Degenerate Quasilinear Parabolic Equations Research Articles

Curve shortening flow on Riemann surfaces with conical singularities

AbstractWe study the curve shortening flow on Riemann surfaces with finitely many conformal conical singularities. If the initial curve is passing through the singular points, then the evolution is governed by a degenerate quasilinear parabolic equation. In this case, we establish short time existence, uniqueness, and regularity of the flow. We also show that the evolving curves stay fixed at the singular points of the surface and obtain some contracting and convergence results.

Gradient estimates of the solution to the Cauchy problem for degenerate parabolic equations with nonpower nonlinearities

ABSTRACT We prove the uniform gradient estimates of the nonnegative solution to the Cauchy problem for quasilinear degenerate parabolic equations with nonpower nonlinearity. The large-time estimates in terms of initial data mass and nonlinearity structure were obtained. In the particular case, when the nonlinearity of the equation is power like, then our result coincides with well-known optimal bound.

O. A. Ladyzhenskaya’s System of Equations of Symmetric Boundary Layer of Modified Fluid

In this paper, we study the system of equations of a boundary layer for a nonlinearly viscous, electrically conductive liquid described by a rheological law proposed by O. A. Ladyzhenskaya for incompressible media. The boundary-layer equations for the Ladyzhenskaya model were first obtained from Prandtl’s axioms. By the Mises transform, the system of boundary-layer equations can be reduced to a single quasilinear equation. The main method used in this paper is the Crocco transform, which turns the system of boundary-layer equations into a quasilinear degenerate parabolic equation. In contrast to the Mises variables, the Crocco substitution allows one to study both stationary and nonstationary equations.

The Dirichlet Problem for Stochastic Degenerate Parabolic-Hyperbolic Equations

We consider the Dirichlet problem for a quasilinear degenerate parabolic stochastic partial differential equation with multiplicative noise and nonhomogeneous Dirichlet boundary condition. We introduce the definition of kinetic solution for this problem and prove existence and uniqueness of solutions. For the uniqueness of kinetic solutions we prove a new version of the doubling of variables method and use it to deduce a comparison principle between solutions. The proof requires a delicate analysis of the boundary values of the solutions for which we develop some techniques that enable the usage of the existence of normal weak traces for divergence measure fields in this stochastic setting. The existence of solutions, as usual, is obtained through a two-level approximation scheme consisting of nondegenerate regularizations of the equations which we show to be consistent with our definition of solutions. In particular, the regularity conditions that give meaning to the boundary values of the solutions are shown to be inherited by limits of nondegenerate parabolic approximations provided by the vanishing viscosity method.

Gradient estimates for an orthotropic nonlinear diffusion equation

Abstract We consider a quasilinear degenerate parabolic equation driven by the orthotropic p-Laplacian. We prove that local weak solutions are locally Lipschitz continuous in the spatial variable, uniformly in time.

Long-time behaviour of the solution to the Cauchy problem for degenerate parabolic equations with nonpower nonlinearities

We prove the qualitative decay in time estimate of supremum norm of the nonnegative solution to the Cauchy problem for quasilinear degenerate parabolic equations with nonpower nonlinearity. The result obtained is given in terms of the initial data mass and nonlinearity structure. In particular, if the nonlinearity of the equation is power like, then our result coincides with well-known optimal bound.

Dynamics of Non-autonomous Quasilinear Degenerate Parabolic Equations: the Non-compact Case

Dynamics of Non-autonomous Quasilinear Degenerate Parabolic Equations: the Non-compact Case

Regularity for a class of quasilinear degenerate parabolic equations in the Heisenberg group

We extend to the parabolic setting some of the ideas originated with Xiao Zhong's proof in [31] of the Hölder regularity of $p-$harmonic functions in the Heisenberg group $\mathbb{H}^n$. Given a number $p\ge 2$, in this paper we establish the $C^{\infty}$ smoothness of weak solutions of a class of quasilinear PDE in $\mathbb{H}^n$ modeled on the equation $$?_t u = \sum_{i = 1}^{2n} X_i \bigg((1+|\nabla_0 u|^2)^{\frac{p-2}{2}} X_i u\bigg).$$

Local strong solutions to a quasilinear degenerate fourth-order thin-film equation

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 alpha ge 2 the corresponding initial boundary value problem fits into the abstract setting of Amann (Function Spaces, Differential Operators and Nonlinear Analysis, Vieweg Teubner Verlag, Stuttgart, 1993). Due to a lack of regularity this is not true for flow behaviour exponents alpha in (1,2). For this reason we prove an existence theorem for abstract quasilinear parabolic evolution problems with Hölder continuous dependence. This result provides existence of strong solutions to the non-Newtonian thin-film problem in the setting of fractional Sobolev spaces and (little) Hölder spaces. Uniqueness of strong solutions is derived by energy methods and by using the particular structure of the equation.

Linearized stability for degenerate and singular semilinear and quasilinear parabolic problems: the linearized singular equation

We study some linear eigenvalue problems for the Laplacian operator with singular absorption or/and source coefficients arising in the linearization around positive solutions to some quasilinear degenerate parabolic equations and singular semilinear parabolic problems as well. We show that the linearization process applies even if the coefficients behave singularly with the distance to the boundary to the exponent two. This improves previous references in the literature. Applications to the above mentioned nonlinear problems are also presented.

System of Boundary Layer Equations for a Rheologically Complicated Medium: Crocco Variables

The system of boundary-layer equations for a nonlinear generalized Newtonian viscous fluid with the Ladyzhenskaya law is studied. The correct solvability of the problem under study is proved using the Crocco transformation method, which reduces the system of boundary-layer equations to a quasilinear degenerate parabolic equation. Asymptotic estimates for the solution on the boundary of the domain are obtained.

Remarks on global solutions to the initial-boundary value problem for quasilinear degenerate parabolic equations with a nonlinear source term

We give an existence theorem of global solution to the initial-boundary value problem for \(u_{t}-\operatorname{div}\{\sigma(|\nabla u|^2)\nabla u\}=f(u)\) under some smallness conditions on the initial data, where \(\sigma (v^2)\) is a positive function of \(v^2\ne 0\) admitting the degeneracy property \(\sigma(0)=0\). We are interested in the case where \(\sigma(v^2)\) has no exponent \(m \geq 0\) such that \(\sigma(v^2) \geq k_0|v|^m , k_0 \gt 0\). A typical example is \(\sigma(v^2)=\operatorname{log}(1+v^2)\). \(f(u)\) is a function like \(f=|u|^{\alpha} u\). A decay estimate for \(\|\nabla u(t)\|_{\infty}\) is also given.

Smoothing effects of the initial-boundary value problem for logarithmic type quasilinear parabolic equations

We give existence theorems of global solutions in Lloc∞((0,∞);W01,∞) to the initial boundary value problem for quasilinear degenerate parabolic equations of the form ut−div{σ(|∇u|2)∇u}=0, where the class of σ(v2) includes the logarithmic case σ(|∇u|2)= log (1+|∇u|2) for a typical example. We assume that the initial data belong to W01,p0,p0≥2, or Lr,r≥1, and we derive precise estimates for ‖∇u(t)‖∞ near t=0.

Decay of the mass of the solution to the Cauchy problem of the degenerate parabolic equation with nonlinear potential

We prove the precise decay in time estimates of the total mass of a solution to the Cauchy problem for quasi-linear degenerate parabolic equations with absorption terms. The equation may involve additional terms whose behaviour at infinity is described by Wolf potentials. In terms of new critical exponents, we control precise behaviour of the total mass.

Dirichlet Boundary Conditions for Degenerate and Singular Nonlinear Parabolic Equations

We study existence and uniqueness of solutions to a class of quasilinear degenerate parabolic equations, in bounded domains. We show that there exists a unique solution which satisfies possibly inhomogeneous Dirichlet boundary conditions. To this purpose some barrier functions are properly introduced and used.

A new second critical exponent and life span for a quasilinear degenerate parabolic equation with weighted nonlocal sources

In this paper, we consider positive solutions of a Cauchy problem for the following quasilinear degenerate parabolic equation with weighted nonlocal sources: \begin{document}$u_{t}=\Delta_{p}u+ \left(\int_{\mathbb{R}^{N}}K(x)u^{q}(x, t)dx\right)^{\frac{r-1}{q}}u^{s+1}, (x, t) \in \mathbb{R}^{N} \times(0, T), $ \end{document} where \begin{document}$N≥1$\end{document} , \begin{document}$p>2$\end{document} , \begin{document}$q$\end{document} , \begin{document}$r≥1$\end{document} , \begin{document}$s≥0$\end{document} , and \begin{document}$r+s>1$\end{document} . We classify global and non-global solutions of the equation in the coexistence region by finding a new second critical exponent via the slow decay asymptotic behavior of an initial value at spatial infinity, and the life span of non-global solution is studied.

Moser–Nash kernel estimates for degenerate parabolic equations

In this paper we deal with the Cauchy problem associated to a class of nonlinear degenerate parabolic equations, whose prototype is the parabolic p-Laplacian (2<p<∞). In his seminal paper, after stating the Harnack estimates, Moser proved almost optimal estimates for the parabolic kernel by using the so called ‘Harnack chain’ method. In the linear case sharp estimates come by using Nash's approach. Fabes and Stroock proved that Gaussian estimates are equivalent to a parabolic Harnack inequality. In this paper, by using the DiBenedetto–DeGiorgi approach we prove optimal kernel estimates for degenerate quasilinear parabolic equations. To obtain this result we need to prove the finite speed of propagation of the support and to establish optimal estimates. Lastly we use these results to prove existence and sharp pointwise estimates from above and from below for the fundamental solutions.

A time-splitting approach to quasilinear degenerate parabolic stochastic partial differential equations

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.

Degenerate parabolic stochastic partial differential equations: Quasilinear case

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 ingredients that provide simpler and more effective method of the proof: a generalized It\^{o} formula that permits a rigorous derivation of the kinetic formulation even in the case of weak solutions of certain nondegenerate approximations and a direct proof of strong convergence of these approximations to the desired kinetic solution of the degenerate problem.

Interior Harnack Estimates: The State-of-the-Art for Quasilinear Singular Parabolic Equations

In this paper we give some historical information about elliptic and parabolic Harnack inequalities. Then we state the main results known for Harnack inequalities of solutions to quasilinear degenerate parabolic equations. Lastly we focus our attention on Harnack inequalities of solutions to quasilinear singular parabolic equations where the theory did important steps forward in the last few years but still there are some points to be fully understood.