Higher-order Parabolic Equation Research Articles

Complicated asymptotic behavior of solutions for the Cauchy problem of Cahn–Hilliard equation

In this paper, we study the Cauchy problem of the Cahn–Hilliard equation, and first reveal that the complicated asymptotic behavior of solutions can happen in high-order parabolic equation.

Higher-Order Elliptic and Parabolic Equations with VMO Assumptions and General Boundary Conditions

Abstract We prove mixed L p ( L q ) -estimates, with p , q ∈ ( 1 , ∞ ) , for higher-order elliptic and parabolic equations on the half space R + d + 1 with general boundary conditions which satisfy the Lopatinskii–Shapiro condition. We assume that the elliptic operators A have leading coefficients which are in the class of vanishing mean oscillations both in the time variable and the space variable. In the proof, we apply and extend the techniques developed by Krylov [24] as well as Dong and Kim in [13] to produce mean oscillation estimates for equations on the half space with general boundary conditions.

Three-dimensional underwater sound pressure sensitivity in oceanic environments

Physical oceanographic processes, marine geological features, sub-bottom geoacoustic structure, sea surface disturbances, etc., can either individually or jointly affect underwater sound propagation in the ocean and cause significant temporal and spatial variability in the sound pressure field. The primary goal of this study is to develop a numerical scheme to determine the sound pressure sensitivity in response to variations of index of refraction due to changes of environmental conditions. This sensitivity analysis is an exention of the Born approximation which assumes perturbations at infinitesimal points. To handle disturbance within a finite volume, an improved sensitivity kernel is derived from a higher-order parabolic-equation (PE) approximation. With this sensitivity kernel, we can analyze the spatial distribution and the temporal evolution of the acoustic sensitivity field in complex oceanic environments. This paper will present numerical examples of three-dimensional (3D) sound propagation in continental slopes, submarine canyons, and nonlinear internal wave fields. Discussions on other applications, including uncertainty quantification of transmission loss prediction and adjoint models for 3D acoustic inversions, will also be provided. [Work supported by the Office of Naval Research.]

Intermediate regularity results for the solution of a high order parabolic equation

Intermediate regularity results for the solution of a high order parabolic equation

Boundary higher integrability for very weak solutions of quasilinear parabolic equations

We prove boundary higher integrability for the (spatial) gradient of very weak solutions of quasilinear parabolic equations of the formut−divA(x,t,∇u)=0onΩ×R, where the non-linear structure divA(x,t,∇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. These results are new even for linear parabolic equations on domains with smooth boundary and make no assumptions on the smoothness ofA(x,t,∇u). These results are also applicable for systems as well as higher order parabolic equations.

Composite wave models for elastic plates

The long-term challenge of formulating an asymptotically motivated wave theory for elastic plates is addressed. Composite two-dimensional models merging the leading or higher-order parabolic equations for plate bending and the hyperbolic equation for the Rayleigh surface wave are constructed. Analysis of numerical examples shows that the proposed approach is robust not only at low- and high-frequency limits but also over the intermediate frequency range.

Higher-Order Parabolic Equations with VMO Assumptions and General Boundary Conditions with Variable Leading Coefficients

AbstractWe prove weighted mixed $L_{p}(L_{q})$-estimates, with $p,q\in (1,\infty )$, and the corresponding solvability results for higher-order elliptic and parabolic equations on the half space ${\mathbb{R}}^{d+1}_{+}$ and on general $C^{2m-1,1}$ domains with general boundary conditions, which satisfy the Lopatinskii–Shapiro condition. We assume that the elliptic operators A have leading coefficients that are in the class of vanishing mean oscillations both in the time and the space variables and that the boundary operators have variable leading coefficients. The proofs are based on and generalize the estimates recently obtained by the authors in [6].

Stochastic m-Point Cauchy Problem for Parabolic Equation with Semi-Wiener Perturbations

The authors consider the problem of the existence, with probability 1, of Green’s function of a stochastic m-point Cauchy problem for a higher-order parabolic equation with white noise perturbations taken with negative values only. Estimates are obtained in spaces of functions whose norm contains the mathematical expectation.

INVERSE PROBLEMS IN THE HEAT AND MASS TRANSFER THEORY

This article is a survey of the recent results obtained preferably by the author and its coauthors and devoted to the study of inverse problem for some mathematical models, in particular those describing heat and mass transfer and convection-diffusion processes. They are defined by second and higher order parabolic equations and systems. We examine the following two types of overdetermination conditions: a solution is specified on some collection of spatial manifolds (or at separate points) or some collection of integrals of a solution with weight is prescribed. We study an inverse problem of recovering a right-hand side (the source function) or the coefficients of equations characterizing the medium. The unknowns (coefficients and the right-hand side) depend on time and a part of the space variables. We expose existence and uniqueness theorems, stability estimates for solutions. The main results in the linear case, i.e., we recover the source function, are global in time while they are local in time in the general case. The main function spaces used are the Sobolev spaces.

Boundary value problems for parabolic equations of high order with a changing time direction

We analyze the solvability of boundary value problems for differential equations of high order with a changing direction of time. It is shown that a generalized solution exists in Sobolev spaces. Correctness of boundary value problems with a complete matrix of conjugation conditions is investigated. For such problems, it is shown that the Hölder classes of their solutions essentially depend on both the type of gluing conditions and on the non-integer index Hölder space.

On Some Lag Synchronization and Higher Order Parabolic Systems

Chaos synchronization is a topic of great interest, due to its observation in a huge variety of phenomena of different nature. We study synchronization of two chaotic oscillators in a Master- Slave configuration. The two dynamic systems are coupled via a directed feedback that randomly switches among a finite set of given constant function at a prescribed time rate. And we use Lyapunov stability theory. This paper discussed the using of lag synchronization approach, and provided the equilibrium solutions of a new class of higher order parabolic partial differential equations to be applicable for Lorenz chaotic system in order to minimize the dynamical error of large Lorenz chaotic system

Fujita phenomenon in higher-order parabolic equation with nonlocal term

This paper considers the Cauchy problem for a higher-order semilinear parabolic equation with nonlocal term in , where , , , and . We obtain the critical Fujita exponent to the problem, in the sense that every nontrivial solution blows up in finite time whenever , and there are both global and non-global solutions if .

Perfectly matched layer for an elastic parabolic equation model in ocean acoustics

The perfectly matched layer (PML) is an effective technique for truncating unbounded domains with minimal spurious reflections. A fluid parabolic equation (PE) model applying PML technique was previously used to analyze the sound propagation problem in a range-dependent waveguide (Lu and Zhu, 2007). However, Lu and Zhu only considered a standard fluid PE to demonstrate the capability of the PML and did not take improved one-way models into consideration. They applied a [1/1] Pade approximant to the parabolic equation. The higher-order PEs are more accurate than standard ones when a very large angle propagation is considered. As for range-dependent problems, the techniques to handle the vertical interface between adjacent regions are mainly energy conserving and single-scattering. In this paper, the PML technique is generalized to the higher order elastic PE, as is to the higher order fluid PE. The correction of energy conserving is used in range-dependent waveguides. Simulation is made in both acoustic cases and seismo-acoustic cases. Range-independent and range-dependent waveguides are both adopted to test the accuracy and efficiency of this method. The numerical results illustrate that a PML is much more effective than an artificial absorbing layer (ABL) both in acoustic and seismo-acoustic sound propagation modeling.

Transparent boundary conditions for iterative high-order parabolic equations

Recently a new approach to the construction of high-order parabolic approximations for the Helmholtz equation was developed. These approximations have the form of the system of iterative parabolic equations, where the solution of the n-th equation is used as an input term for the ( n + 1 ) -th equation. In this study the transparent boundary conditions for such systems of coupled parabolic equations are derived. The existence and uniqueness of the solution of the initial boundary value problem for the system of iterative parabolic equations with the derived boundary conditions are proved. The well-posedness of this problem is also established and an unconditionally stable finite difference scheme for its solution is proposed.

Solvability and spectral properties of the boundary value problem for degenerating higher order parabolic equation

In the present paper we study the boundary value problem for degenerating higher order parabolic equation. A priori estimate for the solution of the boundary value problem is obtained. The existence of unique regular and strong solutions of the problem is proved. The adjoint problem is also investigated.

Critical O(d)-equivariant biharmonic maps

We study $O(d)$-equivariant biharmonic maps in the critical dimension. A major consequence of our study concerns the corresponding heat flow. More precisely, we prove that blowup occurs in the biharmonic map heat flow from $B^4(0, 1)$ into $S^4$. To our knowledge, this was the first example of blowup for the biharmonic map heat flow. Such results have been hard to prove, due to the inapplicability of the maximum principle in the biharmonic case. Furthermore, we classify the possible $O(4)$-equivariant biharmonic maps from $\mathbf{R}^4$ into $S^4$, and we show that there exists, in contrast to the harmonic map analogue, equivariant biharmonic maps from $B^4(0,1)$ into $S^4$ that wind around $S^4$ as many times as we wish. We believe that the ideas developed herein could be useful in the study of other higher-order parabolic equations.

Resolution of a high-order parabolic equation in conical time-dependent domains of [formula omitted

New results on the existence, uniqueness and maximal regularity of a solution are given for a two-space dimensional high-order parabolic equation set in conical time-dependent domains. The study is performed in the framework of anisotropic weighted Sobolev spaces. Our method is based on the technique of decomposition of domains.

Estimates on front propagation for nonlinear higher-order parabolic equations: An algorithmic approach

We present an algorithm for the derivation of lower bounds on support propagation for a certain class of nonlinear parabolic equations. We proceed by combining the ideas in some recent papers by the author with the algorithmic construction of entropies due to Jüngel and Matthes, reducing the problem to a quantifier elimination problem. Due to its complexity, the quantifier elimination problem cannot be solved by present exact algorithms. However, by tackling the quantifier elimination problem numerically, in the case of the thin-film equation we are able to improve recent results by the author in the regime of strong slippage n \in (1,2) . For certain second-order doubly nonlinear parabolic equations, we are able to extend the known lower bounds on free boundary propagation to the case of irregular oscillatory initial data. Finally, we apply our method to a sixth-order quantum drift-diffusion equation, resulting in an upper bound on the time which it takes for the support to reach every point in the domain.

Extinction of solutions of higher order parabolic equations with double nonlinearity and degenerate absorption potential

We study the extinction property of solutions to the Cauchy–Dirichlet problem for nonlinear parabolic equations of the order 2m with absorption potential in a semibounded cylinder (0,+∞) × Ω, where Ω is a bounded domain in ℝN, N ≥ 1. The sufficient conditions ensuring the extinction of a solution in a finite time, which depend on N,m, and q (where q is a parameter of the homogeneous nonlinearity in the main part of the equation), are obtained.

The high-order parabolic equation to solve propagation problems in aeroacoustics

The parabolic equation (PE) has proved its capability to deal with the long range sound propagation in the atmosphere. It also represents an attractive alternative to the ray model to handle duct propagation in high frequencies, for the noise radiated by the nacelle of aero-engines. It was recently shown that the High-Order Parabolic Equation (HOPE), based on a Padé expansion with an order of 5, significantly increases the aperture angle of propagation compared to the standard and the Wide-Angle PEs, allowing prediction close to cut-off frequency of the duct. This paper concerns the propagation using the HOPE in heterogeneous flows, including boundary layers above a wall and in shear layers. The thickness of the boundary layer is about dozens of centimeters while outside it, the Mach number reaches 0.5. The boundary layer effects are investigated showing the refraction effects on a range propagation of 30 m, up to 4 kHz. In the shear layer, discontinuities in the directivity patterns occur significant differences of the directivity patterns occur. Comparisons with the Euler solutions are considered, establishing the domain of application of the HOPE on a set of flow configurations, including beyond its theoretical limits. [Work supported by Airbus-France.]