Fausto Ferrari
We prove Hölder continuous regularity of bounded, uniformly continuous, viscosity solutions of degenerate fully nonlinear equations defined in all of $\mathbb{R}^n$ space. In particular the result applies also to some operators in Carnot groups.
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In this work, we study some properties of the viscosity solutions to a degenerate parabolic equation involving the non-homogeneous infinity-Laplacian.
On equivalence of entropy and viscosity solutions to degenerate parabolic equations and applications
该研究证明了各向异性退化抛物-双曲方程与退化拟线性Hamilton-Jacobi方程的熵解和粘性解的等价性，并应用该结论得到周期情形下两类方程对应解的大时间行为。
This paper is concerned with semiconcavity of viscosity solutions for a class of degenerate elliptic integro-differential equations in $\mathbb R^n$. This class of equations includes Bellman equations containing operators of Lévy-Itô type. Hölder and Lipschitz continuity of viscosity solutions for a more general class of degenerate elliptic integro-differential equations are also provided.
We introduce a notion of viscosity solutions for a nonlinear degenerate diffusion equation with a drift potential. We show that our notion of solutions coincide with the weak solutions defined via integration by parts. As an application of the viscosity solutions theory, we show that the free boundary uniformly converges to the equilibrium as time grows. In the case of a convex potential, an exponential rate of free boundary convergence is obtained.
We study Phragmén-Lindelöf properties of viscosity solutions to a class of doubly nonlinear parabolic equations in $\mathbb{R}^n\times (0,T)$. We also include an application to some doubly nonlinear equations.
We are concerned with fully nonlinear possibly degenerate elliptic partial differential equations (PDEs) with superlinear terms with respect to $Du$. We prove several comparison principles among viscosity solutions which may be unbounded under some polynomial-type growth conditions. Our main result applies to PDEs with convex superlinear terms but we also obtain some results in nonconvex cases. Applications to monotone systems of PDEs are given.
We consider a scalar, possibly degenerate parabolic equation with a source term, in several space dimensions. For initial data with bounded variation we prove the existence of solutions to the initial-value problem. Then we show that these solutions converge, in the vanishing-viscosity limit, to the Kruzhkov entropy solution of the corresponding hyperbolic equation. The proof exploits the H-measure compactness in several space dimensions.
The paper is concerned with the vanishing viscosity limit of the two-dimensional degenerate viscous lake equations when the Navier slip conditions are prescribed on the impermeable boundary of a simply connected bounded regular domain. When the initial vorticity is in the Lebesgue space $L^q$ with $2<q\le\infty$, we show the degenerate viscous lake equations possess a unique global solution and the solution converges to a corresponding weak solution of the inviscid lake equations. In the special case when the vorticity is in $L^\infty$, an explicit convergence rate is obtained.
We consider the functional $$J(v) = \int_Ω[f(|\nabla v|) - v] dx,$$ where $Ω$ is a bounded domain and $f:[0,+\infty)\to \mathbb{R}$ is a convex function vanishing for $s\in [0,σ]$, with $σ>0$. We prove that a minimizer $u$ of $J$ satisfies an equation of the form $$\min(F(\nabla u, D^2 u), |\nabla u|-σ)=0$$ in the viscosity sense.
We are concerned about the barotropic compressible Navier-Stokes equations with density-dependent viscosities which may degenerate in vacuum. We show that any classical solution to barotropic compressible Navier-Stokes equations in bounded domains will blow up, when the initial density admits an isolated mass group and the viscousity coefficients satisfy some conditions. A new condition on viscosities is first put forward in this paper.
In this paper, we investigate the asymptotic behavior of solutions toward a multiwave pattern of the Cauchy problem for the scalar viscous conservation law where the far field states are prescribed. Especially, we deal with the case when the flux function is convex or concave but linearly degenerate on some interval, and also the viscosity is a nonlinearly degenerate one (p-Laplacian type viscosity). When the corresponding Riemann problem admits a Riemann solution which consists of rarefaction waves and contact discontinuity, it is proved that the solution of the Cauchy problem tends toward th…
We deal with the equations of a planar magnetohydrodynamic compressible flow with the viscosity depending on the specific volume of the gas and the heat conductivity proportional to a positive power of the temperature. Under the same conditions on the initial data as those of the constant viscosity and heat conductivity case ([Kazhikhov (1987)], we obtain the global existence and uniqueness of strong solutions which means no shock wave, vacuum, or mass or heat concentration will be developed in finite time, although the motion of the flow has large oscillations and the interaction between the …
In this paper, we study a free boundary problem for compressible spherically symmetric Navier-Stokes equations with a gravitational force and degenerate viscosity coefficients. Under certain assumptions that imposed on the initial data, we obtain the global existence and uniqueness of the weak solution and give some uniform bounds (with respect to time) of the solution. Moreover, we obtain some stabilization rate estimates in $L^\infty$-norm and weighted $H^1$-norm of the solution. The results show that such system is stable under the small perturbations, and could be applied to the astrophysi…
Global Well-Posedness for Planar Compressible Magnetohydrodynamics with Degenerate Viscosity on Unbounded Domains
本文研究无界域上一维平面可压缩磁流体动力学方程，在黏性退化且热导率幂律下，对任意大初值建立全局强解的存在唯一性，并证明解随时间趋于平衡态。
We consider an initial value problem for a Hamilton--Jacobi equation with a quadratic and degenerate Hamiltonian. Our Hamiltonian comes from the dynamics of $N$-peakon in the Camassa--Holm equation. It is given by a quadratic form with a singular positive semi-definite matrix. Such a problem does not fall into the standard theory of viscosity solutions. Also viability related results, sometimes used to deal with degenerate Hamiltonians, do not seem applicable in our case. We prove the global existence of a viscosity solution by looking at the associated optimal control problem and showing that…
We consider the compressible Navier-Stokes system where the viscosity depends on density and the heat conductivity is proportional to a positive power of the temperature under stress-free and thermally insulated boundary conditions. Under the same conditions on the initial data as those of the constant viscosity and heat conductivity case ([Kazhikhov-Shelukhin. J. Appl. Math. Mech. 41 (1977)], we obtain the existence and uniqueness of global strong solutions. Our result can be regarded as a natural generalization of the Kazhikhov's theory for the constant heat conductivity case to the degenera…
The objective of this paper is twofold. First, we show the existence of global classical solutions to the degenerate inviscid lake equations. This result is achieved after revising the elliptic regularity for a degenerate equation on the associated stream-function, and adapting the method used for construction of classical solutions to the incompressible Euler equations. Second, we show that the weak solutions of the viscous lake equations converge to classical solutions of the inviscid lake equations when the viscosity coefficient goes to zero, which constitutes an important physical validati…
We study a version of the strong minimum principle, and large time asymptotics of positive viscosity solutions to classes of doubly nonlinear parabolic equations of the form $$ H(Du,D^2u)-u^{k-1}u_t=0,\;\;k\geq 1,\quad\mbox{in $Ω\times [0,T)$},$$ where $Ω\subset \mathbb{R}^n$ is a bounded domain and $0<T\leq \infty$. The spatial operator $H$ is homogeneous with power $k$.
In this paper we propose a new type of viscosity solutions for fully nonlinear path dependent PDEs. By restricting to certain pseudo Markovian structure, we remove the uniform non- degeneracy condition imposed in our earlier works [9, 10]. We establish the comparison principle under natural and mild conditions. Moreover, as applications we apply our results to two important classes of PPDEs: the stochastic HJB equations and the path dependent Isaacs equations, induced from the stochastic optimization with random coefficients and the path dependent zero sum game problem, respectively.
Nonlinear Stability of Linearly Expanding Goldreich-Weber Solutions for the Navier-Stokes-Poisson System with Degenerate Viscosity Under Radial Perturbations
该研究针对带退化黏性的引力纳维-斯托克斯-泊松系统，在特定剪切黏性条件下，证明了线性膨胀Goldreich-Weber解在径向扰动下的非线性稳定性。
We use Velocity Averaging lemma to show that the almost everywhere limit of quasilinear viscous approximations is the unique entropy solution (in the sense of {\it F. Otto}) of the corresponding scalar conservation laws on a bounded domain in $\mathbb{R}^{d}$, where the viscous term is of the form $\varepsilon\,div\left(B(u^{\varepsilon})\nabla u^{\varepsilon}\right)$ and $B\geq 0$.
Existence and nonexistence of viscosity solutions for a class of degenerate/singular eigenvalue type equations
本文研究退化/奇异本征值类型方程粘性解的存在性与不存在性，通过距离函数分析并推导全局估计。
The long-time regularity and asymptotic of weak solutions are studied for compressible Navier-Stokes equations with degenerate viscosity in a bounded periodic domain in two and three dimensions. It is shown that the density keeps strictly positive from below and above after a finite period of time. Moreover, higher velocity regularity is obtained via a parabolic type iteration technique. Since then the weak solution conserves its energy equality, and decays exponentially to the equilibrium in $L^{2}$-norm as time goes to infinity. In addition, assume that the initial momentum is zero, the expo…
We prove weak-strong uniqueness results for the compressible Navier-Stokes system with degenerate viscosity coefficient and with vacuum in one dimension. In other words, we give conditions on the weak solution constructed in \cite{Jiu} so that it is unique. The novelty consists in dealing with initial density $ρ_0$ which contains vacuum. To do this we use the notion of relative entropy developed recently by Germain, Feireisl et al and Mellet and Vasseur (see \cite{PG,Fei,15}) combined with a new formulation of the compressible system (\cite{cras,CPAM,CPAM1,para}) (more precisely we introduce a…
In this work, we show existence and uniqueness of positive solutions of $H(Du, D^2u)+χ(t)|Du|^Γ-f(u)u_t=$ in $Ω\times(0, T)$ and $u=h$ on its parabolic boundary. The operator $H$ satisfies certain homogeneity conditions, $Γ>0$ and depends on the degree of homogeneity of $H$, $f>0$, increasing and meets a concavity condition. We also consider the case $f\equiv 1$ and prove existence of solutions without sign restrictions.
Stability of viscous shock profile for convective porous-media flow with degenerate viscosity
本文研究退化粘性对流多孔介质方程的粘性激波大时间行为，通过利用扰动符号并分解积分区域，证明激波在小扰动下的非线性稳定性并得到 $L^\infty$ 衰减率。
We consider Navier-Stokes equations for compressible viscous fluids in one dimension. We prove the existence of global strong solution with large initial data for the shallow water system. The key ingredient of the proof relies to a new formulation of the compressible equations involving a new effective velocity $v$ (see \cite{cras,para,CPAM,CPAM1}) such that the density verifies a parabolic equation. We estimate $v$ in $L^\infty$ norm which enables us to control the vacuum on the density.
The earlier paper [LK] (arXiv:0910.3432) contains a lower bound of the solution in terms of its $L^1$ norm, which is incorrect. In this note we explain the mistake and present a correction to it under the restriction that the permeability constant $m$ satisfies $1< m <2$. As a consequence, the quantitative estimates on the convergence rate (Main Theorem (c) and Theorem 3.6 in [LK] only hold for $1<m<2$. For $m\geq 2$ a partial convergence rate is obtained.
Boundary Regularity for viscosity solutions of Fully nonlinear degenerate/singular parabolic equations
本文研究了完全非线性退化/奇异抛物型方程粘性解的边界正则性，证明了边界 $C^{1,α}$ 和 $C^{2,α}$-正则性，并导出了全局正则性结果及 Cauchy-Dirichlet 问题的可解性。
In this paper, we study the decay rate in time to solutions of the Cauchy problem for the one-dimensional viscous conservation law where the far field states are prescribed. Especially, we deal with the case that the flux function which is convex and also the viscosity is a nonlinearly degenerate one (p-Laplacian type viscosity). As the corresponding Riemann problem admits a Riemann solution as the constant state or the single rarefaction wave, it has already been proved by Matsumura-Nishihara that the solution to the Cauchy problem tends toward the constant state or the single rarefaction wav…
Convergence Rates for Vanishing Viscosity Approximations of Possibly Degenerate Viscous Hamilton--Jacobi Equations
研究平坦环面上可能退化的粘性哈密顿 - 雅可比方程的消失粘性逼近的收敛速度，在哈密顿量标准假设下，先证逐点$O(\epsilon|\log\epsilon|)$收敛速度，后证测试误差时对数 Loss 可去，有平均$O(\epsilon)$速度，基于非线性伴随方法等证明。
In this paper, we are concerned with the Cauchy problem for one-dimensional compressible isentropic Navier-Stokes equations with density-dependent viscosity $μ(ρ)=ρ^α(α>0)$ and pressure $P(ρ)=ρ^γ\ (γ>1)$. We will establish the global existence and asymptotic behavior of weak solutions for any $α>0$ and $γ>1$ under the assumption that the density function keeps a constant state at far fields. This enlarges the ranges of $α$ and $γ$ and improves the previous results presented by Jiu and Xin. As a result, in the case that $0<α<\frac12$, we obtain the large time behavior of the strong solution obt…
In this paper, we study the precise decay rate in time to solutions of the Cauchy problem for the one-dimensional conservation law with a nonlinearly degenerate viscosity where the far field states are prescribed. Especially, we deal with the case when the flux function is convex or concave but linearly degenerate on some interval. As the corresponding Riemann problem admits a Riemann solution as a multiwave pattern which consists of the rarefaction waves and the contact discontinuity, it has already been proved by Yoshida that the solution to the Cauchy problem tends toward the linear combina…
On blow-up to the one-dimensional Navier-Stokes equations with degenerate viscosity and vacuum
本文研究一维具有退化黏性和真空的等熵可压缩Navier-Stokes方程，证明当初始数据含孤立质量群时正则解有限时间爆破，并利用内蕴变量输运结构获得密度L∞界以去除δ≤γ限制。
On the Global Strong Solutions to Magnetohydrodynamics with Density-Dependent Viscosity and Degenerate Heat-Conductivity in Unbounded Domains
该研究针对一维无界区域中具有密度依赖粘性和退化热传导的平面可压缩MHD方程，证明了柯西与初边值问题唯一强解的全局存在性，推广了经典Navier-Stokes系统的相关结论，明确有限时间内不会出现激波、真空及质量或热集中。
In the first main result of this paper we prove that one can approximate discontinious solutions of the 1d Navier Stokes system with solutions of the 1d Navier-Stokes-Korteweg system as the capilarity parameter tends to 0. Moreover, we allow the viscosity coefficients $μ$ = $μ$ ($ρ$) to degenerate near vaccum. In order to obtain this result, we propose two main technical novelties. First of all, we provide an upper bound for the density verifing NSK that does not degenerate when the capillarity coefficient tends to 0. Second of all, we are able to show that the positive part of the effective v…
Using a generalized assumption of Osgood type, we prove a comparison result between viscosity sub and supersolutions of fully nonlinear, possibly strongly degenerate, parabolic equations under rather generale assumtpions. The principle allows to consider dependence on the gradient witha quadratic growth and also on complex zeroth-order terms. The result it applies in particular to a differential model for pricing Mortgage-Backed Securities.
We prove a comparison result for viscosity solutions of (possibly degenerate) parabolic fully nonlinear path-dependent PDEs. In contrast with the previous result in Ekren, Touzi & Zhang, our conditions are easier to check and allow for the degenerate case, thus including first order path-dependent PDEs. Our argument follows the regularization method as introduced by Jensen, Lions & Souganidis in the corresponding finite-dimensional PDE setting. The present argument significantly simplifies the comparison proof in Ekren, Touzi & Zhang, but requires an $L^p-$type of continuity (with respect to t…
Inviscid Incompressible Limit for Degenerate Compressible Navier-Stokes Equations on Expanding Domains
研究三维退化可压缩Navier-Stokes方程在扩展域上的无粘和低马赫数极限，从非准备充分数据出发，证明极限系统是不可压缩欧拉系统，扩展了前人结果，首次同时建立依赖密度粘性的无粘和不可压缩极限。
A viscosity approach is introduced for the Dirichlet problem associated to complex Hessian type equations on domains in $\C^n$. The arguments are modelled on the theory of viscosity solutions for real Hessian type equations developed by Trudinger. As consequence we solve the Dirichlet problem for the Hessian quotient and special Lagrangian equations. We also establish basic regularity results for the solutions.
We introduce the concept of k-strictly convexity to describe the accurate convexity of convex domains some directions of which boundary may be flat. Basing this accurate convexity, we construct sub-solutions the Dirichlet problem for some degenerate-singular Monge-Ampère type equations and prove the sharp boundary estimates for convex viscosity solutions of the problem. As a result, we obtain the optimal global Hölder regularity of the convex viscosity solutions.
In this paper we prove the Hölder regularity of bounded, uniformly continuous, viscosity solutions of some degenerate fully nonlinear equations in the Heisenberg group.
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…
Existence results for nonlinear cone degenerate Laplace equations
本文研究由锥退化拉普拉斯算子驱动的一类非散度型非线性椭圆方程，通过证明Alexandrov-Bakelman-Pucci估计和Hölder估计建立粘性解的存在性，并通过证明弱解与粘性解的等价性获得弱解的存在性。
We present comparison principles, Lipschitz estimates and study state constraints problems for degenerate, second-order Hamilton-Jacobi equations.
This is a truncated version of the paper "Degenerate diffusion with a drift potential: a viscosity solutions approach", co-authored with I. C. Kim. The purpose of this version is to withdraw the claim of quantitative rate of convergence of the free boundary on the part of H. K. Lei. The difference from the previous version lies in Section 3 where 1) the quantitative version of the convergence of the free boundary statement has been removed and 2) the more basic version of some convergence of the free boundary given uniform convergence of the function has been rewritten. It is emphasized that w…
Large time behaviour of solutions to the $N-$dimensional scalar conservation law under periodic perturbations with nonlinear degenerate viscosity
研究含非线性退化粘性的 $N$ 维标量守恒律，证明周期初值下解的存在唯一性、正则性与衰减率，并分析周期扰动趋近平面稀疏波的长时间行为。
Existence and uniqueness are investigated for a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions. This system aims to model two-species phase segregation on an atomic lattice; in the balance equations of microforces and microenergy, the two unknowns are the order parameter and the chemical potential. A simpler version of the same system has recently been discussed in arXiv:1103.4585v1. In this paper, a fairly more general phase-field equation is…
Existence and uniqueness for cone degenerate p-Laplace equation
针对锥退化 p-Laplace 方程，该研究借助 ABP 与 Hölder 估计、等价变换及弱解—粘性解关系，建立粘性解和弱解的存在性及比较原理。
Gradient regularity for degenerate fully nonlinear free transmission problems with Hamiltonian terms
研究含哈密顿项的退化完全非线性自由传输问题，通过置于粘性不等式背景建立梯度局部赫尔德正则性，基于新估计与局部分析获逐点\(C^{1,\alpha}\)正则性。
Gradient regularity for viscosity solutions to quasilinear parabolic equations with mixed singular-degenerate structure
本文研究了一类具有混合退化-奇异结构的拟线性抛物型方程粘性解的正则性，通过Jensen-Ishii方法和内在尺度技术，证明了解的Lipschitz正则性和梯度的Hölder连续性。
We establish that a viscosity solution to a multidimensional Hamilton-Jacobi equation with a convex non-degenerate hamiltonian and Bohr almost periodic initial data decays to its infimum as time $t\to+\infty$.
We study cavitating self-similar solutions to compressible Navier-Stokes equations with degenerate density-dependent viscosity. We prove both existence of expanders and non-existence of small shrinkers.
We simplify and give an alternative proof of hypoellipticity for generalizations of the singular sum of squares of complex vector fields studied by Kohn, with an appendix by Derridj and Tartakoff, in the Annals of Mathematics, vol. 162 no. 2, 2005, pp. 943-986. The main generalization is that the complex vector fields now come from domains of finite type. We also prove real analytic hypoellipticity and the optimality of our estimates.
Gradient regularity for fully nonlinear equations with degenerate coefficients
针对在超曲面上退化的完全非线性方程，推导了粘性解的$C^{1,α}$正则性估计。
In this paper, we investigate the compressible Navier-Stokes equations with degenerate, density-dependent, viscosity coefficient driven by multiplicative stochastic noise. We consider three-dimensional periodic domain and prove that the family of weak martingale solutions is sequentially compact.
Improved regularity estimates for degenerate or singular fully nonlinear dead-core systems and Hénon-type equations
本文研究退化或奇异全非线性死核系统与强吸收项的耦合问题，提出粘性解在自由边界处的正则性提升、非退化性、自由边界测度估计、Liouville型结果及解的发散行为，同时推导了退化权重和强吸收下Hénon型方程粘性解的精确正则性估计。
We study a free transmission problem driven by degenerate fully nonlinear operators. Our first result concerns the existence of solutions to the associated Dirichlet problem. By framing the equation in the context of viscosity inequalities, we produce optimal regularity results for viscosity solutions and certain strong solutions to the problem. Our findings include regularity in $ C^{1,α}$ spaces, and an explicit characterization of $α$ in terms of the degeneracy rates. We argue by perturbation methods, relating our problem to a homogeneous, fully nonlinear uniformly elliptic equation.
This paper concerns the existence of global weak solutions to the barotropic compressible Navier-Stokes equations with degenerate viscosity coefficients. We construct suitable approximate system which has smooth solutions satisfying the energy inequality, the BD entropy one, and the Mellet-Vasseur type estimate. Then, after adapting the compactness results due to Mellet-Vasseur [Comm. Partial Differential Equations 32 (2007)], we obtain the global existence of weak solutions to the barotropic compressible Navier-Stokes equations with degenerate viscosity coefficients in two or three dimensiona…
This is the content of the lectures given by the author at the winter school KAWA3 held at the University of Barcelona in 2012 from January 30 to February 3. The main goal was to give an account of viscosity techniques and to apply them to degenerate Complex Monge-Ampère equations following recent works of P. Eyssidieux, V. Guedj and the author. We will survey the main techniques used in the viscosity approach and show how to adapt them to degenerate complex Monge-Ampère equations. The heart of the matter in this approach is the "Comparison Principle" which allows us to prove uniqueness of sol…
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