我们研究了在最大$ L^p $-正则性框架下，偏抛物型拟线性演化方程解的连续依赖性。对于形式为$$ \frac{dϕ}{dt} + A(t,ϕ)ϕ= f(t,ϕ), $$的方程，我们建立了强解对初始数据以及非线性算子$ A $和$ f $的合适近似值的连续依赖性。证明主要结果的一个重要步骤是，当$ t $和$ ϕ $固定时，算子$ A(t,ϕ) $的最大正则性常数在相关巴拿赫空间的紧子集上允许统一界。作为应用，我们考虑了一类具有Carreau型粘度和混合边界条件的非牛顿流体模型。我们证明，当非线性贡献在粘度中消失且初始数据收敛时，非牛顿流体模型的解收敛到经典纳维-斯托克斯方程的解。
英文摘要：
We study continuous dependence of solutions to quasilinear evolution equations of parabolic-type in the framework of maximal $L^p$-regularity. For equations of the form \[ \frac{dϕ}{dt} + A(t,ϕ)ϕ= f(t,ϕ), \] we establish continuous dependence of strong solutions on initial data, and suitable approximations of the nonlinear operators $A$ and $f$. An important step for proving the main result is the fact that the maximal regularity constant of the operator $A(t,ϕ)$, with $t$ and $ϕ$ fixed, admits a uniform bound over compact subsets of the relevant Banach spaces.
As an application, we consider a class of non-Newtonian fluid models with a Carreau-type viscosity and mixed boundary conditions. We show that, as the nonlinear contribution in the viscosity vanishes and the initial data converge, solutions of the non-Newtonian fluid model converge to those of the classical Navier--Stokes equations.
In this paper, we prove existence and regularity of positive solutions for singular quasilinear elliptic systems involving gradient terms. Our approach is based on comparison properties, a priori estimates and Schauder's fixed point theorem.
We consider a class of abstract quasilinear parabolic problems with lower--order terms exhibiting a prescribed singular structure. We prove well--posedness and Lipschitz continuity of associated semiflows. Moreover, we investigate global existence of solutions and we extend the generalized principle of linearized stability to settings with initial values in critical spaces. These general results are applied to the surface diffusion flow in various settings.
Existence and uniqueness of weak solutions to quasilinear PDEs with critical data
Université Paris-Saclay（巴黎萨克雷大学）；France-Australia Mathematical Sciences and Interactions ANU - CNRS International Research Laboratory（法澳数学科学互动ANU-CNRS国际研究实验室）；CNRS Laboratoire de Mathématiques d’Orsay（法国国家科学研究中心奥赛数学实验室）
We find a solution of a quasilinear elliptic equation with Dirichlet's boundary condition on a smooth bounded domain and involving an unbounded continuous nonlinearity with oscillatory behavior near the origin.
In this paper study the regularity of continuous casting problem \begin{equation} \hbox{div}(|\nabla u|^{p-2}\nabla u-{\bf v} β(u))=0\tag{$\sharp$} \end{equation} for prescribed constant velocity $\bf v$ and enthalpy $β(u)$ with jump discontinuity at $u=0$. We establish the following estimates: local log-Lipschitz $p>2$ for $u$ (and BMO for $\nabla u$) for two phase, Lipschitz $p>1$ for one phase and linear growth up-to boundary near the contact points. We also prove that the free boundary is continuous curve in the direction of $\bf v$ in two spatial dimensions. The proof is based on a delica…
We prove a superposition principle in the spirit of Crandall-Zhang and Lindqvist-Manfredi for a class of second order quasilinear equations. Riesz potentials of nonnegative and compactly supported continuous functions are either subsolutions or supersolutions for the operators associated to the stationary form of the doubly nonlinear diffusion equation. This class of operators includes both the p-Laplacian as well as the stationary porous medium equation.
In this paper, we establish existence and multiplicity of solutions for the following class of quasilinear field equation $$ -Δu+V(x)u-Δ_{p}u+W'(u)=0, \,\,\, \mbox{in} \,\,\, \mathbb{R}^{N}, \eqno{(P)} $$ where $u=(u_1,u_2,...,u_{N+1})$, $p>N \geq 2$, $W$ is a singular function and $V$ is a positive continuous function.
Optimal estimates on asymptotic behaviors of weak solutions both at the origin and at the infinity are obtained to the following quasilinear elliptic equations \begin{eqnarray*} -Δ_{p}u-\fracμ{|x|^{p}}|u|^{p-2}u+m|u|^{p-2}u=f(u), & & x\in\R^{N}, \end{eqnarray*} where $1<p<N$, $0\leqμ<\left((N-p)/p\right)^{p}$, $m>0$ and $f$ is a continuous function.
We establish a comprehensive local wellposedness theory for the quasilinear Maxwell system with interfaces in the space of piecewise $H^m$-functions for $m \geq 3$. The system is equipped with instantaneous and piecewise regular material laws and perfectly conducting interfaces and boundaries. We also provide a blow-up criterion in the Lipschitz norm and prove the continuous dependence on the data. The proof relies on precise a priori estimates and the regularity theory for the corresponding linear problem also shown here.
Multiple solutions for a quasilinear Schrödinger equation on $\mathbb{R}^{N}$}
建立了拟线性薛定谔方程$-L_p u +(λA(x)+1)|u|^{p-2}u= h(u)$在$\mathbb{R}^N$中正弱解的多重性，其中$L_p u\doteq ε^{p}Δ_p u +ε^{p}Δ_p (u^2)u$，$A$是非负连续函数，非线性项$h$具有次临界增长.我们利用极大极小方法和Lusternik-Schnirelman的临界点理论得到了这些结果。
Existence of positive solution for a class of quasilinear Schrödinger equations with potential vanishing at infinity on nonreflexive Orlicz-Sobolev spaces
In this article we provide a local wellposedness theory for quasilinear Maxwell equations with absorbing boundary conditions in $\mathcal{H}^m$ for $m \geq 3$. The Maxwell equations are equipped with instantaneous nonlinear material laws leading to a quasilinear symmetric hyperbolic first order system. We consider both linear and nonlinear absorbing boundary conditions. We show existence and uniqueness of a local solution, provide a blow-up criterion in the Lipschitz norm, and prove the continuous dependence on the data. In the case of nonlinear boundary conditions we need a smallness assumpti…
This paper is concerned with the nodal set of weak solutions to a broken quasilinear partial differential equation, \begin{equation*} \mbox{div} (a_+ \nabla u^+ - a_- \nabla u^-) = \mbox{div} f, \end{equation*} where $a_+$ and $a_-$ are uniformly elliptic, Dini continuous coefficient matrices, subject to a strong correlation that $a_+$ and $a_-$ are a multiple of some scalar function to each other. Under such a structural condition, we develop an iteration argument to achieve higher-order approximation of solutions at a singular point, which is also new for standard elliptic PDEs below Hölder …
We study global regularity for solutions of quasilinear elliptic equations of the form $÷\A(x,u,\nabla u) = ÷\F $ in rough domains $Ω$ in $\R^n$ with nonhomogeneous Dirichlet boundary condition. The vector field $\A$ is assumed to be continuous in $u$, and its growth in $\nabla u$ is like that of the $p$-Laplace operator. We establish global gradient estimates in weighted Morrey spaces for weak solutions $u$ to the equation under the Reifenberg flat condition for $Ω$, a small BMO condition in $x$ for $\A$, and an optimal condition for the Dirichlet boundary data.
In this work, we prove the existence and multiplicity of positive solutions for the following class of quasilinear elliptic equations $$ \left \{ \begin{array}{lll} -ε^{N}Δ_{N} u + \left(1+μA(x) \right)\left| u\right|^{N-2}u= f(u)\,\,\,\, \mbox{ in } \, \RR^{N},\\ u > 0 \,\,\,\, \mbox{ in } \, \RR^{N}, \end{array} \right. $$ \\ where $Δ_{N}$ is the N-Laplacian operator, $N \geq 2$, $f$ is a function with exponential critical growth, $μ$ and $ε$ are positive parameters and $A$ is a nonnegative continuous function verifying some hypotheses. To obtain our results, we combine variational arguments…
In the sub-Riemannian setting of Carnot groups, this work investigates a-priori estimates and Liouville type theorems for solutions of coercive, quasilinear differential inequalities of the type $$ Δ_{\mathbb{G}}^φu \ge b(x) f(u) l(|\nabla u|). $$ Prototype examples of $Δ_{\mathbb{G}}^φ$ are the (subelliptic) $p$-Laplacian and the mean curvature operator. The main novelty of the present paper is that we allow a dependence on the gradient $l(t)$ that can vanish both as $t \rightarrow 0^+$ and as $t \rightarrow +\infty$. Our results improve on the recent literature and, by means of suitable coun…
We investigate quasilinear discrete PDEs $\partial_t u = Δ^N φ(u)+ Kf(u)$ of reaction-diffusion type with nonlinear diffusion term defined on an $n$-dimensional unit torus discretized with mesh size $\tfrac1N$ for $N\in {\mathbb N}$, where $Δ^N$ is the discrete Laplacian, $φ$ is a strictly increasing $C^5$ function and $f$ is a $C^1$ function. We establish $L^\infty$ bounds and space-time Hölder estimates, both uniform in $N$, of the first and second spatial discrete derivatives of the solutions. In the equation, $K>0$ is a large constant and we show how these estimates depend on $K$. The moti…
In this paper, we study the existence and regularity of the quasilinear parabolic equations: $$u_t-\operatorname{div}(A(x,t,\nabla u))=B(u,\nabla u)+μ,$$ in either $\mathbb{R}^{N+1}$ or $\mathbb{R}^N\times(0,\infty)$ or on a bounded domain $Ω\times (0,T)\subset\mathbb{R}^{N+1}$ where $N\geq 2$. In this paper, we shall assume that the nonlinearity $A$ fulfills standard growth conditions, the function $B$ is a continuous and $μ$ is a radon measure. Our first task is to establish the existence results with $B(u,\nabla u)=\pm|u|^{q-1}u$, for $q>1$. We next obtain global weighted-Lorentz, Lorentz-M…
We study quasilinear elliptic equations of the form $\text{div} \mathbf{A}(x,u,\nabla u) = \text{div}\mathbf{F} $ in bounded domains in $\mathbb{R}^n$, $n\geq 1$. The vector field $\mathbf{A}$ is allowed to be discontinuous in $x$, Lipschitz continuous in $u$ and its growth in the gradient variable is like the $p$-Laplace operator with $1<p<\infty$. We establish interior $W^{1,q}$-estimates for locally bounded weak solutions to the equations for every $q>p$, and we show that similar results also hold true in the setting of {\it Orlicz} spaces. Our regularity estimates extend results which are …
This paper concerns the existence of a solution for the following class of semipositone quasilinear problems \begin{equation*} \left \{ \begin{array}{rclcl} -Δ_p u = h(x)(f(u)-a),\ & u > 0 & \mbox{in} & \mathbb{R}^N, \end{array} \right. \end{equation*} where $1<p<N$, $a>0$, $ f:[0,+\infty) \to [0,+\infty)$ is a function with subcritical growth and $f(0)=0$, while $h:\mathbb{R}^N \to (0,+\infty)$ is a continuous function that satisfies some technical conditions. We prove via nonsmooth critical points theory and comparison principle, that a solution exists for $a$ small enough. We also provide a…
Symmetry of bounded solutions to quasilinear elliptic equations in a half-space
Faculty of Economic Mathematics, University of Economics and Law, Ho Chi Minh City, Vietnam（经济数学学院，经济与法律大学）；Vietnam National University, Ho Chi Minh City, Vietnam（越南国立大学）
We study the problem of existence and uniqueness of strong solutions to a degenerate quasilinear parabolic non-Newtonian thin-film equation. Originating from a non-Newtonian Navier--Stokes system the equation is derived by lubrication theory and under the assumption that capillarity is the only driving force. The fluid's shear-thinning rheology is described by the so-called Ellis constitutive law. For flow behaviour exponents $α\geq 2$ the corresponding initial boundary value problem fits into the abstract setting of [4,Thm. 12.1]. Due to a lack of regularity this is not true for flow behaviou…
We consider the elliptic quasilinear equation --$Δ$ m u = u p |$\nabla$u| q in R N with q $\ge$ m and p > 0, 1 < m < N. Our main result is a Liouville-type property, namely, all the positive C 1 solutions in R N are constant. We also give their asymptotic behaviour : all the solutions in an exterior domain R N \B r0 are bounded. The solutions in B r0 \ {0} can be extended as a continuous functions in B r0. The solutions in R N \ {0} has a finite limit l $\ge$ 0 as |x| $\rightarrow$ $\infty$. Our main argument is a Bernstein estimate of the gradient of a power of the solution, combined with a p…
We prove boundary higher integrability for the (spatial) gradient of \emph{very weak} solutions of quasilinear parabolic equations of the form $$u_t - \text{div}\,\mathcal{A}(x,t, \nabla u)=0 \quad \text{on} \ Ω\times \mathbb{R},$$ where the non-linear structure $\text{div}\,\mathcal{A}(x, t,\nabla u)$ is modelled after the $p$-Laplace operator. To this end, we prove that the gradients satisfy a reverse Hölder inequality near the boundary. In order to do this, we construct a suitable test function which is Lipschitz continuous and preserves the boundary values. \emph{These results are new even…
We prove existence and uniqueness of global-in-time solutions in the $W^{-1,p}_D$-$W^{1,p}_D$-setting for abstract quasilinear parabolic PDEs with nonsmooth data and mixed boundary conditions, including a nonlinear source term with at most linear growth. Subsequently, we use a bootstrapping argument to achieve improved regularity of these global-in-time solutions within the functional-analytic setting of the interpolation scale of Bessel-potential dual spaces $H^{θ-1,p}_D = [W^{-1,p}_D,L^p]_θ$ with $θ\in [0,1]$ for the abstract equation under suitable additional assumptions. This is done by me…
In this paper we prove that the equation $ -( r^αϕ(|u'(r)|)u'(r))' = λr^γf(u(r)), ~0<r<R$, where $α, γ, {\bf{R}}$ are given real numbers, $ϕ: (0, \infty) \to (0, \infty)$ is a suitable twice differentiable function, $λ> 0$ is a real parameter and $f:{\bf{R}}\to{\bf{R}}$ is continuous, admits an infinite sequence of sign-changing solutions satisfying $u'(0) =u(R) =0$. The function $f$ is required to satisfy $tf(t)>0$ for $ t\neq 0$. Our technique explores fixed point arguments applied to suitable integral equations and shooting arguments. Our main result extends earlier ones in the case $ϕ$ is …
In this paper, we consider the quasilinear Schrödinger equation \begin{equation*} -Δu+V(x)u-uΔ(u^2)=g(u),\ \ x\in \mathbb{R}^{3}, \end{equation*} where $V$ and $g$ are continuous functions. Without the coercive condition on $V$ or the monotonicity condition on $g$, we show that the problem above has a least energy sign-changing solution and infinitely many sign-changing solutions. Our results especially solve the problem above in the case where $g(u)=|u|^{p-2}u$ ($2<p<4$) and complete some recent related works on sign-changing solutions, in the sense that, in the literature only the case $g(u)…
We study regularity for solutions of quasilinear elliptic equations of the form $÷\A(x,u,\nabla u) = ÷\F $ in bounded domains in $\R^n$. The vector field $\A$ is assumed to be continuous in $u$, and its growth in $\nabla u$ is like that of the $p$-Laplace operator. We establish interior gradient estimates in weighted Morrey spaces for weak solutions $u$ to the equation under a small BMO condition in $x$ for $\A$. As a consequence, we obtain that $\nabla u$ is in the classical Morrey space $\calM^{q,λ}$ or weighted space $L^q_w$ whenever $|\F|^{\frac{1}{p-1}}$ is respectively in $\calM^{q,λ}$ o…
In this note we complete a previous study, where we got existence results for the quasilinear elliptic equation \begin{equation*} -Δw+ V\left( \left| x\right| \right) w - w \left( Δw^2 \right)= K(|x|) g(w) \quad \text{in }\mathbb{R}^{N}, \end{equation*} with singular or vanishing continuous radial potentials $V(r)$, $K(r)$. In our previuos study we assumed, for technical reasons, that $K(r)$ was vanishing as $r \rightarrow 0$, while in the present paper we remove this obstruction. To face the problem we apply a suitable change of variables $w=f(u)$ and we find existence of non negative solutio…
We prove the uniqueness property for a class of entire solutions to the equation \begin{equation*} \left\{ \begin{array}{ll} -{\rm div}\, \mathcal{A}(x,\nabla u) = σ, \quad u\geq 0 \quad \text{in } \mathbb{R}^n, \\ \displaystyle{\liminf_{|x|\rightarrow \infty}}\, u = 0, \end{array} \right. \end{equation*} where $σ$ is a nonnegative locally finite measure in $\mathbb{R}^n$, absolutely continuous with respect to the $p$-capacity, and ${\rm div}\, \mathcal{A}(x,\nabla u)$ is the $\mathcal{A}$-Laplace operator, under standard growth and monotonicity assumptions of order $p$ ($1<p<\infty$) on $\mat…
Locally bounded, local weak solutions to a special class of quasilinear, anisotropic, $p$-Laplacian type elliptic equations, are shown to be locally Hölder continuous. Homogeneous local upper bounds are established for local weak solutions to a general class of quasilinear anisotropic equations.
Given two continuous functions $V\left(r \right)\geq 0$ and $K\left(r\right)> 0$ ($r>0$), which may be singular or vanishing at zero as well as at infinity, we study the quasilinear elliptic equation \[ -Δw+ V\left( \left| x\right| \right) w - w \left( Δw^2 \right)= K(|x|) g(w) \quad \text{in }\mathbb{R}^{N}, \] where $N\geq3$. To study this problem we apply a change of variables $w=f(u)$, already used by several authors, and find existence results for nonnegative solutions by the application of variational methods. The main features of our results are that they do not require any compatibilit…
The main goal of this paper is to show how some monotonicity methods related with the subdifferential of suitable convex functions and its extensions as m-accretive operators in Banach spaces lead to new and unexpected results showing, for instance, the continuous and monotone dependence of solutions with the respect to the data (and coefficients) of the problem. In this way, this paper offers `a common roof' to several methods and results concerning monotone and non-monotone frameworks. Besides to present here some new results, this paper offers also a peculiar survey to some topics which att…
