Parabolic Equations In Divergence Form Research Articles

Degenerate linear parabolic equations in divergence form on the upper half space

We study a class of second-order degenerate linear parabolic equations in divergence form in ( − ∞ , T ) × R + d (-\infty , T) \times {\mathbb {R}}^d_+ with homogeneous Dirichlet boundary condition on ( − ∞ , T ) × ∂ R + d (-\infty , T) \times \partial {\mathbb {R}}^d_+ , where R + d = { x ∈ R d : x d > 0 } {\mathbb {R}}^d_+ = \{x \in {\mathbb {R}}^d: x_d>0\} and T ∈ ( − ∞ , ∞ ] T\in {(-\infty , \infty ]} is given. The coefficient matrices of the equations are the product of μ ( x d ) \mu (x_d) and bounded uniformly elliptic matrices, where μ ( x d ) \mu (x_d) behaves like x d α x_d^\alpha for some given α ∈ ( 0 , 2 ) \alpha \in (0,2) , which are degenerate on the boundary { x d = 0 } \{x_d=0\} of the domain. Our main motivation comes from the analysis of degenerate viscous Hamilton-Jacobi equations. Under a partially VMO assumption on the coefficients, we obtain the well-posedness and regularity of solutions in weighted Sobolev spaces. Our results can be readily extended to systems.

Stabilization in degenerate parabolic equations in divergence form and application to chemotaxis systems

This paper presents a stabilization result for weak solutions of degenerate parabolic equations in divergence form. More precisely, the result asserts that the global-in-time weak solution converges to the average of the initial data in some topology as time goes to infinity. It is also shown that the result can be applied to a degenerate parabolic-elliptic Keller-Segel system.

Parabolic homogenization with an interface

This paper considers a family of second-order parabolic equations in divergence form with rapidly oscillating and time-dependent periodic coefficients and an interface between two periodic structures. Following a framework initiated by Blanc, Le Bris and Lions and a generalized two-scale expansion in divergence form of elliptic homogenization with an interface by Josien, we can determine the effective (or homogenized) equation with the coefficient matrix being piecewise constant and discontinuous across the interface. Moreover, we obtain the $O(\varepsilon)$ convergence rates in $L^{2(d+2)/{d}}_{x,t}$ with $\varepsilon$-smoothing method and the uniform interior Lipschitz estimates via compactness argument.

Application of weak stabilization theory for degenerate parabolic equations in divergence form to a chemotaxis model for tumor invasion

This paper deals with the chemotaxis model for tumor invasion, \begin{document}$ \begin{align*} \begin{cases} u_t=\nabla\cdot(D(u,w)\nabla u -u\nabla v), &x\in\Omega,\ t>0,\\ v_t=\Delta v+wz,&x\in\Omega,\ t>0,\\ w_t=-wz,&x\in\Omega,\ t>0,\\ z_t=\Delta z-z+u,&x\in\Omega,\ t>0 \end{cases} \end{align*} $\end{document} under the conditions that $ (D(u, w)\nabla u -u\nabla v)\cdot \nu = \nabla v\cdot\nu = \nabla z\cdot \nu = 0 $ on $ \partial\Omega $ and that $ (u, v, w, z)|_{t = 0} = (u_0, v_0, w_0, z_0) $, where $ \Omega \subset \mathbb{R}^N $ ($ N = 1, 2, 3 $) is a bounded domain with smooth boundary $ \partial \Omega $ and $ \nu $ is the outward normal vector to $ \partial\Omega $. The function $ D $ is supposed to satisfy $ D(0, \sigma _2) = 0 $ and $ D(\sigma _1, \sigma _2)\geq \widetilde{D}(\sigma _1) $, where $ \widetilde{D}(\sigma _1) = k\sigma _1^{m-1} $ $ (\sigma _1\leq \sigma _0) $, $ \widetilde{D}(\sigma _1) = k\sigma _0^{m-1} $ $ (\sigma _1\geq \sigma _0) $ for some $ k>0 $, $ m>1 $ and $ \sigma_0>0 $, which means that the first equation is a degenerate parabolic equation. The previous paper [4] with Fujie and Ito proved global existence and boundedness of weak solutions to the same system, however, an imperfect stabilization property was established. The purpose of this paper is to apply weak stabilization theory in [8] to obtain that$ u(\cdot, t) \to \overline{u_0}\quad \text{weakly}^*\ \text{in}\ L^\infty(\Omega) \quad \text{as}\ t \to \infty, $where $ \overline{u_0}: = \frac{1}{|\Omega|}\int_\Omega u_0 $.

A Marcinkiewicz Type Interpolation Theorem for Orlicz Spaces and Its Application

In this paper, we show a Marcinkiewicz type interpolation theorem for Orlicz spaces. As an application, we obtain an existence result for a parabolic equation in divergence form.

Solvability Issues for Some Noncoercive and Nonmonotone Parabolic Equations Arising in the Image Denoising Problems

This paper is devoted to the solvability of an initial-boundary value problem for second-order parabolic equations in divergence form with variable order of nonlinearity. The characteristic feature of the considered class of Cauchy-Neumann parabolic problem is the fact that the variable exponent p(t, x) and the anisotropic diffusion tensor D(t, x) are not well predefined a priori, but instead these characteristic depend on a solution of this problem, i.e., p = p(t, x, u) and D = D(t, x, u). Recently, it has been shown that the similar models appear in a natural way as the optimality conditions for some variational problems related to the image restoration technique. However, from practical point of view, in this case some principle difficulties can appear because of the absence of the corresponding rigorous mathematically substantiation. Thus, in this paper, we study the solvability issues of the Cauchy-Neumann parabolic boundary value problem for which the corresponding principle operator is strongly non-linear, non-monotone, has a variable order of nonlinearity, and satisfies a nonstandard coercivity and boundedness conditions that do not fall within the scope of the classical method of monotone operators. To construct a weak solution, we apply the technique of passing to the limit in a special approximation scheme and the Schauder fixed-point theorem.

Weak stabilization in degenerate parabolic equations in divergence form: application to degenerate Keller–Segel systems

Weak stabilization in degenerate parabolic equations in divergence form: application to degenerate Keller–Segel systems

Parabolic equations with unbounded lower-order coefficients in Sobolev spaces with mixed norms

We prove the \(L_{p,q}\)-solvability of parabolic equations in divergence form with full lower-order terms. The coefficients and non-homogeneous terms belong to mixed Lebesgue spaces with the lowest integrability conditions. In particular, the coefficients for the lower-order terms are not necessarily bounded. We study both the Dirichlet and conormal derivative boundary value problems on irregular domains. We also prove embedding results for parabolic Sobolev spaces, the proof of which is of independent interest.

On $$\varvec{C^{1/2,1}}$$, $$\varvec{C^{1,2}}$$, and $$\varvec{C^{0,0}}$$ estimates for linear parabolic operators

We show that weak solutions to parabolic equations in divergence form with zero Dirichlet boundary conditions are continuously differentiable up to the boundary when the leading coefficients have Dini mean oscillation and the lower order coefficients verify certain conditions. Similar results are obtained for non-divergence form parabolic operators and their adjoint operators. Under similar conditions, we also establish a Harnack inequality for nonnegative adjoint solutions, together with upper and lower Gaussian bounds for the global fundamental solution.

Regularity for parabolic equations with singular or degenerate coefficients

We study parabolic equations in divergence form with coefficients which are singular or degenerate as Muckenhoupt weight functions in one spatial variable. We establish weighted reverse Holder’s inequalities, and Lipschitz estimates for weak solutions of homogeneous equations with coefficients depending only on one spatial variable. We then use these results to prove interior, boundary, and global weighted estimates of Calderon-Zygmund type for weak solutions, assuming that the coefficients are partially vanishing mean oscillations with respect to the considered weights. The solvability in weighted Sobolev spaces is also achieved. Such results are new even for elliptic equations and our results can be readily extended to systems.

Regularity of weak supersolutions to elliptic and parabolic equations: Lower semicontinuity and pointwise behavior

We demonstrate a measure theoretical approach to the local regularity of weak supersolutions to elliptic and parabolic equations in divergence form. In the first part, we show that weak supersolutions become lower semicontinuous after redefinition on a set of measure zero. The proof relies on a general principle, i.e. the De Giorgi type lemma, which offers a unified approach for a wide class of elliptic and parabolic equations, including an anisotropic elliptic equation, the parabolic p-Laplace equation, and the porous medium equation. In the second part, we shall show that for parabolic problems the lower semicontinuous representative at an instant can be recovered pointwise from the “ess lim inf” of past times. We also show that it can be recovered by the limit of certain integral averages of past times. The proof hinges on the expansion of positivity for weak supersolutions. Our results are structural properties of partial differential equations, independent of any kind of comparison principle.

On conormal derivative problem for parabolic equations with Dini mean oscillation coefficients

<p style='text-indent:20px;'>We show that weak solutions to parabolic equations in divergence form with conormal boundary conditions are continuously differentiable up to the boundary when the leading coefficients have Dini mean oscillation and the lower order coefficients verify certain integrability conditions.

Weighted Lq(⋅)-estimates for the nonlinear parabolic equations with non-standard growth

We establish a global Calderón–Zygmund type estimate for the nonlinear parabolic equation in divergence form with variable exponent growth. The weighted Lq(⋅)-estimates are proved under some precise conditions on the variable exponent, the nonlinearity and the boundary of the domain. Our results extend the existing regularity estimates in variable Lebesgue spaces to the weighted case.

On the Boundary Point Principle for divergence-type equations

We provide some versions of the Zaremba–Hopf–Oleinik boundary point lemma for general elliptic and parabolic equations in divergence form under the sharp requirements on the coefficients of equations and on the boundaries of domains.

Lp-estimates for time fractional parabolic equations in divergence form with measurable coefficients

In this paper, we establish Lp-estimates and solvability for time fractional divergence form parabolic equations in the whole space when leading coefficients are merely measurable in one spatial variable and locally have small mean oscillations with respect to the other variables. The corresponding results for equations on a half space are also derived.

An [formula omitted] regularity condition that implies the [formula omitted] property of parabolic measure

We prove that a condition of Lq-regularity of a Dirichlet problem associated to second order divergence form parabolic equations implies the A∞ property of the corresponding parabolic measure, over a class of non-cylindrical domains.

Optimal Morrey estimate for parabolic equations in divergence form via Green's functions

This paper presents a local Morrey regularity with the optimal exponents for linear parabolic equations in divergence form under the assumption that the leading coefficient is independent of $t$ and not necessarily symmetric based on a rather different approach. Here, we achieve it by applying natural growth properties of Green's functions through the use of parabolic operators and the hole-filling technique.

Quasi-linear SPDEs in divergence form

We develop a solution theory in Hölder spaces for a quasi-linear stochastic PDE driven by an additive noise. The key ingredients are two deterministic PDE lemmas which establish a priori Hölder bounds for a parabolic equation in divergence form with irregular right-hand-side term. We apply these bounds to the case of a right-hand-side noise term which is white in time and trace class in space, to obtain stretched exponential bounds for the Hölder semi-norms of the solution.

L2 Solvability of boundary value problems for divergence form parabolic equations with complex coefficients

We consider parabolic operators of the form∂t+L,L=−divA(X,t)∇, in R+n+2:={(X,t)=(x,xn+1,t)∈Rn×R×R:xn+1>0}, n≥1. We assume that A is a (n+1)×(n+1)-dimensional matrix which is bounded, measurable, uniformly elliptic and complex, and we assume, in addition, that the entries of A are independent of the spatial coordinate xn+1 as well as of the time coordinate t. For such operators we prove that the boundedness and invertibility of the corresponding layer potential operators are stable on L2(Rn+1,C)=L2(∂R+n+2,C) under complex, L∞ perturbations of the coefficient matrix. Subsequently, using this general result, we establish solvability of the Dirichlet, Neumann and Regularity problems for ∂t+L, by way of layer potentials and with data in L2, assuming that the coefficient matrix is a small complex perturbation of either a constant matrix or of a real and symmetric matrix.

Boundedness of single layer potentials associated to divergence form parabolic equations with complex coefficients

We consider parabolic operators of the form $$\partial_t+\mathcal{L}, \mathcal{L}:=-\mbox{div}\, A(X,t)\nabla,$$ in$\mathbb R_+^{n+2}:=\{(X,t)=(x,x_{n+1},t)\in \mathbb R^{n}\times \mathbb R\times ...