We consider strongly degenerate parabolic operators of the form \[
\mathcal{L}:=\nabla_X\cdot(A(X,Y,t)\nabla_X)+X\cdot\nabla_Y-\partial_t \] in unbounded domains \[
Ω=\{(X,Y,t)=(x,x_{m},y,y_{m},t)\in\mathbb R^{m-1}\times\mathbb R\times\mathbb R^{m-1}\times\mathbb R\times\mathbb R\mid x_m>ψ(x,y,t)\}. \] We assume that $A=A(X,Y,t)$ is bounded, measurable and uniformly elliptic (as a matrix in $\mathbb R^{m}$) and concerning $ψ$ and $Ω$ we assume that $Ω$ is what we call an (unbounded) Lipschitz domain: $ψ$ satisfies a uniform Lipschitz condition adapted to the dilation structure and the (non-Euclidean) Lie group underlying the operator $\mathcal{L}$. We prove, assuming in addition that $ψ$ is independent of the variable $y_m$, that $ψ$ satisfies an additional regularity condition formulated in terms of a Carleson measure, and additional conditions on $A$, that the associated parabolic measure is absolutely continuous with respect to a surface measure and that the associated Radon-Nikodym derivative defines an $A_\infty$-weight with respect to the surface measure.
The goal of the present paper is to present a new approach to the construction of asymptotic (approximating) solutions to parabolic PDE by using the characteristics.
We prove the local boundedness of the solutions to degenerate second order partial differential equations of Kolmogorov type with measurable coefficients in divergence form, under minimal integrability assumption on the lower order coefficients.
The aim of this work is to prove the existence of a fundamental solution associated to the Kolmogorov equation L u = f with measurable coefficients in the dilation invariant case. Moreover, we prove Gaussian upper and lower bounds for it, and other related properties.
We consider weak solutions of degenerate second order partial differential equations of Kolmogorov-Fokker-Planck type with measurable coefficients in divergence form. We give a geometric statement of the Harnack inequality recently proven by Golse, Imbert, Mouhot and Vasseur. As a corollary we obtain a strong maximum principle.
We prove Gaussian upper and lower bounds for the fundamental solutions of a class of degenerate parabolic equations satisfying a weak Hormander condition. The bound is independent of the smoothness of the coefficients and generalizes classical results for uniformly parabolic equations
We study a class of elliptic operators $A$ with unbounded coefficients defined in $I\times\CR^d$ for some unbounded interval $I\subset\CR$. We prove that, for any $s\in I$, the Cauchy problem $u(s,\cdot)=f\in C_b(\CR^d)$ for the parabolic equation $D_tu=Au$ admits a unique bounded classical solution $u$. This allows to associate an evolution family $\{G(t,s)\}$ with $A$, in a natural way. We study the main properties of this evolution family and prove gradient estimates for the function $G(t,s)f$. Under suitable assumptions, we show that there exists an evolution system of measures for $\{G(t,…
We consider a class of possibly degenerate second order elliptic operators $\cal A$ on $\R^n$. This class includes hypoelliptic Ornstein-Uhlenbeck type operators having an additional first order term with unbounded coefficients. We establish global Schauder estimates in Hölder spaces both for elliptic equations and for parabolic Cauchy problems involving ${\cal A}$. The Hölder function spaces are defined with respect to a non-euclidean metric related to the operator $\cal A$.
Two essential quantities for the analysis of approximation schemes of evolution equations are stability and convergence. We derive stability and convergence of fully discrete approximation schemes of solutions to linear parabolic evolution equations governed by time dependent coercive operators. We consider abstract Galerkin approximations in space combined with theta-schemes in time. The level of generality of our analysis comprises both a large class of time-dependent operators and a large choice of approximating Galerkin spaces. In particular the results apply to partial integro differentia…
In this paper we develop a potential theory for strongly degenerate parabolic operators of the form \[ \mathcal{L}:=\nabla_X\cdot(A(X,Y,t)\nabla_X)+X\cdot\nabla_{Y}-\partial_t, \] in unbounded domains of the form \[ Ω=\{(X,Y,t)=(x,x_{m},y,y_{m},t)\in\mathbb R^{m-1}\times\mathbb R\times\mathbb R^{m-1}\times\mathbb R\times\mathbb R\mid x_m>ψ(x,y,y_m,t)\}, \] where $ψ$ is assumed to satisfy a uniform Lipschitz condition adapted to the dilation structure and the (non-Euclidean) Lie group underlying the operator $\mathcal{L}$. Concerning $A=A(X,Y,t)$ we assume that $A$ is bounded, measurable, symme…
We establish spatial a priori estimates for the solution u to a class of dilation invariant Kolmogorov equation, where u is assumed to only have a certain amount of regularity in the diffusion's directions. The result is that u is also regular with respect to the remaining directions. The approach we propose is based on the commutators identities and allows us to obtain a Sobolev exponent that does not depend on the integrability assumption of the right-hand side. Lastly, we provide an alternative proof to that of Theorem 1.5 of [9] for the optimal spatial regularity.
Under a precise nonlinearity-diffusivity condition we establish the decay of space-periodic entropy solutions of a multidimensional degenerate nonlinear parabolic equation.
A (2+1)-dimensional linear ultra-parabolic Fokker--Planck--Kolmogorov equation is investigated from the group-theoretical point of view. By using the Berest--Aksenov approach, an algebra of invariance of fundamental solutions of the equation is found. A fundamental solution of the equation under study is computed in an explicit form as a weak invariant solution.
In this paper, we prove Wiener's criterion for parabolic equations with singular and degenerate coefficients. To be precise, we study the problem of the regularity of boundary points for the Dirichlet problem for degenerate parabolic equations, and give a geometric characterization of those boundary points that are regular.
In this paper we prove the existence of finite traveling-wave type solutions to the nonlinear double degenerate parabolic equation of turbulent filtration with absorption.
We prove global existence of a nonnegative weak solution to a degenerate parabolic system, which models the spreading of insoluble surfactant on a thin liquid film.
In this paper, we analyze main properties of the volume and layer potentials as well as the Poisson integral for a multi-dimensional degenerate parabolic equation. As consequences, we obtain trace formulae of the heat volume potential and the Poisson integral which solve Kac's problem for degenerate parabolic equations in cylindrical domains.
In this work, we prove a Carleman estimate for a parabolic problem which has a dissipative degenerate term. The prove relies on choose a suitable weight function that change of sign inside the control domain.
We prove existence of strong traces at $t=0$ for quasi-solutions to (multidimensional) degenerate parabolic equations with no non-degeneracy conditions. In order to solve the problem, we combine the blow up method and a strong precompactness result for quasi-solutions to degenerate parabolic equations with the induction argument with respect to the space dimension.
We prove locally in time the existence of a smooth solution for multidimensional two-phase Stefan problem for degenerate parabolic equations of the porous medium type. We establish also natural Hölder class for the boundary conditions in the Cauchy-Dirichlet problem for a degenerate parabolic equation.
Existence, uniqueness and stability of kinetic and entropy solutions to the boundary value problem for the Kolmogorov-type genuinely nonlinear ultra-parabolic equation with a smooth source term is established. After this, we consider the case when the source term contains a small positive parameter and collapses to the Dirac delta-function, as this parameter tends to zero. In this case, the limiting passage from the original equation with the smooth source to the impulsive ultra-parabolic equation is fulfilled and rigorously justified. The proofs rely on the method of kinetic equation and on t…
Degenerate abstract parabolic equations with variable coefficients are studied. Here the boundary conditions are nonlocal. The maximal regularity properties of solutions for elliptic and parabolic problems and Strichartz type estimates in mixed $L_{p}$ spaces are obtained. Moreover, the existence and uniqueness of optimal regular solution of mixed problem for nonlinear parabolic equation is established. Note that, these problems arise in fluid mechanics and environmental engineering.
The aim of this work is to prove a Harnack inequality and the Hölder continuity for weak solutions to the Kolmogorov equation $\mathscr{L} u = f$ with measurable coefficients, integrable lower order terms and nonzero source term. We introduce a functional space $\mathcal{W}$, suitable for the study of weak solutions to $\mathscr{L}u = f$, that allows us to prove a weak Poincaré inequality. More precisely, our goal is to prove a weak Harnack inequality for non-negative super-solutions by considering their Log-transform and following S. N. Kruzkov (1963). Then this functional inequality is combi…
We study existence and uniqueness of solutions to a class of nonlinear 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.
We give sufficient conditions under which the convergence of finite difference approximations in the space variable of possibly degenerate second order parabolic and elliptic equations can be accelerated to any given order of convergence by Richardson's method.
We show Carleman estimates, observability inequalities and null controllability results for parabolic equations with non smooth coefficients degenerating at an interior point.
Conditions for the existence and uniqueness of weak solutions for a class of nonlinear nonlocal degenerate parabolic equations are established. The asymptotic behaviour of the solutions as time tends to infinity are also studied. In particular, the finite time extinction and polynomial decay properties are proved.
For a general class of divergence type quasi-linear degenerate parabolic equations with measurable coeffcients and lower order terms from non-linear Kato-type classes, we prove local boundedness and continuity of solutions, and the intrinsic Harnack inequality for positive solutions.
We study the boundary behavior of non-negative solutions to a class of degenerate/singular parabolic equations, whose prototype is the parabolic $p$-Laplacian. Assuming that such solutions continuously vanish on some distinguished part of the lateral part $S_T$ of a Lipschitz cylinder, we prove Carleson-type estimates, and deduce some consequences under additional assumptions on the equation or the domain. We then prove analogous estimates for non-negative solutions to a class of degenerate/singular parabolic equations, of porous medium type.
This paper deals with a class of nonlinear anisotropic parabolic equations with degenerate coercivity. Using the anisotropic Gagliardo-Nirenberg-type inequality, we prove some existence and regularity results for the solutions under the framework of anisotropic Sobolev spaces, which generalize the previous results of [10,18,23].
We find an explicit form of entropy solutions to a Riemann problem for a degenerate nonlinear parabolic equation with piecewise constant velocity and diffusion coefficients. It is demonstrated that this solution corresponds to the minimum point of some strictly convex function of a finite number of variables.
Under a precise nonlinearity-diffusivity assumption we establish the decay of entropy solutions of a degenerate nonlinear parabolic equation with initial data being a sum of periodic function and a function vanishing at infinity (in the sense of measure).
We study uniqueness of solutions to degenerate parabolic problems, posed in bounded domains, where no boundary conditions are imposed. Under suitable assumptions on the operator, uniqueness is obtained for solutions that satisfy an appropriate integral condition; in particular, such condition holds for possibly unbounded solutions belonging to a suitable weighted $L^1$ space.
We prove global existence of nonnegative weak solutions for a strongly coupled, fourth order degenerate parabolic system governing the motion of two thin fluid layers in a porous medium when capillarity is the sole driving mechanism.
Bounded weak solutions to a particular class of degenerate parabolic cross-diffusion systems are shown to coincide with the unique strong solution determined by the same initial condition on the maximal existence interval of the latter. The proof relies on an estimate established for a relative entropy associated to the system.
We prove an estimate on the modulus of continuity at a boundary point of a cylindrical domain for local weak solutions to degenerate parabolic equations of $p$-laplacian type. The estimate is given in terms of a Wiener-type integral, defined by a proper elliptic $p$-capacity.
In this paper the well-posedness of some degenerate parabolic equations with a dynamic boundary condition is considered. To characterize the target degenerate parabolic equation from the Cahn-Hilliard system, the nonlinear term coming from the convex part of the double-well potential is chosen using a suitable maximal monotone graph. The main topic of this paper is the existence problem under an assumption for this maximal monotone graph for treating a wider class. The existence of a weak solution is proved.
We establish weak Harnack inequalities for positive, weak supersolutions to certain doubly degenerate parabolic equations. The prototype of this kind of equations is $$\partial_tu-\operatorname{div}|u|^{m-1}|Du|^{p-2}Du=0,\quad p>2,\quad m+p>3.$$ Our proof is based on Caccioppoli inequalities, De Giorgi's estimates and Moser's iterative method.
We prove that solutions to Cauchy problems related to the $p$-parabolic equations are stable with respect to the nonlinearity exponent $p$. More specifically, solutions with a fixed initial trace converge in an $L^q$-space to a solution of the limit problem as $p>2$ varies.
We discuss the well-posedness and decay of Besicovitch almost periodic solutions for a class of nonlinear degenerate anisotropic hyperbolic-parabolic equations. In our definition of weak entropy solution the initial data is only assumed in a weak sense, and in this connection the strong trace of the solution in the initial time hyperplane is also established.
This paper studies a fourth-order, nonlinear, doubly-degenerate parabolic equation derived from the thin film equation in spherical geometry. A regularization method is used to study the equation and several useful estimates are obtained. The main result of this paper is to prove the existence of a weak solution of the equation in a weighted Sobolev space.
This article presents a relationship between the sharp constant of the Sobolev inequality and the initial criterion to the global existence of degenerate parabolic-parabolic Keller-Segel system with the diffusion exponent $\frac{2n}{2+n}<m<2-\frac{2}{n}$. The global weak solution obtained in this article does not need any smallness assumption on the initial data. Furthermore, a uniform in time $L^{\infty}$ estimate of the weak solutions is obtained via the Moser iteration, where the constant in $L^p$ estimate for the gradient of the chemical concentration has been exactly formulated in order t…
We prove $L^p$-parabolic a-priori estimates for $\partial_t u + \sum_{i,j=1}^d c_{ij}(t)\partial_{x_i x_j}^2 u = f $ on $R^{d+1}$ when the coefficients $c_{ij}$ are locally bounded functions on $R$. We slightly generalize the usual parabolicity assumption and show that still $L^p$-estimates hold for the second spatial derivatives of $u$. We also investigate the dependence of the constant appearing in such estimates from the parabolicity constant. Finally we extend our estimates to parabolic equations involving non-degenerate Ornstein-Uhlenbeck type operators.
We determine sufficient conditions for the occurrence of a pointwise gradient estimate for the evolution operators associated to nonautonomous second order parabolic operators with (possibly) unbounded coefficients. Moreover we exhibit a class of operators which satisfy our conditions.
This paper deals with different characterizations of sets of nonlinear parabolic capacity zero, with respect to the parabolic p-Laplace equation. Specifically we prove that certain interior polar sets can be characterized by sets of zero nonlinear parabolic capacity. Furthermore we prove that zero capacity sets are removable for bounded supersolutions and that sets of zero capacity have a relation to a certain parabolic Hausdorff measure.
We study the generation property for a fourth order operator in divergence or in non divergence form with suitable Neumann boundary conditions. As a consequence we obtain the well posedness for the parabolic equations governed by these operators. The novelty of this paper is that the operators depend on a function $a: [0,1] \rightarrow \R_+$ that degenerates somewhere in the interval.
We study a Neumann type initial-boundary value problem for strongly degenerate parabolic-hyperbolic equations under the nonlinearity-diffusivity condition. We suggest a notion of entropy solution for this problem and prove its uniqueness. The existence of entropy solutions is also discussed.
For a general class of divergence type quasi-linear degenerate parabolic equations with differentiable structure and lower order coefficients form bounded with respect to the Laplacian we obtain $L^q$-estimates for the gradients of solutions, and for the lower order coefficients from a Kato-type class we show that the solutions are Lipschitz continuous with respect to the space variable.
We study the asymptotic behavior of Lipschitz continuous solutions of nonlinear degenerate parabolic equations in the periodic setting. Our results apply to a large class of Hamilton-Jacobi-Bellman equations. Defining S as the set where the diffusion vanishes, i.e., where the equation is totally degenerate, we obtain the convergence when the equation is uniformly parabolic outside S and, on S, the Hamiltonian is either strictly convex or satisfies an assumption similar of the one introduced by Barles-Souganidis (2000) for first-order Hamilton-Jacobi equations. This latter assumption allows to …
