We investigate a weak space-time formulation of the heat equation and its use for the construction of a numerical scheme. The formulation is based on a known weak space-time formulation, with the difference that a pointwise component of the solution, which in other works is usually neglected, is now kept. We investigate the role of such a component by first using it to obtain a pointwise bound on the solution and then deploying it to construct a numerical scheme. The scheme obtained, besides being quasi-optimal in the $L^2$ sense, is also pointwise superconvergent in the temporal nodes. We prove a priori error estimates and we present numerical experiments to empirically support our findings.
A mild formulation for stochastic parabolic Anderson model with time-homogeneous Gaussian potential suggests a way of defining a solution to obtain its optimal regularity. Two different interpretations in the equation or in the mild formulation are possible with usual pathwise product and the Wick product: the usual pathwise interpretation is mainly discussed. We emphasize that a modified version of parabolic Schauder estimates is a key idea for the existence and uniqueness of a mild solution. In particular, the mild formulation is crucial to investigate a relation between the equation with us…
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.
We establish the pointwise continuity of bounded weak solutions to of a class of scalar parabolic equations and strongly coupled parabolic systems. Our approach to the regularity theory of parabolic scalar equations is quite elementary and its applications to strongly coupled systems does not require higher $L^p$ integrability of derivatives.
This article proves a weak Harnack inequality with a tail term for sign changing supersolutions of a mixed local and nonlocal parabolic equation. Our argument is purely analytic. It is based on energy estimates and the Moser iteration technique. Instead of the parabolic John-Nirenberg lemma, we adopt a lemma of Bombieri to the mixed local and nonlocal parabolic case. To this end, we prove an appropriate reverse Hölder inequality and a logarithmic estimate for weak supersolutions.
Via abstract results on maximal monotone operators and compactness property of Nemickii operator, existence of a weak solution for a class of nonlinear parabolic systems of partial differential equations is proven.
We present a new stability result for viscosity solutions of fully nonlinear parabolic equations which allows to pass to the limit when one has only weak convergence in time of the nonlinearities.
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.
This paper study the well--posedness of the entropy formulation given by Plotnikov in [{Differential Equations}, 30 (1994), pp. 614--622] for forward-backward parabolic problem obtained as singular limit of a proper pseudoparabolic approximation. It was proved in [C. Mascia, A. Terracina, and A. Tesei, {Arch.\ Ration.\ Mech.\ Anal.}, 194 (2009), pp. 887--925] that such formulation gives uniqueness when the solution takes values in the stable phases. Here we consider the situation in which unstable phase is taken in account, proving that, in general, uniqueness does not hold.
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 uniqueness for weak solutions to abstract parabolic equations with the fractional Marchaud or Caputo time derivative. We consider weak solutions in time for divergence form equations when the fractional derivative is transferred to the test function.
We prove global existence of nonnegative weak solutions to a degenerate parabolic system which models the interaction of two thin fluid films in a porous medium. Furthermore, we show that these weak solutions converge at an exponential rate towards flat equilibria.
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 paper deals with two separate but related results. First we consider weak solutions to a parabolic operator with Hörmander vector fields. Adapting the iteration scheme of Jürgen Moser for elliptic and parabolic equations in $\mathbb{R}^n$ we show a parabolic Harnack inequality. Then, after proving the Harnack inequality for weak solutions to equations of the form $u_t = \sum X_i (a_{ij} X_j u)$ we use this to show Hölder continuity. We assume the coefficients are bounded and elliptic. The iteration scheme is a tool that may be adapted to many settings and we extend this to nonlinear parab…
We establish the existence of non-negative global weak solutions for a strongly couple degenerated parabolic system which was obtained as an approximation of the two-phase Stokes problem driven solely by capillary forces. Moreover, the system under consideration may be viewed as a two-phase generalization of the classical Thin Film equation.
We establish global regularity for weak solutions to quasilinear divergence form elliptic and parabolic equations over Lipschitz domains with controlled growth conditions on low order terms. The leading coefficients belong to the class of BMO functions with small mean oscillations with respect to $x$.
In this paper we study the regularity of weak solutions to an elliptic-parabolic system modeling natural network formation. The system is singular and involves cubic nonlinearity. Our investigation reveals that weak solutions are Hölder continuous when the space dimension $N$ is $2$. This is achieved via an inequality associated with the Stummel-Kato class of functions and refinement of a lemma originally due to S. Campanato and C. B. Morrey (\cite{G}, p. 86).
We study the relationship of viscosity and weak solutions to the equation \[ \smash{\partial_{t}u-Δ_{p}u=f(Du)} \] where $p>1$ and $f\in C(\mathbb{R}^{N})$ satisfies suitable assumptions. Our main result is that bounded viscosity supersolutions coincide with bounded lower semicontinuous weak supersolutions. Moreover, we prove the lower semicontinuity of weak supersolutions when $p\geq2$.
We investigate weighted Sobolev regularity of weak solutions of non-homogeneous parabolic equations with singular divergence-free drifts. Assuming that the drifts satisfy some mild regularity conditions, we establish local weighted $L^p$-estimates for the gradients of weak solutions. Our results improve the classical one to the borderline case by replacing the $L^\infty$-assumption on solutions by solutions in the John-Nirenberg \textup{BMO} space. The results are also generalized to parabolic equations in divergence form with small oscillation elliptic symmetric coefficients and therefore imp…
We discuss the existence and uniqueness of the weak solution of the following quasilinear parabolic equation $u_t-Δ_{p(x)}u = f(x,u)$ in $ (0,T)\timesΩ$; $u = 0$ on $(0,T)\times\partialΩ$; $u(0,x)=u_0(x)$ in $Ω$; involving the $p(x)$-Laplacian operator. Next, we discuss the global behaviour of solutions and in particular some stabilization properties.
In this thesis, a unified approach to prove the boundedness of gradients of solutions to degenerate and singular elliptic and parabolic phi-Laplacian systems is presented. At first, a Cacciopoli-type energy inequality with an additional function f which can be chosen freely is proven. Then, Di Giorgi's method is applied using level sets which will lead to L-infinity-estimates on the gradient of the weak solution.
We consider Kirchhoff equations with a small parameter epsilon in front of the second-order time-derivative, and a dissipative term whose coefficient may tend to 0 as t -> + infinity (weak dissipation). In this note we present some recent results concerning existence of global solutions, and their asymptotic behavior both as t -> + infinity and as epsilon -> 0. Since the limit equation is of parabolic type, this is usually referred to as a hyperbolic-parabolic singular perturbation problem. We show in particular that the equation exhibits hyperbolic or parabolic behavior depending on the value…
We study the weak solvability of a nonlinearly coupled system of parabolic and pseudo-parabolic equations describing the interplay between mechanics, chemical reactions, diffusion and flow in a mixture theory framework. Our approach relies on suitable discrete-in-time energy-like estimates and discrete Gronwall inequalities. In selected parameter regimes, these estimates ensure the convergence of the Rothe method for the discretized partial differential equations.
In this paper we study a rather wide class of quasilinear parabolic problems with nonlinear boundary condition and nonstandard growth terms. It includes the important case of equations with a $p(t,x)$-Laplacian. By means of the localization method and De Giorgi's iteration technique we derive global a priori bounds for weak solutions of such problems. Our results seem to be new even in the constant exponent case.
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 pro…
We introduce a notion of quasilinear parabolic equations over metric measure spaces. Under sharp structural conditions, we prove that local weak solutions are locally bounded and satisfy the parabolic Harnack inequality. Applications include the parabolic maximum principle and pointwise estimates for weak solutions.
We study a nonlocal evolution equation that involves a pseudo-parabolic third-order term. The equation models almost uni-directional two-phase flow in Brinkman regimes. We prove the existence of weak solutions for this equation. We also give a series of numerical examples that demonstrate the ability of the equation to support overshooting and explore the behavior of solutions in various limit regimes.
In this paper, we prove the Lorentz space $L^{q,p}$-estimates for gradients of very weak solutions to the linear parabolic equations with $\mathbf{A}_q$-weights $$u_t-\operatorname{div}(A(x,t)\nabla u)=\operatorname{div}(F),$$ in a bounded domain $Ω\times (0,T)\subset\mathbb{R}^{N+1}$, where $A$ has a small mean oscillation, and $Ω$ is a Lipchistz domain with a small Lipschitz constant.
We establish a new regularity property for weak solutions of parabolic systems with coefficients depending measurably on time as well as on all spatial variables. Namely, weak solutions are locally H{ö}lder continuous Lp valued functions for some p > 2.
We obtain local boundedness of weak solutions of double phase quasilinear parabolic equations of the form \[u_t-\text{div} \left(|\nabla u|^{p-2}\nabla u+a(x,t)|\nabla u|^{q-2}\nabla u\right)=0,\] where, we have imposed the restrictions $\frac{2N}{N+2}<p<\infty$, $0\leq a(x,t)\leq M$ is measurable and $q < p\frac{N+1}{N-1}$.
We consider the heat equation on a bounded domain subject to an inhomogeneous forcing in terms of a rate-independent (hysteresis) operator and a control variable. The aim of the paper is to establish a functional analytical setting which allows to prove weak differentiability properties of the control-to-state mapping. Using results of [BK] and [B] on the weak differentiability of scalar rate-independent operators, we prove Bouligand and Newton differentiability in suitable Bochner spaces of the control-to-state mapping in a parabolic problem.
We study the regularity of weak solutions to a certain class of second order parabolic system under the only assumption of continuous coefficients. By using the $A-$caloric approximation argument, we claim that the weak solution $u$ to such system is locally Hölder continuous with any exponent $α\in(0,1)$ outside a singular set with zero parabolic measure. In particular, we prove that the regularity point in $Q_T$ is an open set with full measure, and we obtain a general criterion for a weak solution to be regular in the neighborhood of a given point. Finally, we deduce the fractional time and…
Hyperbolic-parabolic systems have spatially homogenous stationary states. When the dissipation is weak, one can derive weakly nonlinear-dissipative approximations that govern perturbations of these constant states. These approximations are quadratically nonlinear. When the original system has an entropy, the approximation is formally dissipative in a natural Hilbert space. We show that when the approximation is strictly dissipative it has global weak solutions for all initial data in that Hilbert space. We also prove a weak-strong uniqueness theorem for it. In addition, we give a Kawashima typ…
We consider mildly degenerate Kirchhoff equations with a small parameter and a weak dissipation term. We prove the existence of global solutions when the parameter is small with respect to the size of initial data. Then we provide global-in-time error estimates on the difference between the solution of our problem and the solution of the corresponding first order problem.
The weak Harnack inequality for $L^p$-viscosity supersolutions of fully nonlinear second-order uniformly parabolic partial differential equations with unbounded coefficients and inhomogeneous terms is proved. It is shown that Hölder continuity of $L^p$-viscosity solutions is derived from the weak Harnack inequality for $L^p$-viscosity supersolutions. The local maximum principle for $L^p$-viscosity subsolutions and the Harnack inequality for $L^p$-viscosity solutions are also obtained. Several further remarks are presented when equations have superlinear growth in the first space derivatives.
In this work we consider a class of stochastic parabolic equations with singular space depending potential, random driving force and random initial condition. For the analysis of these equations we combine the chaos expansion method from the white noise analysis and the concept of very weak solutions. For given stochastic parabolic equation we introduce the notion of a stochastic very weak solution, prove the existence and uniqueness of the very weak solution to corresponding stochastic initial value problem and show its independence of a regularization on given singular potential. In addition…
We are concerned in this paper with the degenerate fractional diffusion advection equations posed in bounded domains. Due to a suitable formulation, we show the existence of weak entropy solutions for measurable and bounded initial and Dirichlet boundary data. Moreover, we prove a $L^1-$type contraction property for weak entropy solutions obtained via parabolic perturbation. This is a weak selection principle which means that the weak entropy solutions are stable in this class.
We show that local weak solutions to parabolic systems of p-Laplace type are H{ö}lder continuous in time with values in a spatial Lebesgue space and H{ö}lder continuous on almost every time line. We provide an elementary and self-contained proof building on the local higher integrability result of Kinnunen and Lewis.
The aim of this paper is to establish a higher integrability result for very weak solutions of certain parabolic systems whose model is the parabolic $p(x,t)$-Laplacian system. Under assumptions on the exponent function $p:Ω_T=Ω\times (0,T)\to\left(\frac{2n}{n+2},2\right]$, it is shown that any very weak solution $u:Ω_T\rightarrow\mathbb{R}^N$ with $|Du|^{p(\cdot)(1-\varepsilon)}\in L^1(Ω_T)$ belongs to the natural energy spaces, i.e. $|Du|^{p(\cdot)}\in L^1_{\operatorname{loc}}(Ω_T)$, provided $ε>0$ is small enough. This extends the main result of [V. Bögelein and Q. Li, Nonlinear Anal., 98 (…
The aim of this paper is to prove existence of weak solutions of hyperbolic-parabolic evolution inclusions defined on Lipschitz domains with mixed boundary conditions describing, for instance, damage processes and elasticity with inertia terms. To this end, a suitable weak formulation to deal with such evolution inclusions in a non-smooth setting is presented. Then, existence of weak solutions is proven by utilizing time-discretization, $H^2$-regularization of the displacement variable and variational techniques from [C. Heinemann, C. Kraus: Existence results of weak solutions for Cahn-Hilliar…
We consider a model Venttsel type problem for linear parabolic systems of equations. The Venttsel type boundary condition is fixed on the flat part of the lateral surface of a given cylinder. It is defined by parabolic operator (with respect to the tangential derivatives) and the conormal derivative. The Hölder continuity of a weak solution of the problem is proved under optimal assumptions on the data. In particular, only boundedness in the time variable of the principal matrices of the system and the boundary operator is assumed. All results are obtained by so-called A(t)-caloric method.
In this article, we provide existence results for a general class of nonlocal and nonlinear second-order parabolic equations. The main motivation comes from front propagation theory in the cases when the normal velocity depends on the moving front in a nonlocal way. Among applications, we present level-set equations appearing in dislocations' theory and in the study of Fitzhugh-Nagumo systems.
We discuss pointwise behavior of weak supersolutions for a class of doubly nonlinear parabolic fractional $p$-Laplace equations which includes the fractional parabolic $p$-Laplace equation and the fractional porous medium equation. More precisely, we show that weak supersolutions have lower semicontinuous representative. We also prove that the semicontinuous representative at an instant of time is determined by the values at previous times. This gives a pointwise interpretation for a weak supersolution at every point. The corresponding results hold true also for weak subsolutions. Our results …
We consider mixed local and nonlocal quasilinear parabolic equations of $p$-Laplace type and discuss several regularity properties of weak solutions for such equations. More precisely, we establish local boundeness of weak subsolutions, lower semicontinuity of weak supersolutions as well as upper semicontinuity of weak subsolutions. We also discuss the pointwise behavior of the semicontinuous representatives. Our main results are valid for sign changing solutions. Our approach is purely analytic and is based on energy estimates and the De Giorgi theory.
In this paper, we introduce a weak maximum principle-based approach to input-to-state stability (ISS) analysis for certain nonlinear partial differential equations (PDEs) with boundary disturbances. Based on the weak maximum principle, a classical result on the maximum estimate of solutions to linear parabolic PDEs has been extended, which enables the ISS analysis for certain {}{nonlinear} parabolic PDEs with boundary disturbances. To illustrate the application of this method, we establish ISS estimates for a linear reaction-diffusion PDE and a generalized Ginzburg-Landau equation with {}{mixe…
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 study nonlinear parabolic PDEs with Orlicz-type growth conditions. The main result gives the existence of a unique solution to the obstacle problem related to these equations. To achieve this we show the boundedness of weak solutions and that a uniformly bounded sequence of weak supersolutions converges to a weak supersolution. Moreover, we prove that if the obstacle is continuous, so is the solution.
We prove the existence of global-in-time weak solutions to a version of the parabolic-parabolic Keller-Segel system in one spatial dimension. If the coupling of the system is suitably weak, we prove convergence of those solutions to the unique equilibrium with an exponential rate. Our proofs are based on an underlying gradient flow structure with respect to a mixed Wasserstein-$L^2$ distance.
