Heat-type Equation Research Articles

Heat-type equations on manifolds with fibered boundaries I: Schauder estimates

Heat-type equations on manifolds with fibered boundaries I: Schauder estimates

Unsteady Magnetohydrodynamics PDE of Monge–Ampère Type: Symmetries, Closed-Form Solutions, and Reductions

The paper studies an unsteady equation with quadratic nonlinearity in second derivatives, that occurs in electron magnetohydrodynamics. In mathematics, such PDEs are referred to as parabolic Monge–Ampère equations. An overview of the Monge–Ampère type equations is given, in which their unusual qualitative features are noted. For the first time, the Lie group analysis of the considered highly nonlinear PDE with three independent variables is carried out. An eleven-parameter transformation is found that preserves the form of the equation. Some one-dimensional reductions allowing to obtain self-similar and other invariant solutions that satisfy ordinary differential equations are described. A large number of new additive, multiplicative, generalized, and functional separable solutions are obtained. Special attention is paid to the construction of exact closed-form solutions, including solutions in elementary functions (in total, more than 30 solutions in elementary functions were obtained). Two-dimensional symmetry and non-symmetry reductions leading to simpler partial differential equations with two independent variables are considered (including stationary Monge–Ampère type equations, linear and nonlinear heat type equations, and nonlinear filtration equations). The obtained results and exact solutions can be used to evaluate the accuracy and analyze the adequacy of numerical methods for solving initial boundary value problems described by highly nonlinear partial differential equations.

Stable distributions and pseudo-processes related to fractional Airy functions

In this article, we study pseudo-processes related to odd-order heat-type equations composed with Lévy stable subordinators. The aim of the article is twofold. We first show that the pseudo-density of the subordinated pseudo-process can be represented as an expectation of damped oscillations with generalized gamma distributed parameters. This stochastic representation also arises as the solution to a fractional diffusion equation, involving a higher-order Riesz-Feller operator, which generalizes the odd-order heat-type equation. We then prove that, if the stable subordinator has a suitable exponent, the time-changed pseudo-process is identical in distribution to a genuine Lévy stable process. This result permits us to obtain a power series representation for the probability density function of an arbitrary asymmetric stable process of exponent ν > 1 and skewness parameter β, with 0 < | β | < 1 . The methods we use in order to carry out our analysis are based on the study of a fractional Airy function which emerges in the investigation of the higher-order Riesz-Feller operator.

Harnack Estimation for Nonlinear, Weighted, Heat-Type Equation along Geometric Flow and Applications

The method of gradient estimation for the heat-type equation using the Harnack quantity is a classical approach used for understanding the nature of the solution of these heat-type equations. Most of the studies in this field involve the Laplace–Beltrami operator, but in our case, we studied the weighted heat equation that involves weighted Laplacian. This produces a number of terms involving the weight function. Thus, in this article, we derive the Harnack estimate for a positive solution of a weighted nonlinear parabolic heat equation on a weighted Riemannian manifold evolving under a geometric flow. Applying this estimation, we derive the Li–Yau-type gradient estimation and Harnack-type inequality for the positive solution. A monotonicity formula for the entropy functional regarding the estimation is derived. We specify our results for various different flows. Our results generalize some works.

Heat- and Wave-Type Equations with Nonlocal Operators, I. Compact Lie Groups

Abstract We prove existence and uniqueness and give the analytical solution of heat and wave type equations on a compact Lie group $G$ by using a nonlocal (in time) differential operator and a positive left invariant operator (maybe unbounded) acting on the group. For heat type equations, solutions are given in $L^q(G)$ for data in $L^p(G)$ with $1&amp;lt;p\leqslant 2\leqslant q&amp;lt;+\infty $. We also provide some asymptotic estimates (large-time behavior) for the solutions. Some examples are given. Also, for wave-type equations, we give the solution on some suitable Sobolev spaces over $L^2(G)$. We complement our results, by studying a multi-term heat-type equation as well.

Remarks on entropy formulae for linear heat equation

In this note, we prove some new entropy formula for linear heat equation on static Riemannian manifold with nonnegative Ricci curvature. The results are analogies of Cao and Hamilton's entropies for Ricci flow coupled with heat-type equations.

Stochastic fractional heat equation perturbed by general Gaussian and non-Gaussian noise

In this work, we study the stochastic heat equation with Caputo derivative perturbed by a class of noises that include Gaussian noises and non-Gaussian noises. We have deduced the existence, uniqueness and pathwise spatial–temporal regularity properties of mild solutions to the corresponding non-linear time-fractional stochastic heat type equation.

Stochastic solutions for time-fractional heat equations with complex spatial variables

We deal with complex spatial diffusion equations with time-fractional derivative and study their stochastic solutions. In particular, we complexify the integral operator solution to the heat-type equation where the time derivative is replaced with the convolution-type generalization of the regularized Caputo derivative. We prove that this operator is solution of a complex time-fractional heat equation with complex spatial variable. This approach leads to a wrapped Brownian motion on a circle time-changed by the inverse of the related subordinator. This time-changed Brownian motion is analyzed and, in particular, some results on its moments, as well as its construction as weak limit of continuous-time random walks, are obtained. The extension of our approach to the higher dimensional case is also provided.

On the Reformulation of the Classical Stefan Problem as a Shape Optimization Problem

This article is concerned with the multi-dimensional one-phase Stefan problem, which belongs to the class of moving boundary problems. We suggest to reformulate the classical Stefan problem as a shape optimization problem, consisting of an objective functional for the moving boundary and a partial differential equation corresponding to a heat type equation. Minimizing the objective functional subject to the differential equation under consideration is equivalent to solving the Stefan problem. In order to apply gradient-based optimization algorithms, we analytically compute the shape gradient of the objective functional. A numerical example justifies our approach.

OBSERVABILITY OF BAOUENDI–GRUSHIN-TYPE EQUATIONS THROUGH RESOLVENT ESTIMATES

AbstractIn this article, we study the observability (or equivalently, the controllability) of some subelliptic evolution equations depending on their step. This sheds light on the speed of propagation of these equations, notably in the ‘degenerated directions’ of the subelliptic structure.First, for any $\gamma \geq 1$ , we establish a resolvent estimate for the Baouendi–Grushin-type operator $\Delta _{\gamma }=\partial _x^2+\left \lvert x\right \rvert ^{2\gamma }\partial _y^2$ , which has step $\gamma +1$ . We then derive consequences for the observability of the Schrödinger-type equation $i\partial _tu-\left (-\Delta _{\gamma }\right )^{s}u=0$ , where $s\in \mathbb N$ . We identify three different cases: depending on the value of the ratio $(\gamma +1)/s$ , observability may hold in arbitrarily small time or only for sufficiently large times or may even fail for any time.As a corollary of our resolvent estimate, we also obtain observability for heat-type equations $\partial _tu+\left (-\Delta _{\gamma }\right )^su=0$ and establish a decay rate for the damped wave equation associated with $\Delta _{\gamma }$ .

Direct and inverse Cauchy problems for generalized space-time fractional differential equations

In this paper, explicit solutions of a class of generalized space-time fractional Cauchy problems with time-variable coefficients are given. The representation of a solution involves kernels given by convergent infinite series of fractional integro-differential operators, which can be extensively and efficiently applied for analytic and computational goals. Time-fractional operators of complex orders with respect to a given function are used. Further, we study inverse Cauchy problems of finding time dependent coefficients for fractional wave and heat type equations, which involve the explicit representation of the solution of the direct Cauchy problem and a recent method to recover variable coefficients for the considered inverse problems. Concrete examples and particular cases of the obtained results are discussed.

Existence of strong solutions for a system of interaction between a compressible viscous fluid and a wave equation * *Debayan Maity was partially supported by INSPIRE faculty fellowship (IFA18-MA128) and by Department of Atomic Energy, Government of India, under Project No. 12-R&D-TFR-5.01-0520. Arnab Roy was supported by the Czech Science Foundation (GAČR) Project GA19-04243S. The Institute of Mathematics, CAS is

In this article, we consider a fluid–structure interaction system where the fluid is viscous and compressible and where the structure is a part of the boundary of the fluid domain and is deformable. The fluid is governed by the barotropic compressible Navier–Stokes system, whereas the structure displacement is described by a wave equation. We show that the corresponding coupled system admits a unique, strong solution for an initial fluid density and an initial fluid velocity in H 3 and for an initial deformation and an initial deformation velocity in H 4 and H 3 respectively. The reference configuration for the fluid domain is a rectangular cuboid with the elastic structure being the top face. We use a modified Lagrangian change of variables to transform the moving fluid domain into the rectangular cuboid and then analyze the corresponding linear system coupling a transport equation (for the density), a heat-type equation, and a wave equation. The corresponding results for this linear system and estimations of the coefficients coming from the change of variables allow us to perform a fixed point argument and to prove the existence and uniqueness of strong solutions for the nonlinear system, locally in time.

Heat and Laplace type equations with complex spatial variables in weighted Fock spaces

In a recent book co-authored by the authors of this article, we studied by semigroup theory methods several classical evolution equations, including the heat and Laplace equations, with real time variable and complex spatial variable, under the hypothesis that the boundary function belongs to the space of analytic functions in the open unit disk and continuous in the closed unit disk, endowed with the uniform norm. Also, in a subsequent paper, the authors have extended the results for the heat and Laplace equations in weighted Bergman spaces on the unit disk. The purpose of this article is to show that the semigroup theory methods work for these two evolution equations of complex spatial variables, under the hypothesis that the boundary function belongs to the weighted Fock space on \(\mathbb{C}\), \(F_{\alpha }^p(\mathbb{C})\), with \(1\leq p&lt;+\infty \), endowed with The \(L^p\)-norm. Also, the case of several complex variables is considered. The proofs use the Jensen's inequality, Fubini's theorem for integrals and the \(L^p\)-integral modulus of continuity.&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2020/109/abstr.html

Local parabolic and elliptic gradient estimates for a generalized heat-type equation under the Yamabe flow

In this paper, we investigate local parabolic (Li-Yau's type and J. Li's type) and local elliptic (Hamilton's type Shouplet-Zhang's type) gradient estimates for positive solutions to a generalized heat-type equation of (Δϕ−∂t)u=huq under the Yamabe flow. As applications of local parabolic gradient estimates, we establish associated Harnack inequalities.

Probabilistic representation formula for the solution of fractional high-order heat-type equations

We propose a probabilistic construction for the solution of a general class of fractional high-order heat-type equations in the one-dimensional case, by using a sequence of random walks in the complex plane with a suitable scaling. A time change governed by a class of subordinated processes allows to handle the fractional part of the derivative in space. We first consider evolution equations with space fractional derivatives of any order, and later we show the extension to equations with time fractional derivative (in the sense of Caputo derivative) of order $$\alpha \in (0,1)$$ .

Talagrand concentration inequalities for stochastic heat-type equations under uniform distance

In this paper, we established a quadratic transportation cost inequality under the uniform/maximum norm for solutions of stochastic heat equations driven by multiplicative space-time white noise. The proof is based on a new inequality we obtained for the moments of the stochastic convolution with respect to space-time white noise, which is of independent interest. The solutions of such stochastic partial differential equations are typically not semimartingales on the state space.

Numerical solutions to time-fractional stochastic partial differential equations

This study is concerned with numerical approximations of time-fractional stochastic heat-type equations driven by multiplicative noise, which can be used to model the anomalous diffusion in porous media with random effects with thermal memory. A standard finite element approximation is used in space as well as a spatial-temporal discretization which is achieved by a new algorithm in time direction. The strong convergence error estimates for both semidiscrete and fully discrete schemes in a semigroup framework are proved, which are based on the error estimates for the corresponding deterministic equations, and together with the optimal regularity results of mild solutions to the time-fractional SPDEs. Numerical results are finally reported to confirm our theoretical findings.

Generalized Feynman path integrals and applications to higher-order heat-type equations

Techniques of infinite dimensional integrations providing the mathematical definition of Feynman path integrals are shortly reviewed and generalized, and then applied to the construction of a functional integral representation for solutions of a general class of higher-order heat-type equations.

Estimates for Functionals of Solutions to Higher-Order Heat-Type Equations with Random Initial Conditions

In the present paper we continue the investigation of solutions to higher-order heat-type equations with random initial conditions, which play the important role in many applied areas. We consider the random initial conditions given by harmonizable $\varphi$-sub-Gaussian processes. The main results are the bounds for the distributions of the suprema over bounded and unbounded domains for solutions of such equations. The results obtained in the paper hold, in particular, for the case of Gaussian initial condition.

Anomalous Diffusion Limit of Kinetic Equations in Spatially Bounded Domains

This paper is devoted to the anomalous diffusion limit of kinetic equations with a fractional Fokker–Planck collision operator in a spatially bounded domain. We consider two boundary conditions at the kinetic scale: absorption and specular reflection. In the absorption case, we show that the long time/small mean free path asymptotic dynamics are described by a fractional diffusion equation with homogeneous Dirichlet-type boundary conditions set on the whole complement of the spatial domain. On the other hand, specular reflections will give rise to a new operator which we call specular diffusion operator and write $${(-\Delta)_{{\rm SR}}^{s}}$$ . This non-local diffusion operator strongly depends on the geometry of the domain and includes in its definition the interaction between the diffusion and the boundary. We consider two types of domains: half-spaces and balls in $${\mathbb{R}^d}$$ . In these domains, we prove properties of the specular diffusion operator and establish existence and uniqueness of weak solutions to the associated heat-type equation.