We give a unified approach to weighted mixed-norm estimates and solvability for both the usual and time fractional parabolic equations in nondivergence form when coefficients are merely measurable in the time variable. In the spatial variables, the leading coefficients locally have small mean oscillations. Our results extend the previous result in [6] for unmixed $L_p$-estimates without weights.
We prove the boundedness of the time derivative in the parabolic Signorini problem, as well as establish its Hölder continuity at regular free boundary points.
We consider an oblique derivative problem in a wedge for nondivergence parabolic equations with discontinuous in $t$ coefficients. We obtain weighted coercive estimates of solutions in anisotropic Sobolev spaces.
We study a nonlocal nonlinear parabolic problem with a fractional time derivative. We prove a Krylov-Safonov type result; mainly, we prove Holder regularity of solutions. Our estimates remain uniform as the order of the fractional time derivative approaches 1.
We study the Evolutionary p-Laplace Equation in the singular case 1 < p < 2. We prove that a weak solution has a time derivative in Sobolev's sense and that the time derivative is locally summable to some power > 1.
We study regularity for a parabolic problem with fractional diffusion in space and a fractional time derivative. Our main result is a De Giorgi-Nash-Moser Holder regularity theorem for solutions in a divergence form equation. We also prove results regarding existence, uniqueness, and higher regularity in time.
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 two maximal regularity results in spaces of continuous and Hölder continuous functions, for a mixed linear Cauchy-Dirichlet problem with a fractional time derivative $\mathbb{D}_t^α$. This derivative is intended in the sense of Caputo and $α$ is taken in $(0, 2)$. In case $α= 1$, we obtain maximal regularity results for mixed parabolic problems already known in mathematica literature.
We prove regularity estimates for time derivatives of a large class of nonlinear parabolic partial differential systems. This includes the instationary (symmetric) p-Laplace system and models for non Newtonien fluids of powerlaw or Carreau type. By the use of special weak different quotients, adapted to the variational structure we bound fractional derivatives of $u_t$ in time and space direction. Although the estimates presented here are valid under very general assumptions they are a novelty even for the parabolic p-Laplace equation.
We prove Hölder regularity results for a class of nonlinear parabolic problem with fractional-time derivative with nonlocal and Mittag-Leffler nonsingular kernel. Existence of weak solutions via approximating solutions is proved. Moreover, the Hölder continuity of viscosity solutions is obtained. We get the similar results as those obtained by Allen (see {\url{https://arxiv.org/abs/1610.10073}})
We study integrability of the derivative of solutions to a singular one-dimensional parabolic equation with initial data in $W^{1,1}$. In order to avoid additional difficulties we consider only the periodic boundary conditions. The problem we study is a gradient flow of a convex, linear growth variational functional. We also prove a similar result for the elliptic companion problem, i.e. the time semidiscretization.
We consider the decay of solution to fractional diffusion equation with the distributed order Caputo derivative. We assume that the elliptic operator is time-dependent and that the weight function contained in the definition of the distributed order Caputo derivative is just integrable. We establish the relation between behavior of weight function near zero and the decay rate of solution.
We consider an oblique derivative problem for non-divergence parabolic equations with discontinuous in $t$ coefficients in a half-space. We obtain weighted coercive estimates of solutions in anisotropic Sobolev spaces. We also give an application of this result to linear parabolic equations in a bounded domain. In particular, if the boundary is of class ${\cal C}^{1,δ}$, $δ\in (0,1]$, then we present a coercive estimate of solutions in weighted anisotropic Sobolev spaces, where the weight is a power of the distance to the boundary.
In the present work we investigate the Tricomi problem with integral gluing condition for parabolic-hyperbolic equation with the Caputo fractional order derivative. Using the method of energy integrals we prove the uniqueness of the solution for considered problem. The existence will be proved using methods of ordinary differential equations, Fredholm integral equations and solution will be represented in an explicit form.
We present an abstract framework for treating the theory of well-posedness of solutions to abstract parabolic partial differential equations on evolving Hilbert spaces. This theory is applicable to variational formulations of PDEs on evolving spatial domains including moving hypersurfaces. We formulate an appropriate time derivative on evolving spaces called the material derivative and define a weak material derivative in analogy with the usual time derivative in fixed domain problems; our setting is abstract and not restricted to evolving domains or surfaces. Then we show well-posedness to a …
In this paper, we establish the gradient and Pogorelov estimates for $k$-convex-monotone solutions to parabolic $k$-Hessian equations of the form $-u_tσ_k(λ(D^2u))=ψ(x,t,u)$. We also apply such estimates to obtain a Liouville type result, which states that any $k$-convex-monotone and $C^{4,2}$ solution $u$ to $-u_tσ_k(λ(D^2u))=1$ in $\mathbb{R}^n\times(-\infty,0]$ must be a linear function of $t$ plus a quadratic polynomial of $x$, under some growth assumptions on $u$.
This paper proves Hölder continuity of viscosity solutions to certain nonlocal parabolic equations that involve a generalized fractional time derivative of Marchaud or Caputo type. As a necessary and preliminary result, this paper first shows that viscosity solutions to certain nonlinear ordinary differential equations involving the generalized fractional time derivative are Hölder continuous.
In this work we investigate a boundary problem with non-local conditions for mixed parabolic-hyperbolic type equation with three lines of type-changing with Caputo fractional derivative in the parabolic part. We equivalently reduce considered problem to the system of second kind Volterra integral equations. In the parabolic part we use solution of the first boundary problem with appropriate Green's function and in hyperbolic parts we use corresponding solutions of the Cauchy problem.
We show that for any uniformly parabolic fully nonlinear second-order equation with bounded measurable "coefficients" and bounded "free" term in any cylindrical smooth domain with smooth boundary data one can find an approximating equation which has a unique continuous solution with the first derivatives bounded and the second spacial derivatives locally bounded. The approximating equation is constructed in such a way that it modifies the original one only for large values of the unknown function and its spacial derivatives.
We consider a Cauchy problem for a fractional anisotropic parabolic equation in anisotropic Hölder spaces. The equation generalizes the heat equation to the case of fractional power of the Laplace operator and the power of this operator can be different with respect to different groups of space variables. The time derivative can be either fractional Caputo - Jrbashyan derivative or usual derivative. Under some necessary conditions on the order of the time derivative we show that the operator of the whole problem is an isomorphism of appropriate anisotropic Hölder spaces. Under some another con…
We derive global analytic representations of fundamental solutions for a class of linear parabolic systems with full coupling of first order derivative terms where coefficient may depend on space and time. Pointwise convergence of the global analytic expansion is proved. This leads to analytic representations of solutions of initial-boundary problems of first and second type in terms of convolution integrals or convolution integrals and linear integral equations. The results have both analytical and numerical impact. Analytically, our representations of fundamental solutions of coupled parabol…
In the class of the so called non-dynamic Fractional Obstacle Problems of parabolic type, it is shown how to obtain higher regularity as well as optimal regularity of the space derivatives of the solution. Furthermore, at free boundary points of positive parabolic density, it is proven that the time derivative of the solution is Hölder continuous. Finally, at regular free boundary points, space-time regularity of the corresponding free boundary is obtained for any fraction $s\in(0,1)$.
We prove gradient estimates for solutions of the oblique derivative problem for a large class of elliptic and parabolic quasilinear PDEs. In particular, we expand on previous work of the author using a maximum principle argument. In addition, we prove estimates for the parabolic problem of evolution via mean curvature with nonzero Neumann data, which seem to be new.
We prove that boundary value problems for fully nonlinear second-order parabolic equations admit $L_{p}$-viscosity solutions, which are in $C^{1+α}$ for an $α\in(0,1)$. The equations have a special structure that the "main" part containing only second-order derivatives is given by a positive homogeneous function of second-order derivatives and as a function of independent variables it is measurable in the time variable and, so to speak, VMO in spatial variables.
We establish Hölder estimates for the time derivative of solutions of fully non-linear parabolic equations that does not necessarily have $C^{2,α}$ estimates.
Hölder estimates for second derivatives are proved for solutions of fully nonlinear parabolic equations in two space variables. Related techniques extend the regularity theory for fully nonlinear parabolic equations in higher dimensions.
The parabolic normalized p-Laplace equation is studied. We prove that a viscosity solution has a time derivative in the sense of Sobolev belonging locally to $L^2$.
We show that for any uniformly parabolic fully nonlinear second-order equation with bounded measurable "coefficients" and bounded "free" term in the whole space or in any cylindrical smooth domain with smooth boundary data one can find an approximating equation which has a continuous solution with the first and the second spatial derivatives under control: bounded in the case of the whole space and locally bounded in case of equations in cylinders. The approximating equation is constructed in such a way that it modifies the original one only for large values of the second spatial derivatives o…
We study existence and regularity properties of solutions to the singular $p$-Laplacean parabolic system in a bounded domain $Ω$. The main purpose is to prove global $L^r(\varepsilon,T;L^q(Ω))$, $\varepsilon\geq0$, integrability properties of the second spatial derivatives and of the time derivative of the solutions. Hence, for suitable $p$ and exponents $r,\,q$, by Sobolev embedding theorems, we deduce global regularity of $u$ and $\nabla u$ in Hölder spaces. Finally we prove a global pointwise bound for the solution under the assumption $p>\frac{2n}{n+2}$.
We prove that the spatial gradient of (variational) solutions to parabolic obstacle problems of p-Laplacian type enjoys the same regularity of the data and of the derivatives of the obstacle in the scale of Lorentz spaces.
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.
We establish Hölder estimates for the time derivative of solutions of non-local parabolic equations under mild assumptions for the boundary data. As a consequence we are able to extend the Evans-Krylov estimate for rough kernels to parabolic equations.
We study the parabolic free boundary problem of obstacle type $$ \lap u-\frac{\partial u}{\partial t}= fχ_{u\ne 0}. $$ Under the condition that $f=Hv$ for some function $v$ with bounded second order spatial derivatives and bounded first order time derivative, we establish the same regularity for the solution $u$. Both the regularity and the assumptions are optimal. Using this result and assuming that $f$ is Dini continuous, we prove that the free boundary is, near so called low energy points, a $C^1$ graph. Our result completes the theory for this type of problems for the heat operator.
We prove optimal pointwise Schauder estimates in the spatial variables for solutions of linear parabolic integro-differential equations. Optimal Hölder estimates in space-time for those spatial derivatives are also obtained.
In this paper there are estimated the derivatives of the solution of an initial boundary value problem for a nonlinear uniformly parabolic equation in the interior with the total variation of the boundary data and the L^{infinity}-norm of the initial condition.
We derive regularity estimates for viscosity solutions to the parabolic normalized p-Laplace. By using approximation methods and scaling arguments for the normalized p-parabolic operator, we show that the gradient of bounded viscosity solutions is locally asymptotically Lipschitz continuous when $p$ is sufficiently close to 2. In addition, we establish regularity estimates in Sobolev spaces.
We give an optimal in mixed (anisotropic) Strichartz type Lebesgue space-time norm estimates for the solution of linear parabolic inhomogeneous initial problem, with are exact or exact up to multiplicative constant coefficient evaluation.
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.
In this article we prove results concerning upper and lower decay estimates for homogeneous Sobolev norms of solutions to a rather general family of parabolic equations. Following the ideas of Kreiss, Hagstrom, Lorenz and Zingano, we use eventual regularity of solutions to directly work with smooth solutions in physical space, bootstrapping decay estimates from the $L^2$ norm to higher order derivatives. Besides obtaining upper and lower bounds through this method, we also obtain reverse results: from higher order derivatives decay estimates, we deduce bounds for the $L^2$ norm. We use these g…
We study weak solutions ${\bf v}:U\times (0,T)\rightarrow \mathbb{R}^m$ of the nonlinear parabolic system $$ Dψ({\bf v}_t)=\text{div}DF(D{\bf v}), $$ where $ψ$ and $F$ are convex functions. This is a prototype for more general doubly nonlinear evolutions which arise in the study of structural properties of materials. Under the assumption that the second derivatives of $F$ are Hölder continuous, we show that $D^2{\bf v}$ and ${\bf v}_t$ are locally Hölder continuous except for possibly on a lower dimensional subset of $U\times (0,T)$. Our approach relies on two integral identities, decay of the…
Via Carleman estimates we prove uniqueness and continuous dependence results for lateral Cauchy problems for linear integro-differential parabolic equations without initial conditions. The additional information supplied prescribes the conormal derivative of the temperature on a relatively open subset of the lateral boundary of the space-time domain.
We prove weighted and mixed-norm Sobolev estimates for fully nonlinear elliptic and parabolic equations in the whole space under a relaxed convexity condition with almost VMO dependence on space-time variables. The corresponding interior and boundary estimates are also obtained.
A heat equation with uncertain domains is thoroughly investigated. Statistical moments of the solution is approximated by the counterparts of the shape derivative. A rigorous proof for the existence of the shape derivative is presented. Boundary integral equation methods are used to compute statistical moments of the shape derivative.
Classical results of second order parabolic quasi-linear equations always require that the nonlinear terms are controlled by a power of the unknown functions and their first derivatives. We improve the previous results. More precisely, in the present article the upper bound of nonlinear term can be extended to a power series.
The unique solvability of parabolic equations in Sobolev spaces with mixed norms is presented. The second order coefficients (except $a^{11}$) are assumed to be only measurable in time and one spatial variable, and VMO in the other spatial variables. The coefficient $a^{11}$ is measurable in one spatial variable and VMO in the other variables.
We consider time-fractional parabolic equations with a Caputo time derivative of order $α\in(0,1)$. For such equations, we give an elementary proof of the weak maximum principle under no assumptions on the sign of the reaction coefficient. This proof is also extended for weak solutions, as well as for various types of boundary conditions and variable-coefficient variable-order multiterm time-fractional parabolic equations.
In order to extend the blow-up criterion of solutions to the Euler equations, Kozono and Taniuchi have proved a logarithmic Sobolev inequality by means of isotropic (elliptic) $BMO$ norm. In this paper, we show a parabolic version of the Kozono-Taniuchi inequality by means of anisotropic (parabolic) $BMO$ norm. More precisely we give an upper bound for the $L^{\infty}$ norm of a function in terms of its parabolic $BMO$ norm, up to a logarithmic correction involving its norm in some Sobolev space. As an application, we also explain how to apply this inequality in order to establish a long-time …
