Krerley Oliveira, Eduardo Teixeira
In this paper we study the existence, regularity and geometric properties of an optimal configuration to a free boundary optimization problem governed by the $p$-Laplacian.
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We prove radial symmetry of singular solutions to an overdetermined boundary value problem for a class of degenerate quasilinear elliptic equations.
We consider a quasilinear Schrödinger equation on $\mathbb R$ for which the dispersive effects degenerate when the solution vanishes. We first prove local well-posedness for sufficiently smooth, spatially localized, degenerate initial data. As a corollary in the focusing case we obtain a short time stability result for the energy-minimizing compact breather.
The main result of the paper is on the continuity of weak solutions of infinitely degenerate quasilinear second order equations. Namely, we show that every weak solution to a certain class of degenerate quasilinear equations is continuous. More precisely, we show that it is Hölder continuous with respect to a certain metric associated to the operator. One of the essential features of this metric is that the metric balls are non doubling with respect to Lebesgue measure. The proof of the continuity together with a recent result by Rios et al. completes the result on hypoellipticity of a class o…
Regimes with a singular peaking for a wide class of quasilinear second order parabolic equations are studied. On the basis of energy methods, precise estimates of a final profile of a weak solution in a neighborhood of the peaking time are established depending on the rate of increase of a global energy of this solution. Key words: quasilinear parabolic equations, peaking regimes, energy solutions.
We prove Harnack inequality and local regularity results for weak solutions of a quasilinear degenerate equation in divergence form under natural growth conditions. The degeneracy is given by a suitable power of a strong $A_\infty$ weight. Regularity results are achieved under minimal assumptions on the coefficients and, as an application, we prove $C^{1,α}$ local estimates for solutions of a degenerate equation in non divergence form.
We consider a quasilinear KdV equation that admits compactly supported traveling wave solutions (compactons). This model is one of the most straightforward instances of degenerate dispersion, a phenomenon that appears in a variety of physical settings as diverse as sedimentation, magma dynamics and shallow water waves. We prove the existence and uniqueness of solutions with sufficiently smooth, spatially localized initial data.
The aim of this paper is to establish regularity for weak solutions to the nondiagonal quasilinear degenerate elliptic systems related to Hörmander's vector fields, where the coefficients are bounded with vanishing mean oscillation. We first prove $L^p$($p \ge 2$) estimates for gradients of weak solutions by using a priori estimates and a known reverse Hölder inequality, and consider regularity to the corresponding nondiagonal homogeneous degenerate elliptic systems. Then we get higher Morrey and Campanato estimates for gradients of weak solutions to original systems and Hölder estimates for w…
On a class of nonautonomous quasilinear systems with general time-gradually-degenerate damping
本文研究具有时变系数和一般时间渐近退化阻尼的Zeldovich近似与无压气体系统，通过相似变量法和非线性粘性法构造出经典Riemann解与delta激波解。
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 $B\geq 0$.
In a previous paper we considered a class of infinitely degenerate quasilinear equations and derived a priori bounds for high order derivatives of solutions in terms of the Lipschitz norm. We now show that it is possible to obtain bounds just in terms of the supremum norm for a further subclass of such equations, and we apply the resulting estimates to prove that continuous weak solutions are necessarily smooth. We also obtain existence, uniqueness and interior regularity of solutions for the Dirichlet problem with continuous boundary data.
We consider local weak large solutions with its blow-up rate near the boundary to certain class of degenerate and/or singular quasilinear elliptic equation\\ ${\rm div}(d^α(x,\partial{}B)Φ_p(\nabla u)) = b(x)f(u)$ in a ball B, where $f$ is normalized regularly varying at infinity with index $σ+1>p-1,\ p>1$. In particular, how the asymptotic behavior of the solution changes over the varying index and degeneracy and/ or singularity present in the equation. We also include the second order blow-up rate for the corresponding semilinear problem.
We extend to the parabolic setting some of the ideas originated with Xiao Zhong's proof in \cite{Zhong} of the Hölder regularity of $p-$harmonic functions in the Heisenberg group $\Hn$. Given a number $p\ge 2$, in this paper we establish the $C^{\infty}$ smoothness of weak solutions of quasilinear pde's in $\Hn$ modelled on the equation $$\p_t u= \sum_{i=1}^{2n} X_i \bigg((1+|\nabla_0 u|^2)^{\frac{p-2}{2}} X_i u\bigg).$$
We derive local boundedness estimates for weak solutions of a large class of second order quasilinear equations. The structural assumptions imposed on an equation in the class allow vanishing of the quadratic form associated with its principal part and require no smoothness of its coefficients. The class includes second order linear elliptic equations as studied by D. Gilbarg and N. S. Trudinger [1998] and second order subelliptic linear equations as studied by E. Sawyer and R. L. Wheeden [2006 and 2010]. Our results also extend ones obtained by J. Serrin [1964] concerning local boundedness of…
We study quasilinear degenerate parabolic-hyperbolic stochastic partial differential equations with general multiplicative noise within the framework of kinetic solutions. Our results are twofold: First, we establish new regularity results based on averaging techniques. Second, we prove the existence and uniqueness of solutions in a full $L^1$ setting requiring no growth assumptions on the nonlinearities. In addition, we prove a comparison result and an $L^1$-contraction property for the solutions.
Sobolev estimates for singular-degenerate quasilinear equations beyond the $A_2$ class
针对系数可退化或奇异的拟线性椭圆方程，在系数振荡小性假设下建立加权 Sobolev 估计，方法基于微扰和新的 Lipschitz 估计。
We prove a new, universal gradient continuity estimate for solutions to quasilinear equations with varying coefficients at points on its critical singular set of degeneracy $S(u) := \{X : D u(X) = 0 \}$. Our main Theorem reveals that along $S(u)$, $u$ is asymptotically as regular as solutions to constant coefficient equations. In particular, along the critical set $S(u)$, $Du$ enjoys a modulus of continuity much superior than the, possibly low, continuity feature of the coefficients. Our main, leading result fosters a new understanding on smoothness properties of solutions to degenerate or sin…
We consider the optimization problem of minimizing $\int_Ω|\nabla u|^p dx$ with a constrain on the volume of $\{u>0\}$. We consider a penalization problem, and we prove that for small values of the penalization parameter, the constrained volume is attained. In this way we prove that every solution $u$ is locally Lipschitz continuous and that the free boundary, $\partial\{u>0\}\cap Ω$, is smooth.
We prove a weak comparison principle in narrow unbounded domains for solutions to $-Δ_p u=f(u)$ in the case $2<p< 3$ and $f(\cdot)$ is a power-type nonlinearity, or in the case $p>2$ and $f(\cdot)$ is super-linear. We exploit it to prove the monotonicity of positive solutions to $-Δ_p u=f(u)$ in half spaces (with zero Dirichlet assumption) and therefore to prove some Liouville-type theorems.
Convergence to a single steady state is shown for non-negative and radially symmetric solutions to a diffusive Hamilton-Jacobi equation with homogeneous Dirichlet boundary conditions, the diffusion being the $p$-Laplacian operator, $p\ge 2$, and the source term a power of the norm of the gradient of $u$. As a first step, the radially symmetric and non-increasing stationary solutions are characterized.
In this paper, we discuss the Cauchy problem for a degenerate parabolic hyperbolic equation with a multiplicative noise. We focus on the existence of a solution. Using nondegenerate smooth approximations, Debussche, Hofmanová and Vovelle [8] proved the existence of a kinetic solution. On the other hand, we propose to construct a sequence of approximations by applying a time splitting method and prove that this converges strongly in $L^1$ to a kinetic solution. This method will somewhat give us not only a simpler and more direct argument but an improvement over the existence result.
The equation arising from Prandtl boundary layer theory is considered. The existence of the entropy solution can be proved by BV estimate method. The interesting problem is that, since a may be degenerate on the boundary, the usual boundary value condition may be overdetermined. Accordingly, only dependent on a partial boundary value condition, the stability of solutions can be expected. This expectation is turned to reality by Kruzkov's bi-variables method, a reasonable partial boundary value condition matching up with the equation is found first time. Moreover, the stability can be proved ev…
Local behavior for solutions to anisotropic weighted quasilinear degenerate parabolic equations
本文研究各向异性加权退化抛物方程解的局部行为，给出权重奇异点附近衰减率的显式上界，并证明非齐次p-Laplace方程解的局部Hölder正则性。
We continue to study regularity results for weak solutions of the large class of second order degenerate quasilinear equations of the form \begin{eqnarray} \text{div}\big(A(x,u,\nabla u)\big) = B(x,u,\nabla u)\text{ for }x\inΩ\nonumber \end{eqnarray} as considered in our previous paper giving local boundedness of weak solutions. Here we derive a version of Harnack's inequality as well as local Hölder continuity for weak solutions. The possible degeneracy of an equation in the class is expressed in terms of a nonnegative definite quadratic form associated with its principal part. No smoothness …
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…
Lie symmetries, conservation laws and exact solutions of a generalized quasilinear KdV equation with degenerate dispersion
本文对广义拟线性KdV方程进行了点对称性和低阶局部守恒律的完整分类，给出了对称变换群的解释、守恒量的物理解释，并获得了少数行波解。
This paper deals with the quasilinear degenerate chemotaxis system with flux limitation \begin{align*} \begin{cases} u_t = \nabla\cdot\left(\dfrac{u^p \nabla u}{\sqrt{u^2 + |\nabla u|^2}} \right) -χ\nabla\cdot\left( \dfrac{u^q\nabla v}{\sqrt{1 + |\nabla v|^2}}\right), &x\in Ω,\ t>0, \\[1mm] 0 = Δv - μ+ u, &x\in Ω,\ t>0, \end{cases} \end{align*} where $Ω:= B_R(0) \subset \mathbb{R}^n$ ($n \in \mathbb{N}$) is a ball with some $R>0$, and $χ>0$, $p,q\geq1$, $μ:= \frac 1{|Ω|} \int_Ωu_0$ and $u_0$ is an initial data of an unknown function $u$. Bellomo--Winkler (Trans.\ Amer.\ Math.\ Soc.\ Ser.\ B;20…
In historical mathematics and physics, the Kardar-Parisi-Zhang equation or a quasilinear stationary version of a time-dependent viscous Hamilton-Jacobi equation in growing interface and universality classes, is also known by the different name as the quasilinear Riccati type equation. The existence of solutions to this type of equation under some assumptions and requirements, still remains an interesting open problem at the moment. In our previous studies \cite{MP2018, MPT2019}, we obtained the global bounds and gradient estimates for quasilinear elliptic equations with measure data. There hav…
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 deals with the quasilinear degenerate chemotaxis system with flux limitation \begin{equation*} \begin{cases} u_t = \nabla\cdot\left(\dfrac{u^p \nabla u}{\sqrt{u^2 + |\nabla u|^2}} \right) -χ\nabla\cdot\left(\dfrac{u^q\nabla v}{\sqrt{1 + |\nabla v|^2}}\right), \\[1mm] 0 = Δv - μ+ u \end{cases}\end{equation*} under no-flux boundary conditions in balls $Ω\subset\mathbb{R}^n$, and the initial condition $u|_{t=0}=u_0$ for a radially symmetric and positive initial data $u_0\in C^3(\overlineΩ)$, where $χ>0$ and $μ:=\frac{1}{|Ω|}\int_Ωu_0$. Bellomo--Winkler (Comm.\ Partial Differential Equa…
In this paper, we study the Cauchy problem for a quasilinear degenerate parabolic stochastic partial differential equation driven by a cylindrical Wiener process. In particular, we adapt the notion of kinetic formulation and kinetic solution and develop a well-posedness theory that includes also an $L^1$-contraction property. In comparison to the previous works of the authors concerning stochastic hyperbolic conservation laws [J. Funct. Anal. 259 (2010) 1014-1042] and semilinear degenerate parabolic SPDEs [Stochastic Process. Appl. 123 (2013) 4294-4336], the present result contains two new ing…
For a linear nonvariational operator structured on smooth Hörmander's vector fields, with Hölder continuous coefficients, we prove a regularity result in the spaces of Hölder functions. We deduce an analogous regularity result for nonvariational degenerate quasilinear equations.
We study the long-time behaviour of solutions to quasilinear doubly degenerate parabolic problems of fourth order. The equations model for instance the dynamic behaviour of a non-Newtonian thin-film flow on a flat impermeable bottom and with zero contact angle. We consider a shear-rate dependent fluid the rheology of which is described by a constitutive power-law or Ellis-law for the fluid viscosity. In all three cases, positive constants (i.e. positive flat films) are the only positive steady-state solutions. Moreover, we can give a detailed picture of the long-time behaviour of solutions wit…
Well-posedness and qualitative properties of quasilinear degenerate evolution systems
本文研究了生物膜生长等背景下的非线性退化耦合 PDE-PDE 和 PDE-ODE 系统，通过正则化与不动点论证证明了弱解存在性与唯一性，确立了不同边界条件下的全局或局部适定性，并给出了解的有限时间爆破或一致有界性条件及全局正则性。
We introduce a new class of quasilinear nonlocal operators and study equations involving these operators. The operators are degenerate elliptic and may have arbitrary growth in the gradient. Included are new nonlocal versions of p-Laplace, $\infty$-Laplace, mean curvature of graph, and even strongly degenerate operators, in addition to some nonlocal quasilinear operators appearing in the existing literature. Our main results are comparison, uniqueness, and existence results for viscosity solutions of linear and fully nonlinear equations involving these operators. Because of the structure of ou…
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。
Loss of regularity for solutions to 1D degenerate quasilinear wave equations
本文研究一维退化波动方程解的正则性损失，证明当 \\(\beta<\alpha\\) 时，退化拟线性方程的正则性损失条件必要，解的阶从初始数据 \\(u_0\\) 的阶变为 \\(u_1\\) 的阶。
The goal of this article is to establish local Lipschitz continuity of weak solutions for a class of degenerated elliptic equations of divergence form, in the Heisenberg Group. The considered hypothesis for the growth and ellipticity condition is a natural generalization of the p-Laplace equation and more general quasilinear elliptic equations with polynomial or exponential type growth.
In this paper we study the geometry of certain functionals associated to quasilinear elliptic boundary value problems with a degenerate nonlocal term of Kirchhoff type. Due to the degeneration of the nonlocal term it is not possible to directly use classical results such as uniform a-priori estimates and "Sobolev versus Hölder local minimizers" type of results. We prove that results similar to these hold true or not, depending on how degenerate the problem is. We apply our findings in order to show existence and multiplicity of solutions for the associated quasilinear equations, considering se…
We consider a class of quasi-linear anisotropic elliptic equations, possibly degenerate or singular, which are of interest in several applications such as computer vision and continuum mechanics. We prove a Hopf Lemma as well as local and global regularity estimates for positive solutions, generalizing previous results known in the context of p-Laplacian equations.
In this article we establish sharp $C^{1,α}$ estimates for weak solutions of singular and degenerate quasilinear elliptic equation $$-\,div\, a(x, \nabla u) = f,$$ which includes the standard $p$-laplacean equation with varying coefficients as a special case. The sharp exponent $α$ is asymptotically optimal and is determined by the Hölder regularity of the coefficients, the exponent $p$ and the $q$-integrability of the source term $f$.
In this paper, we are interested in obtaining a unified approach for $C^{1,α}$ estimates for weak solutions of quasilinear parabolic equations, the prototype example being \[ u_t - \text{div} (|\nabla u|^{p-2} \nabla u) = 0. \] without having to consider the singular and degenerate cases separately. This is achieved via a new scaling and a delicate adaptation of the covering argument developed by E.~DiBenedetto and A.~Friedman.
A Free Boundary Problem for quasilinear Systems with mixed variable exponent
本文研究单位球中由(p,q)-拉普拉斯型混合变指数算子控制、产生Alt-Phillips型自由边界的半线性方程组，在特定结构条件下建立了非负径向C¹解的存在性，其范数取决于结构数据。
We study the Cauchy problem for the quasilinear wave equation $ \partial^2 _t u = u^{2a} \partial^2_x u + F(u) u_x $ with $a \geq 0$ and show a result for the local in time existence under new conditions. In the previous results, it is assumed that $u(0,x) \geq c_0>0$ for some constant $c_0$ to prove the existence and the uniqueness. This assumption ensures that the equation does not degenerate. In this paper, we allow the equation to degenerate at spacial infinity. Namely we consider the local well-posedness under the assumption that $u(0,x)>0$ and $u(0,x) \rightarrow 0$ as $|x| \rightarrow \…
We establish existence with sharp rates of decay and distance from the Chapman--Enskog approximation of small-amplitude quasilinear relaxation shocks in the general case that the profile ODE may become degenerate. Our method of analysis follows the general approach used by Métivier and Zumbrun in the semilinear case, based on Chapman--Enskog expansion and the macro--micro decomposition of Liu and Yu. In the quasilinear case, however, we find it necessary to apply a parameter-dependent Nash-Moser iteration to close the analysis, whereas, in the semilinear case, a simple contraction-mapping argu…
We consider the large time behavior of solutions to the following nonlinear wave equation: $\partial_{t}^2 u = c(u)^{2}\partial^2_x u + λc(u)c'(u)(\partial_x u)^2$ with the parameter $λ\in [0,2]$. If $c(u(0,x))$ is bounded away from a positive constant, we can construct a local solution for smooth initial data. However, if $c(\cdot )$ has a zero point, then $c(u(t,x))$ can be going to zero in finite time. When $c(u(t,x))$ is going to 0, the equation degenerates. We give a sufficient condition that the equation with $0\leq λ< 2$ degenerates in finite time.
We prove a comparison principle for weak solutions of elliptic quasilinear equations in divergence form whose ellipticity constants degenerate at every point where $\nabla u \in K$, where $K\subset \mathbb{R}^N$ is a Borel set containing the origin.
We investigate $(d+1)$-dimensional quasilinear systems which are integrable by the method of hydrodynamic reductions. In the case $d\geq 3$ we formulate a conjecture that any such system with an irreducible dispersion relation must be linearly degenerate. We prove this conjecture in the 2-component case, providing a complete classification of multi-dimensional integrable systems in question. In particular, our results imply the non-existence of 2-component integrable systems of hydrodynamic type for $d\geq 6$. In the second half of the paper we discuss a numerical and analytical evidence for t…
The regularity and characterization of solutions to degenerate, quasilinear SPDE is studied. Our results are two-fold: First, we prove regularity results for solutions to certain degenerate, quasilinear SPDE driven by Lipschitz continuous noise. In particular, this provides a characterization of solutions to such SPDE in terms of (generalized) strong solutions. Second, for the one-dimensional stochastic mean curvature flow with normal noise we adapt the notion of stochastic variational inequalities to provide a characterization of solutions previously obtained in a limiting sense only. This so…
We consider the asymptotic behavior of quasilinear parabolic equations posed in a family of unbounded domains that degenerates onto a lower dimensional set. Considering an auxiliary family of weighted Sobolev spaces we show the existence of global attractors and we analyze convergence properties of the solutions as well of the attractors.
Hölder regularity of doubly nonlinear nonlocal quasilinear parabolic equations in some mixed singular-degenerate regime
本文研究了非局部拟线性抛物方程在混合奇异-退化情形下的Hölder正则性，通过结合新内在尺度和已有方法，证明了特定范围内的正则性结果。
We study existence of solutions for a boundary degenerate (or singular) quasilinear equation in a smooth bounded domain under Dirichlet boundary conditions. We consider a weighted $p-${L}aplacian operator with a coefficient that is {locally bounded inside the domain and satisfying certain additional integrability assumptions}. Our main result applies for boundary value problems involving continuous non-linearities having no growth restriction, but provided the existence of a sub and a supersolution is guaranteed. As an application, we present an existence result for a boundary value pro\-blem …
We prove that every weak solution to a certain class of infinitely degenerate quasilinear equations is continuous. An essential feature of the operators we consider is that their Fefferman-Phong associated metric may be non doubling with respect to Lebesgue measure.
On the regularity theory for mixed local and nonlocal weighted quasilinear elliptic equations
该研究针对带Muckenhoupt权的混合局部与非局部退化p-拉普拉斯方程，建立了首个系统的局部正则性理论，扩展了相关现有理论框架。
We prove that well posed quasilinear equations of parabolic type, perturbed by bounded nondegenerate random forces, are exponentially mixing for a large class of random forces.
We discuss the asymptotic behavior of positive solutions of the quasilinear elliptic problem $-Δ_p u=a u^{p-1}-b(x) u^q$, $u|_{\partial Ω}=0$ as $q \to p-1+0$ and as $q \to \infty$ via a scale argument. Here $Δ_p$ is the $p$-Laplacian with $1<p<\infty$ and $q>p-1$. If $p=2$, such problems arise in population dynamics. Our main results generalize the results for $p=2$, but some technical difficulties arising from the nonlinear degenerate operator $-Δ_p$ are successfully overcome. As a by-product, we can solve a free boundary problem for a nonlinear $p$-Laplacian equation.
We study random homogenization of second-order, degenerate and quasilinear Hamilton-Jacobi equations which are positively homogeneous in the gradient. Included are the equations of forced mean curvature motion and others describing geometric motions of level sets as well as a large class of viscous, non-convex Hamilton-Jacobi equations. The main results include the first proof of qualitative stochastic homogenization for such equations. We also present quantitative error estimates which give an algebraic rate of homogenization.
We consider the global existence and blow up of solutions of the Cauchy problem of the quasilinear wave equation: $\partial_{t}^2 u = \partial_x(c(u)^2 \partial_x u)$, which has richly physical backgrounds. Under the assumption that $c(u(0,x))\geq δ$ for some $δ>0$, we give sufficient conditions for the existence of global smooth solutions and the occurrence of two types of blow-up respectively. One of the two types is that $L^{\infty}$-norm of $\partial_t u$ or $\partial_x u$ goes up to the infinity. The other type is that $c(u)$ vanishes, that is, the equation degenerates.
Regularity for monotone Operators and applications to homogenization of $p$-Laplace type equations
本研究针对主部缺乏强单调性的拟线性方程解的梯度，给出局部L^q估计，进而推导退化/奇异拟线性方程解梯度的一致大尺度L^q估计与非退化方程解的大尺度Lipschitz估计，为p-Laplace型方程均匀化提供正则性支撑。
We prove Hölder continuity up to the boundary for solutions of quasi-linear degenerate elliptic problems in divergence form, not necessarily of variational type, on Lipschitz domains with Neumann and Robin boundary conditions. This includes the $p$-Laplace operator for all $p \in (1,\infty)$, but also operators with unbounded coefficients. Based on the elliptic result we show that the corresponding parabolic problem is well-posed in the space $C(\barΩ)$ provided that the coefficients satisfy a mild monotonicity condition. More precisely, we show that the realization of the elliptic operator in…
Bifurcation phenomena in one-phase quasilinear equations with nonstandard growth
本文研究由g-拉普拉斯算子控制的单相自由边界问题，证明边界数据低于阈值时出现分岔，存在第三个解，并分析演化问题的渐近稳定性。
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