We prove the solvability in Sobolev spaces for both divergence and non-divergence form higher order parabolic and elliptic systems in the whole space, on a half space, and on a bounded domain. The leading coefficients are assumed to be merely measurable in the time variable and have small mean oscillations with respect to the spatial variables in small balls or cylinders. For the proof, we develop a set of new techniques to produce mean oscillation estimates for systems on a half space.
We establish a connection between the absolute continuity of elliptic measure associated to a second order divergence form operator with bounded measurable coefficients with the solvability of an endpoint $BMO$ Dirichlet problem. We show that these two notions are equivalent. As a consequence we obtain an end-point perturbation result, i.e., the solvability of the $BMO$ Dirichlet problem implies $L^p$ solvability for all $p>p_0$.
We show that BMO-solvability implies scale invariant quantitative absolute continuity (specifically, the weak-$A_\infty$ property) of caloric measure with respect to surface measure, for an open set $Ω\subset \mathbb{R}^{n+1}$, assuming as a background hypothesis only that the essential boundary of $Ω$ satisfies an appropriate parabolic version of Ahlfors-David regularity, entailing some backwards in time thickness. Since the weak-$A_\infty$ property of the caloric measure is equivalent to $L^p$ solvability of the initial-Dirichlet problem, we may then deduce that $BMO$-solvability implies $L^…
By using some recent results for divergence form equations, we study the $L_p$-solvability of second-order elliptic and parabolic equations in nondivergence form for any $p\in (1,\infty)$. The leading coefficients are assumed to be in locally BMO spaces with suitably small BMO seminorms. We not only extend several previous results by Krylov and Kim [14]-[18] to the full range of $p$, but also deal with equations with more general coefficients.
We show that for a uniformly elliptic divergence form operator $L$, defined in an open set $Ω$ with Ahlfors-David regular boundary, BMO-solvability implies scale invariant quantitative absolute continuity (the weak-$A_\infty$ property) of elliptic-harmonic measure with respect to surface measure on $\partial Ω$. We do not impose any connectivity hypothesis, qualitative or quantitative; in particular, we do not assume the Harnack Chain condition, even within individual connected components of $Ω$. In this generality, our results are new even for the Laplacian. Moreover, we obtain a converse, un…
We prove that the $A_\infty$ property of parabolic measure for operators in certain time-varying domains is equivalent to a Carleson measure property of bounded solutions. Kircheim, Kenig, Pipher, and T. Toro established this criterion on bounded solutions in the elliptic case, improving an earlier result of Dindos, Kenig and Pipher for solutions with data in BMO. The extension to the parabolic setting requires an approach to the key estimate that primarily exploits the maximum principle. For various classes of parabolic operators, this criterion makes it easier to establish the solvability of…
It is proved that there exists a local-in-time solution $u\in C([0,T),bmo(\mathbb{R}^d)^d)$ of the Navier-Stokes equations such that every $u(t)$ has an analytic extension on a complex domain whose size only depends on $t$ (and increases with $t$) and the external force $f$, assuming only that the initial velocity $u_0$ is a local $BMO$ function. Our method for proving is a combination and refinement of the work by Grujić and Kukavica [13], Guberović [15] and Kozono et al. [18]. One challenging step is the estimation of the heat and Stokes semigroups from $BMO$-type spaces to $L^\infty$; a res…
We prove the unique solvability of solutions in Sobolev spaces to the stationary Stokes system on a bounded Reifenberg flat domain when the coefficients are partially BMO functions, i.e., locally they are merely measurable in one direction and have small mean oscillations in the other directions. Using this result, we establish the unique solvability in Muckenhoupt type weighted Sobolev spaces for the system with partially BMO coefficients on a Reifenberg flat domain. We also present weighted a priori $L_q$-estimates for the system when the domain is the whole Euclidean space or a half space.
We consider the Dirichlet boundary value problem for divergence form elliptic operators with bounded measurable coefficients. We prove that for uniform domains with Ahlfors regular boundary, the BMO solvability of such problems is equivalent to a quantitative absolute continuity of the elliptic measure with respect to the surface measure, i.e. $ω_L\in A_{\infty}(σ)$. This generalizes a previous result on Lipschitz domains by Dindos, Kenig and Pipher.
In this paper we present a weighted $L_p$-theory of second-order parabolic partial differential equations defined on $C^1$ domains. The leading coefficients are assumed to be measurable in time variable and have VMO (vanishing mean oscillation) or small BMO (bounded mean oscillation) with respect to space variables, and lower order coefficients are allowed to be unbounded and to blow up near the boundary. Our BMO condition is slightly relaxed than the others in the literature.
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 {\em stochastic} partial differential equations with BMO$_x$ coefficients.
In this paper, we prove the Lorentz space $L^{q,p}$-estimates for gradients of very weak solutions to the linear parabolic equations with $\mathbf{A}_q$-weights $$u_t-\operatorname{div}(A(x,t)\nabla u)=\operatorname{div}(F),$$ in a bounded domain $Ω\times (0,T)\subset\mathbb{R}^{N+1}$, where $A$ has a small mean oscillation, and $Ω$ is a Lipchistz domain with a small Lipschitz constant.
In the recent work [DFM1, DFM2] G. David, J. Feneuil, and the first author have launched a program devoted to an analogue of harmonic measure for lower-dimensional sets. A relevant class of partial differential equations, analogous to the class of elliptic PDEs in the classical context, is given by linear degenerate equations with the degeneracy suitably depending on the distance to the boundary. The present paper continues this line of research and focuses on the criteria of quantitative absolute continuity of the newly defined harmonic measure with respect to the Hausdorff measure, $ω\in A_\…
We develop a strategy making extensive use of tent spaces to study parabolic equa-tions with quadratic nonlinearities as for the Navier-Stokes system. We begin with a new proof of the well-known result of Koch and Tataru on the well-posedness of Navier-Stokes equations in \R^n with small initial data in BMO^{-1}(\R^n). We then study another model where neither pointwise kernel bounds nor self-adjointness are available.
Global well-posedness of the time-dependent (degenerate) thermistor problem remains open for many years. In this paper, we solve the problem by establishing a uniform-in-time BMO estimate of inhomogeneous parabolic equations. Applying this estimate to the temperature equation, we derive a BMO bound of the temperature uniform with respect to time, which implies that the electric conductivity is a $A_2$ weight. The Hölder continuity of the electric potential is then proved by applying the De Giorgi--Nash--Moser estimate for degenerate elliptic equations with $A_2$ coefficient. Uniqueness of solu…
We establish the solvability of second order divergence type parabolic systems in Sobolev spaces. The leading coefficients are assumed to be only 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 $W^1_p$-solvability of second order elliptic systems in divergence form. Our results are new even for scalar equations and the proofs simplify the methods used previously in [12]
We prove weighted $L_{p,q}$-estimates for divergence type higher order elliptic and parabolic systems with irregular coefficients on Reifenberg flat domains. In particular, in the parabolic case the coefficients do not have any regularity assumptions in the time variable. As functions of the spatial variables, the leading coefficients are permitted to have small mean oscillations. The weights are in the class of Muckenhoupt weights $A_p$. We also prove the solvability of the systems in weighted Sobolev spaces.
We prove that local and global parabolic BMO spaces are equal thus extending the classical result of Reimann and Rychener. Moreover, we show that functions in parabolic BMO are exponentially integrable in a general class of space-time cylinders. As a corollary, we establish global integrability for positive supersolutions to a wide class of doubly nonlinear parabolic equations.
The solvability in Sobolev spaces is proved for divergence form complex-valued higher order parabolic systems in the whole space, on a half space, and on a Reifenberg flat domain. The leading coefficients are assumed to be merely measurable in one spacial direction and have small mean oscillations in the orthogonal directions on each small cylinder. The directions in which the coefficients are only measurable vary depending on each cylinder. The corresponding elliptic problem is also considered.
Let $L$ be a Neumann operator of the form $L=-Δ_{N}$ acting on $L^2(\mathbb R^n)$. Let ${BMO}_{Δ_{N}}(\mathbb R^n)$ denote the BMO space on $\mathbb R^n$ associated to the Neumann operator $Ł$. In this article we will show that a function $f\in { BMO}_{Δ_{N}}(\mathbb R^n)$ is the trace of the solution of $${\mathbb L}u=u_{t}+L u=0, u(x,0)= f(x),$$ where $u$ satisfies a Carleson-type condition \begin{eqnarray*} \sup_{x_B, r_B} r_B^{-n}\int_0^{r_B^2}\int_{B(x_B, r_B)} |\nabla u(x,t)|^2 {dx dt } \leq C <\infty, \end{eqnarray*} for some constant $C>0$. Conversely, this Carleson condition character…
We study necessary conditions and sufficient conditions for the existence of local-in-time solutions of the Cauchy problem for superlinear fractional parabolic equations. Our conditions are sharp and clarify the relationship between the solvability of the Cauchy problem and the strength of the singularities of the initial measure.
The global existence of classical solutions to strongly coupled parabolic systems is shown to be equivalent to the availability of an iterative scheme producing a sequence of solutions with uniform continuity in the BMO norms. Amann's results on global existence of classical solutions still hold under much weaker condition that their BMO norms do not blow up in finite time. The proof makes use of some new global and local weighted Gagliardo-Nirenberg inequalities involving BMO norms.
In order to extend the blow-up criterion of solutions to the Euler equations, Kozono and Taniuchi have proved a logarithmic Sobolev inequality by means of isotropic (elliptic) $BMO$ norm. In this paper, we show a parabolic version of the Kozono-Taniuchi inequality by means of anisotropic (parabolic) $BMO$ norm. More precisely we give an upper bound for the $L^{\infty}$ norm of a function in terms of its parabolic $BMO$ norm, up to a logarithmic correction involving its norm in some Sobolev space. As an application, we also explain how to apply this inequality in order to establish a long-time …
We study if the parabolic forward-in-time maximal operator is bounded on parabolic BMO. It turns out that for non-negative functions the answer is positive, but the behaviour of sign changing functions is more delicate. The class parabolic BMO and the forward-in-time maximal operator originate from the regularity theory of nonlinear parabolic partial differential equations. In addition to that context, we also study the question in dimension one.
In this paper we study the quasilinear nondiagonal parabolic type systems. We assume that the principal elliptic operator, which is part of the parabolic system, has a divergence structure. Under certain conditions it is proved the well-posedness of classical solutions, which exist globally in time.
In this article we prove the BMO-$L_{\infty}$ estimate $$ \|(-Δ)^{γ/2} u\|_{BMO(\mathbf{R}^{d+1})}\leq N \|\frac{\partial}{\partial t}u-A(t)u\|_{L_{\infty}(\mathbf{R}^{d+1})}, \quad \forall\, u\in C^{\infty}_c(\mathbf{R}^{d+1}) $$ for a wide class of pseudo-differential operators $A(t)$ of order $γ\in (0,\infty)$. The coefficients of $A(t)$ are assumed to be merely measurable in time variable. As an application to the equation $$ \frac{\partial}{\partial t}u=A(t)u+f,\quad t\in \mathbf{R} $$ we prove that for any $u\in C^{\infty}_c(\mathbf{R}^{d+1})$ $$ \|u_t\|_{L_p(\mathbf{R}^{d+1})}+\|(-Δ)^{γ…
We prove locally in time the existence of a smooth solution for multidimensional two-phase Stefan problem for degenerate parabolic equations of the porous medium type. We establish also natural Hölder class for the boundary conditions in the Cauchy-Dirichlet problem for a degenerate parabolic equation.
We consider a linear second order parabolic system with a third order dispersion term. This type of system arises when considering a nonlinear model equation describing the motion of a vortex filament with axial flow immersed in an incompressible and inviscid fluid. We prove the solvability of an initial-boundary value problem of the parabolic-dispersive system which allows application to the motion of a vortex filament. To do so, we propose a new regularization technique by adding a space-time derivative term.
Let $(u,v)$ be a nonnegative solution to the semilinear parabolic system \[ \mbox{(P)} \qquad \cases{ \partial_t u=D_1Δu+v^p, & $x\in{\bf R}^N,\,\,\,t>0,$\\ \partial_t v=D_2Δv+u^q, & $x\in{\bf R}^N,\,\,\,t>0,$\\ (u(\cdot,0),v(\cdot,0))=(μ,ν), & $x\in{\bf R}^N,$ } \] where $D_1$, $D_2>0$, $0<p\le q$ with $pq>1$ and $(μ,ν)$ is a pair of nonnegative Radon measures or nonnegative measurable functions in ${\bf R}^N$. In this paper we study sufficient conditions on the initial data for the solvability of problem~(P) and clarify optimal singularities of the initial functions for the solvability.
We consider the Cauchy problem for the Hardy parabolic equation $\partial_t u-Δu=|x|^{-γ}u^p$ with initial data $u_0$ singular at some point $z$. Our main results show that, if $z\neq 0$, then the optimal strength of the singularity of $u_0$ at $z$ for the solvability of the equation is the same as that of the Fujita equation $\partial_t u-Δu=u^p$. Moreover, if $z=0$, then the optimal singularity for the Hardy parabolic equation is weaker than that of the Fujita equation. We also obtain analogous results for a fractional case $\partial_t u+(-Δ)^{θ/2} u=|x|^{-γ}u^p$ with $0<θ<2$.
A Characterization of Solvability of the Parabolic $L^p$ Dirichlet Problem on Lipschitz Graph Domains Via Carleson Measure Estimates of Bounded Solutions
Liouville theorem and sharp solvability for solutions of the parabolic Monge-Ampère equation with periodic data
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education（北京师范大学数学科学学院，教育部数学与复杂系统重点实验室）
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 measurable in one direction and have 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 …
This paper deals with a parabolic partial differential equation that includes a non-linear nonlocal in time term. This term is the product of a so-called interaction potential and the solution of the problem. The interaction potential depends on a weighted integral of the solution over the entire time interval, where the problem is considered, and satisfies fairly general conditions. Namely, it is assumed to be a continuous bounded from below function that can behave arbitrarily at infinity. This fact implies that the interaction term is not a lower order term in the equation. The weak solvabi…
The existence of strong solutions to general class of strongly coupled parabolic systems will be discussed. These systems can be degenerate or singular as boundedness of theirs solutions are unavailable and not assummed. The results greatly improve those in a recent papers \cite{letrans,dleJFA,dleANS} as the systems can have quadratic growth in gradients. A unified proof for both cases is presented. Most importantly, the VMO assumption in \cite{dleJFA,dleANS} will be replaced by a much versatile one thanks to a new local weighted Gagliardo-Nirenberg involving BMO norms. Degenerate and singular…
Systems of parabolic, possibly degenerate parabolic SPDEs are considered. Existence and uniqueness are established in Sobolev spaces. Similar results are obtained for a class of equations generalizing the deterministic first order symmetric hyperbolic systems.
In this work we investigate a boundary problem with non-local conditions for mixed parabolic-hyperbolic type equation with three lines of type-changing with Caputo fractional derivative in the parabolic part. We equivalently reduce considered problem to the system of second kind Volterra integral equations. In the parabolic part we use solution of the first boundary problem with appropriate Green's function and in hyperbolic parts we use corresponding solutions of the Cauchy problem.
We prove the solvability in Sobolev spaces $W^{1,2}_p$, $p>d+1$, of the terminal-boundary value problem for a class of fully nonlinear parabolic equations, including parabolic Bellman's equations, in bounded cylindrical domains with VMO ``coefficients''. The solvability in $W^{2}_p$, $p>d$, of the corresponding elliptic boundary-value problem is also obtained.
We discuss a parabolic version of the space of functions of bounded mean oscillation related to a doubly nonlinear parabolic partial differential equation. Parabolic John-Nirenberg inequalities, which give exponential decay estimates for the oscillation of a function, are shown in the natural geometry of the partial differential equation. Chaining arguments are applied to change the time lag in the parabolic John-Nirenberg inequality. We also show that the quasihyperbolic boundary condition is a necessary and sufficient condition for a global parabolic John-Nirenberg inequality. Moreover, we c…
In this paper, we consider a doubly nonlinear parabolic equation $ \partial _t β(u) - \nabla \cdot α(x , \nabla u) \ni f$ with the homogeneous Dirichlet boundary condition in a bounded domain, where $β: \mathbb{R} \to 2 ^{ \mathbb{R} }$ is a maximal monotone graph satisfying $0 \in β(0)$ and $ \nabla \cdot α(x , \nabla u )$ stands for a generalized $p$-Laplacian. Existence of solution to the initial boundary value problem of this equation has been investigated in an enormous number of papers for the case where single-valuedness, coerciveness, or some growth condition is imposed on $β$. However…
The solvability in Sobolev spaces $W^{1,2}_p$ is proved for nondivergence form second order parabolic equations for $p>2$ close to 2. The leading coefficients are assumed to be measurable in the time variable and two coordinates of space variables, and almost VMO (vanishing mean oscillation) with respect to the other coordinates. This implies the $W^{2}_p$-solvability for the same $p$ of nondivergence form elliptic equations with leading coefficients measurable in two coordinates and VMO in the others. Under slightly different assumptions, we also obtain the solvability results when $p=2$.
Let $(u,v)$ be a solution to a semilinear parabolic system \[ \mbox{(P)} \qquad \begin{cases} \partial_t u=D_1Δu+v^p\quad & \quad\mbox{in}\quad{\bf R}^N\times(0,T),\\ \partial_t v=D_2Δv+u^q\quad & \quad\mbox{in}\quad{\bf R}^N\times(0,T),\\ u,v\ge 0 & \quad\mbox{in}\quad{\bf R}^N\times(0,T),\\ (u(\cdot,0),v(\cdot,0))=(μ,ν) & \quad\mbox{in}\quad{\bf R}^N, \end{cases} \] where $N\ge 1$, $T>0$, $D_1>0$, $D_2>0$, $0<p\le q$ with $pq>1$ and $(μ,ν)$ is a pair of Radon measures or nonnegative measurable functions in ${\bf R}^N$. In this paper we study qualitative properties of the initial trace of the…
We consider parabolic operators of the form $$\partial_t+\mathcal{L},\ \mathcal{L}=-\mbox{div}\, A(X,t)\nabla,$$ in $\mathbb R_+^{n+2}:=\{(X,t)=(x,x_{n+1},t)\in \mathbb R^{n}\times \mathbb R\times \mathbb R:\ x_{n+1}>0\}$, $n\geq 1$. We assume that $A$ is a $(n+1)\times (n+1)$-dimensional matrix which is bounded, measurable, uniformly elliptic and complex, and we assume, in addition, that the entries of A are independent of the spatial coordinate $x_{n+1}$ as well as of the time coordinate $t$. For such operators we prove that the boundedness and invertibility of the corresponding layer potent…
We prove weighted mixed-norm $L^q_t(W^{2,p}_x)$ and $L^q_t(C^{2,α}_x)$ estimates for $1<p,q<\infty$ and $0<α<1$, weighted mixed weak-type estimates for $q=1$, $L^\infty_{t}(L^p_x)-BMO_t(W^{2,p}_x)$, and $L^\infty_{t}(C^α_x)-BMO_t(C^{2,α}_x)$, and a.e.~pointwise formulas for derivatives, for solutions $u=u(t,x)$ to parabolic equations of the form $$\partial_tu-a^{ij}(t)\partial_{ij}u+u=f\quad t\in\mathbb{R},~x\in\mathbb{R}^n$$ and for the Cauchy problem $$\begin{cases} \partial_tv-a^{ij}(t)\partial_{ij}v+v=f&\hbox{for}~t >0,~x\in\mathbb{R}^n \\ v(0,x)=g&\hbox{for}~x\in\mathbb{R}^n. \end{cases}$…
Let $Ω$ be a Lipschitz domain in $\mathbb R^n,n\geq 3,$ and $L=\divt A\nabla$ be a second order elliptic operator in divergence form. We will establish that the solvability of the Dirichlet regularity problem for boundary data in Hardy-Sobolev space $\HS$ is equivalent to the solvability of the Dirichlet regularity problem for boundary data in $H^{1,p}$ for some $1<p<\infty$. This is a "dual result" to a theorem in \cite{DKP09}, where it has been shown that the solvability of the Dirichlet problem with boundary data in $\text{BMO}$ is equivalent to the solvability for boundary data in $L^p(\pa…
In this paper we present a weighted $L_p$-theory of parabolic systems on a half space. The leading coefficients are assumed to be only measurable in $t$ and have small bounded mean oscillations (BMO) with respect to $x$, and the lower order coefficients are allowed to blow up near the boundary.
We investigate parabolic Muckenhoupt weights and functions of bounded mean oscillation (BMO) related to nonlinear parabolic partial differential equations. The main result gives a full characterization of weak and strong type weighted norm inequalities for parabolic forward in time maximal operators. In addition, we give a Jones type factorization result for the parabolic Muckenhoupt weights and a Coifman-Rochberg type characterization of the parabolic BMO from Moser's seminal paper through parabolic Muckenhoupt weights and maximal functions.
We prove generalized Fefferman-Stein type theorems on sharp functions with $A_p$ weights in spaces of homogeneous type with either finite or infinite underlying measure. We then apply these results to establish mixed-norm weighted $L_p$-estimates for elliptic and parabolic equations/systems with (partially) BMO coefficients in regular or irregular domains.
In this paper, we study an elliptic operator in divergence-form but not necessary symmetric. In particular, our results can be applied to elliptic operator $L=νΔ+u(x,t)\cdot\nabla$, where $u(\cdot,t)$ is a time-dependent vector field in $\mathbb{R}^{n}$, which is divergence-free in distribution sense, i.e. $\nabla\cdot u=0$. Suppose $u\in L_{t}^{\infty}(\textrm{BMO}_{x}^{-1})$. We show the existence of the fundamental solution $\varGamma(x,t;ξ,τ)$ of the parabolic operator $L-\partial_{t}$, and show that $\varGamma$ satisfies the Aronson estimate with a constant depending only on the dimension…
