Well-posedness Of Initial-boundary Value Problem Research Articles

Global well-posedness of initial-boundary value problem of fifth-order KdV equation posed on finite interval

Abstract We have established the existence and uniqueness of the local solution for (0.1) ∂ t u + ∂ x 5 u − u ∂ x u = 0 , 0 < x < 1 , t > 0 , u ( x , 0 ) = φ ( x ) , 0 < x < 1 , u ( 0 , t ) = h 1 ( t ) , u ( 1 , t ) = h 2 ( t ) , ∂ x u ( 1 , t ) = h 3 ( t ) , ∂ x u ( 0 , t ) = h 4 ( t ) , ∂ x 2 u ( 1 , t ) = h 5 ( t ) , t > 0 , \left\{\begin{array}{ll}{\partial }_{t}u+{\partial }_{x}^{5}u-u{\partial }_{x}u=0,& 0\lt x\lt 1,\hspace{1.0em}t\gt 0,\\ u\left(x,0)=\varphi \left(x),& 0\lt x\lt 1,\\ u\left(0,t)={h}_{1}\left(t),u\left(1,t)={h}_{2}\left(t),\hspace{0.33em}{\partial }_{x}u\left(1,t)={h}_{3}\left(t),& \\ {\partial }_{x}u\left(0,t)={h}_{4}\left(t),\hspace{0.33em}{\partial }_{x}^{2}u\left(1,t)={h}_{5}\left(t),& t\gt 0,\end{array}\right. in the study of Zhao and Zhang [Non-homogeneous boundary value problem of the fifth-order KdV equations posed on a bounded interval, J. Math. Anal. Appl. 470 (2019), 251–278]. A question arises naturally: Can the local solution be extended to a global one? This article will address this question. First, through a series of logical deductions, a global a priori estimate is established, and then the local solution is naturally extended to a global solution.

New Thought on Matsumura–Nishida Theory in the L_p–L_q Maximal Regularity Framework

This paper is devoted to proving the global well-posedness of initial-boundary value problem for Navier–Stokes equations describing the motion of viscous, compressible, barotropic fluid flows in a three dimensional exterior domain with non-slip boundary conditions. This was first proved by an excellent paper due to Matsumura and Nishida (Commun Math Phys 89:445–464, 1983). In [10], they used energy method and their requirement was that space derivatives of the mass density up to third order and space derivatives of the velocity fields up to fourth order belong to L_2 in space-time, detailed statement of Matsumura and Nishida theorem is given in Theorem 1 of Sect. 1 of context. This requirement is essentially used to estimate the L_infty norm of necessary order of derivatives in order to enclose the iteration scheme with the help of Sobolev inequalities and also to treat the material derivatives of the mass density. On the other hand, this paper gives the global wellposedness of the same problem as in [10] in L_p (1 <p le 2) in time and L_2cap L_6 in space maximal regularity class, which is an improvement of the Matsumura and Nishida theory in [10] from the point of view of the minimal requirement of the regularity of solutions. In fact, after changing the material derivatives to time derivatives by Lagrange transformation, enough estimates obtained by combination of the maximal L_p (1 <p le 2) in time and L_2cap L_6 in space regularity and L_p–L_q decay estimate of the Stokes equations with non-slip conditions in the compressible viscous fluid flow case enable us to use the standard Banach’s fixed point argument. Moreover, one of the purposes of this paper is to present a framework to prove the L_p–L_q maximal regularity for parabolic-hyperbolic type equations with non-homogeneous boundary conditions and how to combine the maximal L_p–L_q regularity and L_p–L_q decay estimates of linearized equations to prove the global well-posedness of quasilinear problems in unbounded domains, which gives a new thought of proving the global well-posedness of initial-boundary value problems for systems of parabolic or parabolic-hyperbolic equations appearing in mathematical physics.

On the theoretical foundation of overset grid methods for hyperbolic problems: Well-posedness and conservation

We use the energy method to study the well-posedness of initial-boundary value problems approximated by overset mesh methods in one and two space dimensions for linear constant-coefficient hyperbolic systems. We show that in one space dimension, for both scalar equations and systems of equations, the problem where one domain partially oversets another is well-posed when characteristic coupling conditions are used. If a system cannot be diagonalized, as is usually the case in multiple space dimensions, then the energy method does not give proper bounds in terms of initial and boundary data. For those problems, we propose a novel penalty approach. We show, by using a global energy that accounts for the energy in the overlap region of the domains, that under well-defined conditions on the coupling matrices the penalized overset domain problems are energy bounded, conservative, well-posed and have solutions equivalent to the original single domain problem.

Operator Sine Functions and Exponential Trigonometric Pairs

Using the operator functional relation Sh(t+s)+Sh(t−s) = 2[I +2Sh2( $$ \frac{t}{2} $$ )] Sh(s), Sh(0) = 0, we introduce and study the strongly continuous sine function Sh(t), t ∈ (−∞, ∞), of linear bounded transformations acting in a complex Banach space E. Also, we study the cosine function Ch(t) given by the equation Ch(t) = I + 2Sh2( $$ \frac{t}{2} $$ ), where I is the identity operator in E. The pair Ch(t) and Sh(t) is called an exponential trigonometric pair (ETP, in brief). For such pairs, we determine the generating operator (generator) by the equation Sh″ (0)φ = Ch″ (0)φ = Aφ and we give a criterion for A to be the generator of ETP. We find a connection between Sh(t) and the uniform well-posedness of the Cauchy problem with Krein’s condition for the equation $$ \frac{d^2u(t)}{dt^2} $$ = Au(t). This problem is uniformly well-posed if and only if A is the exponent generator of the sine function Sh(t). We introduce the concept of a bundle of several ETPs, which also forms an ETP, and we give a representation for bundle’s generator. The facts obtained significantly expand the applicability of operator methods to study the well-posedness of initial-boundary value problems.

Determination of thermal conductivity of inhomogeneous orthotropic materials from temperature measurements

ABSTRACTWe consider the two-dimensional inverse determination of the thermal conductivity of inhomogeneous orthotropic materials from internal temperature measurements. The inverse problem is general and is classified as a function estimation since no prior information about the functional form of the thermal conductivity is assumed in the inverse calculation. The least-squares functional minimizing naturally the gap between the measured and computed temperature leads to a set of direct, sensitivity and adjoint problems, which have forms of direct well-posed initial boundary value problems for the heat equation, and new formulas for its gradients are derived. The conjugate gradient method employs recursively the solution of these problems at each iteration. Stopping the iterations according to the discrepancy principle criterion yields a stable solution. The employment of the Sobolev -gradient is shown to result in much more robust and accurate numerical reconstructions than when the standard -gradient is used.

On strong well-posedness of initial-boundary value problems for higher order nonlinear hyperbolic equations with two independent variables

Abstract Problems with linear initial-boundary conditions for higher order nonlinear hyperbolic equations are investigated. The concept of strong well-posedness of an initial-boundary value problem is introduced, and conditions guaranteeing solvability and strong well-posedness of the problem under consideration are established.

Evolution PDEs and augmented eigenfunctions. Half-line

The solution of an initial-boundary value problem for a linear evolution partial differential equation posed on the half-line can be represented in terms of an integral in the complex (spectral) plane. This representation is obtained by the unified transform introduced by Fokas in the 90's. On the other hand, it is known that many initial-boundary value problems can be solved via a classical transform pair, constructed via the spectral analysis of the associated spatial operator. For example, the Dirichlet problem for the heat equation can be solved by applying the Fourier sine transform pair. However, for many other initial-boundary value problems there is no suitable transform pair in the classical literature. Here we pose and answer two related questions: Given any well-posed initial-boundary value problem, does there exist a (non-classical) transform pair suitable for solving that problem? If so, can this transform pair be constructed via the spectral analysis of a differential operator? The answer to both of these questions is positive and given in terms of augmented eigenfunctions, a novel class of spectral functionals. These are eigenfunctions of a suitable differential operator in a certain generalised sense, they provide an effective spectral representation of the operator, and are associated with a transform pair suitable to solve the given initial-boundary value problem.

Notion of blowup of the solution set of differential equations and averaging of random semigroups

We propose a unique approach to studying the violation of the well-posedness of initial boundary-value problems for differential equations. The blowup of the set of solutions of a problem for a differential equation is defined as a discontinuity of a multivalued map associating an initial boundary-value problem with the set of solutions of this problem. We show that such a definition not only describes effects of the solution destruction or its nonuniqueness but also permits prescribing a procedure for extending the solution through the singularity origination instant by using an appropriate random process. Considering the initial boundary-value problems whose solution sets admit singularities of the blowup type and a neighborhood of these problems in the space of problems permits associating the initial problem with the set of limit points of a sequence of solutions of the approximating problems. Endowing the space of problems with the structure of a space with measure, we obtain a random semigroup generated by the initial problem. We study the properties of the mathematical expectations (means) of a random semigroup and their equivalence in the sense of Chernoff to semigroups with averaged generators.

A new well-posed vorticity divergence formulation of the shallow water equations

A new vorticity–divergence formulation of the two-dimensional shallow water equations including boundary conditions is derived. The new formulation is necessary since the conventional one does not lead to a well-posed initial boundary value problem for limited-area modelling.The new vorticity–divergence formulation includes four dependent variables instead of three and requires more equations and boundary conditions than the conventional formulation. On the other hand, it forms a hyperbolic set of equations with well-defined boundary conditions that leads to a well-posed problem with bounded energy.

Solution of Terzaghi one-dimensional consolidation equation with general boundary conditions

Boundary conditions for the classical solution of the Terzaghi one-dimensional consolidation equation conflict with the equation’s initial condition. As such, the classical initial-boundary value problem for the Terzaghi one-dimensional consolidation equation is not well-posed. Moreover, the classical boundary conditions of the equation can only be applied to problems with either perfectly pervious or perfectly impervious boundaries. General boundary conditions are proposed to overcome these shortcomings and thus transfer the solution of the Terzaghi one-dimensional consolidation equation to a well-posed initial boundary value problem. The solution for proposed general boundary conditions is validated by comparing it to the classical solution. The actual field drainage conditions can be simulated by adjusting the values of parameters b and c given in the proposed general boundary conditions. For relatively high coefficient of consolidation, just one term in series expansions is enough to obtain results with acceptable accuracy.

INTERMEDIATE MODEL FOR SPATIAL EVOLUTION IN NONLINEAR OPTICS

This paper follows the work of Colin–Gallice–Laurioux6 in which a new model generalizing the Schrödinger (NLS) model of the diffractive optics is derived for the laser propagation in nonlinear media. In particular, it provides good approximate solutions of the Maxwell–Lorentz system for highly oscillating initial data with broad spectrum. In real situations one is given boundary data. We propose to derive a similar evolution model but in the variable associated to the direction of propagation. However, since the space directions for the Maxwell equations are not hyperbolic, the boundary problem is ill-posed and one needs to apply a cutoff defined in the Fourier space, selecting those frequencies for which the operator is hyperbolic. The model we obtain is nearly L2 conservative on its domain of validity. We then give a justification of the derivation. For this purpose we introduce a related well-posed initial boundary value problem. Finally, we perform numerical computations on the example of Maxwell with Kerr nonlinearity in some cases of short or spectrally chirped data where our model outperforms the Schrödinger one.

Disembodied boundary data for Einstein’s equations

A strongly well-posed initial boundary value problem based upon constraint-preserving boundary conditions of the Sommerfeld type has been established for the harmonic formulation of the vacuum Einstein's equations. These Sommerfeld conditions have been previously presented in a four-dimensional geometric form. Here we recast the associated boundary data as three-dimensional tensor fields intrinsic to the boundary. This provides a geometric presentation of the boundary data analogous to the three-dimensional presentation of Cauchy data in terms of three-metric and extrinsic curvature. In particular, diffeomorphisms of the boundary data lead to vacuum spacetimes with isometric geometries. The proof of well-posedness is valid for the harmonic formulation and its generalizations. The Sommerfeld conditions can be directly applied to existing harmonic codes which have been used in simulating binary black holes, thus ensuring boundary stability of the underlying analytic system. The geometric form of the boundary conditions also allows them to be formally applied to any metric formulation of Einstein's equations, although well-posedness of the boundary problem is no longer ensured. We discuss to what extent such a formal application might be implemented in a constraint-preserving manner to $3+1$ formulations, such as the Baumgarte-Shapiro-Shibata-Nakamura system which has been highly successful in binary black hole simulation.

Geometrization of metric boundary data for Einstein’s equations

The principle part of Einstein equations in the harmonic gauge consists of a constrained system of 10 curved space wave equations for the components of the space-time metric. A well-posed initial boundary value problem based upon a new formulation of constraint-preserving boundary conditions of the Sommerfeld type has recently been established for such systems. In this paper these boundary conditions are recast in a geometric form. This serves as a first step toward their application to other metric formulations of Einstein's equations.

Uniqueness and non-uniqueness of bounded solutions to singular nonlinear parabolic equations

We investigate well-posedness of initial-boundary value problems for a class of nonlinear parabolic equations with variable density. At some part of the boundary, called singular boundary, the density can either vanish or diverge or not need to have a limit. We provide simple conditions for uniqueness or non-uniqueness of bounded solutions, depending on the behaviour of the density near the singular boundary.

A note on Burgers' equation with time delay: Instability via finite-time blow-up

Burgers' equation with time delay is considered. Using the Cole–Hopf transformation, the exact solution of this nonlinear partial differential equation (PDE) is determined in the context of a (seemingly) well-posed initial-boundary value problem (IBVP) involving homogeneous Dirichlet data. The solution obtained, however, is shown to exhibit a delay-induced instability, suffering blow-up in finite-time.

WELL-POSED INITIAL-BOUNDARY VALUE PROBLEM FOR A CONSTRAINED EVOLUTION SYSTEM AND RADIATION-CONTROLLING CONSTRAINT-PRESERVING BOUNDARY CONDITIONS

A well-posed initial-boundary value problem is formulated for the model problem of the vector wave equation subject to the divergence-free constraint. Existence, uniqueness and stability of the solution is proved by reduction to a system evolving the constraint quantity statically, i.e. the second time derivative of the constraint quantity is zero. A new set of radiation-controlling constraint-preserving boundary conditions is constructed for the free evolution problem. Comparison between the new conditions and the standard constraint-preserving boundary conditions is made using the Fourier–Laplace analysis and the power series decomposition in time. The new boundary conditions satisfy the Kreiss condition and are free from the ill-posed modes growing polynomially in time.

Well-posed initial-boundary value problem for the harmonic Einstein equations using energy estimates

In recent work, we used pseudo-differential theory to establish conditions that the initial-boundary value problem for second-order systems of wave equations be strongly well-posed in a generalized sense. The applications included the harmonic version of the Einstein equations. Here we show that these results can also be obtained via standard energy estimates, thus establishing strong well-posedness of the harmonic Einstein problem in the classical sense.

Well-posedness of the time-varying linear electromagnetic initial-boundary value problem

The well-posedness of the initial-boundary value problem of the time-varying linear electromagnetic field in a multi-medium region is investigated. Function spaces are defined, with Faraday's law of electromagnetic induction and the initial-boundary conditions considered as constraints. Gauss's formula applied to a multi-medium region is used to derive the energy-estimating inequality. After converting the initial-boundary conditions into homogeneous ones and analysing the characteristics of an operator introduced according to the total current law, the existence, uniqueness and stability of the weak solution to the initial-boundary value problem of the time-varying linear electromagnetic field are proved.

A minimization problem for the lapse and the initial-boundary value problem for Einstein's field equations

We discuss the initial-boundary value problem of general relativity. Previous considerations for a toy model problem in electrodynamics motivate the introduction of a variational principle for the lapse with several attractive properties. In particular, we show that the resulting elliptic gauge condition for the lapse together with a suitable condition for the shift and constraint-preserving boundary conditions controlling the Weyl scalar Ψ0 yields a priori estimates on the metric, connection and curvature fields. These estimates are expected to be useful for obtaining a well-posed initial-boundary value problem for metric formulations of Einstein's field equations which are commonly used in numerical relativity. To present a simple and explicit example, we consider the 3 + 1 decomposition introduced by York of the field equations on a cubic domain with two periodic directions and prove in the weak field limit that our gauge condition for the lapse, an a priori specified shift vector and our boundary conditions lead to a well-posed problem. The method discussed here is quite general and should also yield well-posed problems for different ways of writing the evolution equations, including first-order symmetric hyperbolic or mixed first-order second-order formulations. Well-posed initial-boundary value formulations for the linearization about arbitrary stationary configurations will be presented elsewhere.

The discrete energy method in numerical relativity: towards long-term stability

The energy method can be used to identify well-posed initial boundary value problems for quasi-linear, symmetric hyperbolic partial differential equations with maximally dissipative boundary conditions. A similar analysis of the discrete system can be used to construct stable finite difference equations for these problems at the linear level. In this paper we apply these techniques to some test problems commonly used in numerical relativity and observe that while we obtain convergent schemes, fast growing modes, or ‘artificial instabilities’, contaminate the solution. We find that these growing modes can partially arise from the lack of a Leibnitz rule for discrete derivatives and discuss ways to limit this spurious growth.