|^{g_0 -1}$ and $|Du|^{g_1 -1}$ with $1<g_0 \leq g_1 < \infty$. Through this generalization in the setting from Orlicz spaces, we provide a uniform proof with a single geometric setting that a bounded weak solution is locally Hölder continuous considering $1 < g_0 \leq g_1 \leq 2$ and $2 \leq g_0 \leq g_1 < \infty$ separately. By using…
We establish partial regularity for vector-valued solutions to parabolic systems where the coefficients are possibly discontinuous with respect to (x,t). More precisely, we assume a VMO-condition with respect to the (x,t) and continuity with respect to u and prove Hölder continuity of the solutions outside of singular sets.
Continuity of the spatial gradient of weak solutions to very singular parabolic equations involving the one-Laplacian
本文证明了一类含一-Laplace型算子的极奇异抛物方程弱解空间梯度的时空连续性，通过截断梯度的Hölder估计和抛物近似方法，解决了宾汉流体模型中的长期公开问题。
On the Hölder continuity of signed solutions to doubly nonlinear parabolic equations in the mixed degenerate/singular cases
本文研究了双重非线性抛物方程在混合退化/奇异性情况下的符号解的Hölder连续性，通过新版本的积分Harnack型不等式证明了其连续性。
We bound the modulus of continuity of solutions to quasilinear parabolic equations in one space variable in terms of the initial modulus of continuity and elapsed time. In particular we characterize those equations for which the Lipschitz constants of solutions can be bounded in terms of their initial oscillation and elapsed time.
We consider time-inhomogeneous, second order linear parabolic partial differential equations of the non-divergence type, and assume the ellipticity and the continuity on the coefficient of the second order derivatives and the boundedness on all coefficients. Under the assumptions we show the Hölder continuity of the solution in the spatial component. Furthermore, additionally assuming the Dini continuity of the coefficient of the second order derivative, we have the better continuity of the solution. In the proof, we use a probabilistic method, in particular the coupling method. As a corollary…
Gradient continuity for the parabolic $(1,\,p)$-Laplace equation under the subcritical case
本文研究抛物 $(1,\\,p)$-Laplace 方程在次临界情形下的梯度连续性，在弱解满足特定可积性条件下，通过 Moser 迭代证明解及其梯度的局部有界性，进而确立梯度连续性。
This is a note on \cite{LSU} and \cite{FS}. Using their work line by line, we prove the Hölder-continuity of solutions to linear parabolic equations of mixed type, assuming the coefficient of $\frac{\partial}{\partial t}$ has time-derivative bounded from above. On a Kähler manifold, this Hölder estimate works when the metrics possess conic singularities along a normal crossing divisor.
We study the local Hölder continuity and the asymptotic behaviour of solution, $\mathbf{u}=(u^1,\cdots, u^k)$, of the degenerate system \begin{equation*} u^i_t=\nabla\cdot\left(m\,U^{m-1}\nabla u^i\right) \qquad \text{for $m>1$ and $i=1,\cdots,k$ } \end{equation*} which describes the populations density of $k$-species whose diffusion is determined by their total population density $U=u^1+\cdots+u^k$. For the local Hölder continuity, we adopt the intrinsic scaling and iteration arguments of DeGiorgi, Moser, and Dibenedetto. Under some regularity conditions, we also prove that the population den…
This article studies the partial Hölder continuity of weak solutions to certain degenerate parabolic systems whose model is the differentiable parabolic $p(x,t)$-Laplacian system, \begin{equation*}\partial_t u-\operatorname{div}[μ(z)(1+|Du|^2)^{\frac{p(z)-2}{2}}Du]=0,\qquad p(z)\geq2.\end{equation*} Here, the exponential function $p(z)$ satisfies a logarithmic continuity condition. We show that if $μ(z)$ satisfies a certain VMO-type condition, then $u$ is locally Hölder continuous except for a measure zero set.
Hölder Continuity of the Gradient of Solutions to Doubly Non-Linear Parabolic Equations
本文研究双重非线性抛物方程弱解梯度的Hölder连续性，在超临界快速扩散区域建立正则性估计，并利用时间无关的Harnack不等式和Schauder估计获得消失时间附近的衰减估计。
Extending investigations of Métivier&Zumbrun in the hyperbolic case, we treat stability of viscous shock and boundary layers for viscous perturbations of multidimensional hyperbolic systems with characteristics of variable multiplicity, specifically the construction of symmetrizers in the low-frequency regime where variable multiplicity plays a role. At the same time, we extend the boundary-layer theory to ``real'' or partially parabolic viscosities, Neumann or mixed-type parabolic boundary conditions, and systems with nonconservative form, in addition proving a more fundamental version of the…
The global equi-continuity estimate on $L^p$-viscosity solutions of parabolic bilateral obstacle problems with unbounded ingredients is established when obstacles are merely continuous. The existence of $L^p$-viscosity solutions is established via an approximation of given data. The local Hölder continuity estimate on the space derivative of $L^p$-viscosity solutions is shown when the obstacles belong to $C^{1, β}$, and $p>n+2$.
In this paper we study viscosity solutions of semilinear parabolic equations in the Heisenberg group. We show uniqueness of viscosity solutions with exponential growth at space infinity. We also study Lipschitz and horizontal convexity preserving properties under appropriate assumptions. Counterexamples show that in general such properties that are well-known for semilinear and fully nonlinear parabolic equations in the Euclidean spaces do not hold in the Heisenberg group.
Continuity of attractors of parabolic equations with nonlinear boundary conditions and rapidly varying boundaries. The case of a Lipschitz deformation
本文研究了带非线性边界条件的抛物型方程在边界随参数$ε$趋于零时快速振荡变化下吸引子的连续性，通过引入变化域函数与Banach空间算子族的收敛性，证明了解收敛至含振荡因子的极限方程。
We consider the non-degenerate second-order parabolic partial differential equations of non-divergence form with bounded measurable coefficients (not necessary continuous). Under some assumptions it is known that the fundamental solution to the equations exists uniquely, has the Gaussian bounds and is locally Hölder continuous. In the present paper, we concern the Gaussian bounds and the lower bound of the index of the Hölder continuity with respect to the initial point. We use the pinned diffusion processes for the probabilistic representation of the fundamental solutions and apply the coupli…
Answering a question left open in \cite{MZ2}, we show for general symmetric hyperbolic boundary problems with constant coefficients, including in particular systems with characteristics of variable multiplicity, that the uniform Lopatinski condition implies strong $L^2$ well-posedness, with no further structural assumptions. The result applies, more generally, to any system that is strongly $L^2$ well-posed for at least one boundary condition. The proof is completely elementary, avoiding reference to Kreiss symmetrizers or other specific techniques. On the other hand, it is specific to the con…
By borrowing ideas from the parabolic theory, we use a combination of De Giorgi's and Moser's methods to give some remarks on the proof of Hölder continuity of weak solutions of elliptic equations.
We derive sharp estimates on modulus of continuity for solutions of the heat equation on a compact Riemannian manifold with a Ricci curvature bound, in terms of initial oscillation and elapsed time. As an application, we give an easy proof of the optimal lower bound on the first eigenvalue of the Laplacian on such a manifold as a function of diameter.
Weak parabolic Harnack inequality and Hölder regularity for non-local Dirichlet forms
本文研究一般正则狄利克雷型，通过局部热核估计与增长引理建立弱抛物型哈纳克不等式的等价条件，并在取消上跳跃光滑性假设的情形下证明热函数与调和函数的赫尔德连续性，推广了已有理论。
Optimal in tail Hölder estimates for weak solutions of the nonlocal parabolic p-Laplace equations on the Heisenberg group
本文研究了海森堡群上非局部抛物p-拉普拉斯方程弱解的最优尾Hölder估计问题，通过引入最优尾条件证明了弱解的Hölder连续性。
We give a proof for the Hölder continuity of functions in the parabolic De Giorgi classes in metric measure spaces. We assume the measure to be doubling, to support a weak $(1,p)$-Poincaré inequality and to satisfy the annular decay property.
Locally bounded, local weak solutions to a doubly nonlinear parabolic equation, which models the multi-phase transition of a material, is shown to be locally continuous. Moreover, an explicit modulus of continuity is given. The effect of the $p$-Laplacian type diffusion is also considered.
We study second order parabolic equations on Lipschitz domains subject to inhomogeneous Neumann (or, more generally, Robin) boundary conditions. We prove existence and uniqueness of weak solutions and their continuity up to the boundary of the parabolic cylinder. Under natural assumptions on the coefficients and the inhomogeneity we can also prove convergence to an equilibrium or asymptotic almost periodicity.
On Holder Continuity and Equivalent Formulation of Intrinsic Harnack Estimates for an Anisotropic Parabolic Degenerate Prototype Equation
本文利用内在Harnack估计证明各向异性退化抛物方程有界局部弱解的Hölder连续性，并给出这些估计在内在几何下的等价表述。
We exhibit a singularly perturbed parabolic problems for which the asymptotic behavior can be described by an one-dimensional ordinary differential equation. We estimate the continuity of attractors in the Hausdorff metric by rate of convergence of resolvent operator.
We prove continuous dependence on initial data for a backward parabolic operator whose leading coefficients are Osgodd continuous in time. This result fills the gap between uniqueness and continuity results obtained so far.
We prove the natural weighted Calderón and Zygmund estimates for solutions to elliptic and parabolic obstacle problems in nondivergence form with discontinuous coefficients and irregular obstacles. We also obtain Morrey regularity results for the Hessian of the solutions and Hölder continuity of the gradient of the solutions.
We prove an estimate on the modulus of continuity at a boundary point of a cylindrical domain for local weak solutions to degenerate parabolic equations of $p$-laplacian type. The estimate is given in terms of a Wiener-type integral, defined by a proper elliptic $p$-capacity.
We prove regularity results such as interior Lipschitz regularity and boundary continuity for the Cauchy-Dirichlet problem associated to a class of parabolic equations inspired by the evolutionary $p$-Laplacian, but extending it at a wide scale. We employ a regularization technique of viscosity-type that we find interesting in itself.
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 discuss various issues related to the finite-dimensionality of the asymptotic dynamics of solutions of parabolic equations. In particular, we study the regularity of the vector field on the global attractor associated with these equations. We show that certain dissipative partial differential equations possess a linear term that is log-Lipschitz continuous on the attractor. We then prove that this property implies that the associated global attractor $\mathcal A$ lies within a small neighbourhood of a smooth manifold, given as a Lipschitz graph over a finite number of Fourier modes. Consequ…
We prove an estimate on the modulus of continuity at a boundary point of a cylindrical domain for local weak solutions to singular parabolic equations of p-laplacian type. The estimate is given in terms of a Wiener-type integral, defined by a proper elliptic p-capacity.
We study a parabolic equation for the fractional $p-$Laplacian of order $s$, for $p\ge 2$ and $0<s<1$. We provide space-time Hölder estimates for weak solutions, with explicit exponents. The proofs are based on iterated discrete differentiation of the equation in the spirit of J. Moser.
This note is devoted to continuity results of the time derivative of the solution to the one-dimensional parabolic obstacle problem with variable coefficients. It applies to the smooth fit principle in numerical analysis and in financial mathematics. It relies on various tools for the study of free boundary problems: blow-up method, monotonicity formulae, Liouville's results.
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