The Oleinik inequality for conservation laws and Aronson-Benilan type inequalities for porous medium or p-Laplacian equations are one-sided inequalities that provide the fundamental features of the solution such as the uniqueness and sharp regularity. In this paper such one-sided inequalities are unified and generalized for a wide class of first and second order equations in the form of $$ u_t=σ(t,u,u_x,u_{xx}),\quad u(x,0)=u^0(x)\ge0,\quad t>0,\,x\in\bfR, $$ where the non-strict parabolicity ${\partial\over\partial q} σ(t,z,p,q)\ge0$ is assumed. The generalization or unification of one-sided inequalities is given in a geometric statement that the zero level set $$ A(t;m,x_0):=\{x:\rhom(x-x_0,t)-u(x,t)>0\} $$ is connected for all $t,m>0$ and $x_0\in\bfR$, where $\rhom$ is the fundamental solution with mass $m>0$. This geometric statement is shown to be equivalent to the previously mentioned one-sided inequalities and used to obtain uniqueness and TV boundedness of conservation laws without convexity assumption. Multi-dimensional extension for the heat equation is also given.
We consider viscosity solutions to nonlinear uniformly parabolic equations in nondivergence form on a Riemannian manifold $M$, with the sectional curvature bounded from below by $-κ$ for $κ\geq 0$. In the elliptic case, Wang and Zhang \cite{WZ} recently extended the results of \cite{Ca} to nonlinear elliptic equations in nondivergence form on such $M$, where they obtained the Harnack inequality for classical solutions. We establish the Harnack inequality for nonnegative {\it viscosity solutions} to nonlinear uniformly {\it parabolic equations} in nondivergence form on $M$. The Harnack inequali…
The weak Harnack inequality for $L^p$-viscosity supersolutions of fully nonlinear second-order uniformly parabolic partial differential equations with unbounded coefficients and inhomogeneous terms is proved. It is shown that Hölder continuity of $L^p$-viscosity solutions is derived from the weak Harnack inequality for $L^p$-viscosity supersolutions. The local maximum principle for $L^p$-viscosity subsolutions and the Harnack inequality for $L^p$-viscosity solutions are also obtained. Several further remarks are presented when equations have superlinear growth in the first space derivatives.
Following the ideas of Andrei Lerner in [ A pointwise estimate for the local sharp maximal function with applications to singular integrals" Bull. London Math. Soc. 42 (2010) 843856], we obtain another decomposition of an arbitrary measurable function f in terms of local mean oscillations. This allows us to get new estimates involving one-sided singular integrals and one-sided maximal operator. As an application to this result we obtain two weighted inequality for one-sided singular integrals and a L(w) inequality relating a measurable function f and sharp one-sided operator. These estimates a…
We provide regularity results at the boundary for continuous viscosity solutions to nonconvex fully nonlinear uniformly elliptic equations and inequalities in Euclidian domains. We show that (i) any solution of two sided inequalities with Pucci extremal operators is $C^{1,α}$ on the boundary; (ii) the solution of the Dirichlet problem for fully nonlinear uniformly elliptic equations is $C^{2,α}$ on the boundary; (iii) corresponding asymptotic expansions hold. This is an extension to viscosity solutions of the classical Krylov estimates for smooth solutions.
On a smooth bounded domain Ω\subset R^N we consider the Schrödinger operators -Δ-V, with V being either the critical borderline potential V(x)=(N-2)^2/4 |x|^{-2} or V(x)=(1/4) dist (x,\partialΩ)^{-2}, under Dirichlet boundary conditions. In this work we obtain sharp two-sided estimates on the corresponding heat kernels. To this end we transform the Scrödinger operators into suitable degenerate operators, for which we prove a new parabolic Harnack inequality up to the boundary. To derive the Harnack inequality we have established a serier of new inequalities such as improved Hardy, logarithmic …
Two-sided bounds for the efficiency of the torsion function are obtained in terms of the square of the distance to the boundary function under the hypothesis that the Dirichlet Laplacian satisfies a strong Hardy inequality. Localisation properties of the torsion function are obtained under that hypothesis. An example is analysed in detail.
We prove an abstract result ensuring that one-sided geometric control yields two-sided estimates for functions satisfying general conditions. Our findings resonate in the context of nonlinear elliptic problems, including supersolutions to fully nonlinear elliptic equations and functions in the De Giorgi class. Among the consequences of our abstract results are regularity estimates, and conditions for a continuous function to be in the class of viscosity solutions. We also prove that one-sided geometric control yields $L^pL^\infty$-estimates. It provides a converse to the implication in the De …
We study the existence, uniqueness and regularity of solutions to the $n$-dimensional ($n=2,3$) Camassa-Holm equations with fractional Laplacian viscosity with smooth initial data. It is a coupled system between the Navier-Stokes equations with nonlocal viscosity and a Helmholtz equation. The main difficulty lies in establishing some a priori estimates for the fractional Laplacian viscosity. To achieve this, we need to explore suitable fractional-power Sobolev-type estimates, and bilinear estimates for fractional derivatives. Especially, for the critical case $\displaystyle s=\frac{n}{4}$ with…
In this paper we prove the fractional Gagliardo-Nirenberg inequality on homogeneous Lie groups. Also, we establish weighted fractional Caffarelli-Kohn-Nirenberg inequality and Lyapunov-type inequality for the Riesz potential on homogeneous Lie groups. The obtained Lyapunov inequality for the Riesz potential is new already in the classical setting of $\mathbb{R}^{N}$. As an application, we give two-sided estimate for the first eigenvalue of the Riesz potential. Also, we obtain Lyapunov inequality for the system of the fractional $p$-sub-Laplacian equations and give an application to estimate it…
In this paper, we study fully non-linear elliptic equations in non-divergence form which can be degenerate when "the gradient is small". Typical examples are either equations involving the $m$-Laplace operator or Bellman-Isaacs equations from stochastic control problems. We establish an Alexandroff-Bakelman-Pucci estimate and we prove a Harnack inequality for viscosity solutions of such degenerate elliptic equations.
In this paper, we investigate the global well-posedness of three-dimensional Navier-Stokes equations with horizontal viscosity under a special symmetric structure: helical symmetry. More precisely, by a revised Ladyzhenskaya-type inequality and utilizing the behavior of helical flow, we prove the global existence and uniqueness of weak and strong solution to the three-dimensional helical flows. Our result reveals that for the issue of global well-posedness of the viscous helical fluids, the horizontal viscosity plays the important role. To some extent, our work can be seen as a generalization …
Balanced Viscosity solutions to rate-independent systems arise as limits of regularized rate-independent flows by adding a superlinear vanishing-viscosity dissipation. We address the main issue of proving the existence of such limits for infinite-dimensional systems and of characterizing them by a couple of variational properties that combine a local stability condition and a balanced energy-dissipation identity. A careful description of the jump behavior of the solutions, of their differentiability properties, and of their equivalent representation by time rescaling is also presented. Our tec…
This paper is concerned with existence of a $C^α$ viscosity solution of a second order non-translation invariant integro-PDE. We first obtain a weak Harnack inequality for such integro-PDE. We then use the weak Harnack inequality to prove Hölder regularity and existence of solutions of the integro-PDEs.
We study global well-posedness of strong solutions for the nonhomogeneous Navier-Stokes equations with density-dependent viscosity and initial density allowing vanish in $\mathbb{R}^2$. Applying a logarithmic interpolation inequality and delicate energy estimates, we show the global existence of a unique strong solution provided that $\|\nablaμ(ρ_0)\|_{L^q}$ is suitably small, which improves the previous result of Huang and Wang [SIAM J. Math. Anal. 46, 1771--1788 (2014)] to the whole space case. Moreover, we also derive exponential decay rates of the solution. In particular, there is no need …
In this paper, we consider the initial-boundary value problem of the 3D primitive equations for planetary oceanic and atmospheric dynamics with only horizontal eddy viscosity in the horizontal momentum equations and only horizontal diffusion in the temperature equation. Global well-posedness of strong solution is established for any $H^2$ initial data. An $N$-dimensional logarithmic Sobolev embedding inequality, which bounds the $L^\infty$ norm in terms of the $L^q$ norms up to a logarithm of the $L^p$-norm, for $p>N$, of the first order derivatives, and a system version of the classic Gronwal…
We prove interior $C^{1,α}$-regularity for solutions \[ - Λ\leq F(D^2 u) \leq Λ\] where $Λ$ is a constant and $F$ is fully nonlinear, 1-homogeneous, uniformly elliptic. The proof is based on a reduction to the homogeneous equation $F(D^2u) = 0$ by a blow-up argument -- i.e. just like what is done in the case of viscosity solutions $F(D^2 u) = f$ for $f \in L^\infty$. However it was not clear to us that the above inequality implies $F(D^2 u) = f$ for some bounded $f$ (as would be the case for linear equations in distributional sense by approximation). Nor were we able to find the literature on …
A multi-point maximum principle to prove the global Harnack inequality for the doubly nonlinear heat equation
Mathematical Sciences Institute, Australian National University（澳大利亚国立大学数学科学学院）；Brandenburg University of Technology Cottbus–Senftenberg（勃兰登堡工业大学科特布斯-森夫滕贝格分校）；School of Mathematics and Statistics, The University of Sydney（悉尼大学数学与统计学院）
We prove that convex viscosity solutions to the quadratic Hessian inequality $σ_2(D^2u) \geq 1$ are strictly $2$-convex. As a consequence we obtain short proofs of smoothness and interior $C^2$ estimates for convex viscosity solutions to $σ_2(D^2u) = 1$, which were proven using different methods in recent works of Guan-Qiu \cite{GQ}, McGonagle-Song-Yuan \cite{MSY} and Shankar-Yuan \cite{SY2}.
Global well-posedness and exponential decay of 2D nonhomogeneous Navier-Stokes and magnetohydrodynamic equations with density-dependent viscosity and vacuum
In order to find a better physical model to describe the large-scale cloud-water transformation and rainfall, we consider a moist atmosphere model consisting of the primitive equations with only horizontal viscosity in the dynamic equation and a set of humidity equations describing water vapor, rain water and cloud condensates. To overcome difficulties caused by the absence of vertical viscosity in the dynamic equation, we get the local existence of $v$ in $H^{1}$ space by combining the viscous elimination method and the $z-$weak solution method and using the generalized Bihari-Lasalle inequal…
We study the boundary behavior of viscosity nonnegative solutions of fully nonlinear parabolic Pucci extremal operators. We establish local and global comparison theorems in $C^{1,1} cylinders, along with a backward Harnack inequality.
Gradient-flow characterizations of one-dimensional quasistatic viscoelasticity with Bhattacharya-like viscosity
Weierstraß-Institut für Angewandte Analysis und Stochastik（魏尔斯特拉斯应用分析与随机性研究所）；Maxwell Institute for Mathematical Sciences, Department of Mathematics, Heriot-Watt University（赫瑞-瓦特大学数学系麦克斯韦数学科学研究所）
We show that for a certain family of initial data, there exist non-unique weak solutions to the 3D incompressible Euler equations satisfying the weak energy inequality, whereas the weak limit of every sequence of Leray-Hopf weak solutions for the Navier-Stokes equations, with the same initial data, and as the viscosity tends to zero, is uniquely determined and equals the shear flow solution of the Euler equations. This simple example suggests that, also in more general situations, the vanishing viscosity limit of the Navier-Stokes equations could serve as a uniqueness criterion for weak soluti…
We prove the uniqueness for viscosity solutions of a differential equation involving the infinity-Laplacian with a variable exponent. A version of the Harnack's inequality is derived for this minimax problem.
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 kno…
We give a simple proof of the strong maximum principle for viscosity subsolutions of fully nonlinear elliptic PDEs on the form $$ F(x,u,Du,D^2u) = 0 $$ under suitable structure conditions on the equation allowing for non-Lipschitz growth in the gradient terms. In case of smooth boundaries, we also prove the Hopf lemma, the boundary Harnack inequality and that positive viscosity solutions vanishing on a portion of the boundary are comparable with the distance function near the boundary. Our results apply to weak solutions of an eigenvalue problem for the variable exponent $p$-Laplacian.
We study the compressible quantum Navier-Stokes (QNS) equations with degenerate viscosity in the three dimensional periodic domains. On the one hand, we consider QNS with additional damping terms. Motivated by the recent works [Li-Xin, arXiv:1504.06826] and [Antonelli-Spirito, Arch. Ration. Mech. Anal., 203(2012), 499--527], we construct a suitable approximate system which has smooth solutions satisfying the energy inequality and the BD entropy estimate. Using this system, we obtain the global existence of weak solutions to the compressible QNS equations with damping terms for large initial da…
Two-sided Gaussian estimates for the fundamental solution of a second order linear parabolic differential equation are upper and lower bounds in terms of the fundamental solution of the classical heat conduction equation. In his seminal 1958 paper Nash stated, without proof, two-sided non-Gaussian bounds for the fundamental solution of a uniformly parabolic divergence structure equation assuming only boundedness of the coefficients. In his 1967-1968 papers Aronson derived truly Gaussian estimates for the fundamental solutions of a large class of linear parabolic equations (including the diverg…
We study the Krylov-Safonov type Hölder estimate and Harnack inequality for viscosity solutions satisfying a uniformly parabolic equation only where the gradient with respect to the space-time variables is large.
We give a generalization of a theorem of Bôcher for the Laplace equation to a class of conformally invariant fully nonlinear degenerate elliptic equations. We also prove a Harnack inequality for locally Lipschitz viscosity solutions and a classification of continuous radially symmetric viscosity solutions.
An important set of theorems in geometric analysis consists of constant rank theorems for a wide variety of curvature problems. In this paper, for geometric curvature problems in compact and non-compact settings, we provide new proofs which are both elementary and short. Moreover, we employ our method to obtain constant rank theorems for homogeneous and non-homogeneous curvature equations in new geometric settings. One of the essential ingredients for our method is a generalization of a differential inequality in a viscosity sense satisfied by the smallest eigenvalue of a linear map (Brendle-C…
In this paper, we first define the notion of viscosity solution for the following system of partial differential equations involving a subdifferential operator:\[\{[c]{l}\dfrac{\partial u}{\partial t}(t,x)+\mathcal{L}_tu(t,x)+f(t,x,u(t,x))\in\partialϕ(u(t,x)),\quad t\in[0,T),x\in\mathbb{R}^d, u(T,x)=h(x),\quad x\in\mathbb{R}^d,\] where $\partialϕ$ is the subdifferential operator of the proper convex lower semicontinuous function $ϕ:\mathbb{R}^k\to (-\infty,+\infty]$ and $\mathcal{L}_t$ is a second differential operator given by $\mathcal{L}_tv_i(x)={1/2}\operatorname {Tr}[σ(t,x)σ^*(t,x)\mathrm…
In this paper, we consider the 3D primitive equations of oceanic and atmospheric dynamics with only horizontal eddy viscosities in the horizontal momentum equations and only vertical diffusivity in the temperature equation. Global well-posedness of strong solutions is established for any initial data such that the initial horizontal velocity $v_0\in H^2(Ω)$ and the initial temperature $T_0\in H^1(Ω)\cap L^\infty(Ω)$ with $\nabla_HT_0\in L^q(Ω)$, for some $q\in(2,\infty)$. Moreover, the strong solutions enjoy correspondingly more regularities if the initial temperature belongs to $H^2(Ω)$. The …
We consider a space-continuous and time-discrete polymer model for positive temperature and the associated zero temperature model of last passage percolation type. In our previous work, we constructed and studied infinite-volume polymer measures and one-sided infinite minimizers for the associated variational principle, and used these objects for the study of global stationary solutions of the Burgers equation with positive or zero viscosity and random kick forcing, on the entire real line. In this paper, we prove that in the zero temperature limit, the infinite-volume polymer measures concent…
In this paper, we consider the initial-boundary value problem of the three-dimensional primitive equations for oceanic and atmospheric dynamics with only horizontal viscosity and horizontal diffusivity. We establish the local, in time, well-posedness of strong solutions, for any initial data $(v_0, T_0)\in H^1$, by using the local, in space, type energy estimate. We also establish the global well-posedness of strong solutions for this system, with any initial data $(v_0, T_0)\in H^1\cap L^\infty$, such that $\partial_zv_0\in L^m$, for some $m\in(2,\infty)$, by using the logarithmic type anisot…
