On Some Inequalities and Their Application to the Cauchy Problem
Derivation of inequalities and application to Cauchy problem for infinite system of equations
PROBLEM SOLVING↗
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On the Cauchy problem of the relativistic Boltzmann equation.
Cauchy problem for relativistic Boltzmann equation, discussing initial distribution function and scattering cross section
Bichteler, K.↗
Well-posedness of the Cauchy problem for the stochastic system for the Lorenz model for a baroclinic atmosphere
The paper is concerned with the Cauchy problem for a nonlinear system of partial differential equations with parameters. This system describes the two-layer quasi-solenoidal Lorenz model for a baroclinic atmosphere on a rotating two-dimensional sphere. The right-hand side of the system is perturbed by white noise, and random initial data is considered. This system is shown to be uniquely solvable, and an estimate for the continuous dependence of the solution on the set of random initial data and the right-hand side is established on a finite time interval. In passing, an estimate for the continuous dependence on the set of parameters, the initial data, and the right-hand side is obtained on a finite time interval for the solution of the Cauchy problem with deterministic initial data and deterministic right-hand side. Bibliography: 32 titles.
97 MATHEMATICS AND COMPUTING↗
Uniqueness of the Cauchy problem when the initial surface contains characteristic points.
Uniqueness of Cauchy problem when initial surface contains characteristic points
UNIQUENESS THEOREM↗
A Semi-linear Backward Parabolic Cauchy Problem with Unbounded Coefficients of Hamilton–Jacobi–Bellman Type and Applications to Optimal Control
We obtain weighted uniform estimates for the gradient of the solutions to a class of linear parabolic Cauchy problems with unbounded coefficients. Such estimates are then used to prove existence and uniqueness of the mild solution to a semi-linear backward parabolic Cauchy problem, where the differential equation is the Hamilton–Jacobi–Bellman equation of a suitable optimal control problem. Via backward stochastic differential equations, we show that the mild solution is indeed the value function of the controlled equation and that the feedback law is verified.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
On the Cauchy Problem for the Wave Equation with Data on the Boundary
The Cauchy problem for the wave equation in Ω × ℝ with data given on some part of the boundary ∂Ω × ℝ is considered. A reconstruction algorithm for this problem based on analytic expressions is given. This result is applicable to the problem of determining a nonstationary wave field arising in geophysics, photoacoustic tomography, tsunami wave source recovery.
97 MATHEMATICS AND COMPUTING↗
Cauchy problem in the scalar-tensor gravitational theory.
Cauchy problem for scalar-tensor gravitational theory, splitting field equations into initial value equations and evolution equations
Cocke, W. J.↗
On a Limit Theorem Related to Probabilistic Representation of Solution to the Cauchy Problem for the Schrödinger Equation
A new method of probabilistic approximation of solution to the Cauchy problem for the unperturbed Schrödinger equation by expectations of functionals of a random walk is suggested. In contrast to earlier papers of the authors, the existence of exponential moment for each step of the random walk is not assumed.
97 MATHEMATICS AND COMPUTING↗
The Cauchy problem for a quasilinear parabolic equation with gradient absorption
The qualitative properties of solutions to the Cauchy problem for a degenerate parabolic equation containing a nonlinear operator of Baouendi-Grushin type and with gradient absorption whose density depends on time, as well as the space variables, are investigated. Bounds for the diameter of the support of the solution which are sharp with respect to time are obtained, together with its maximum. A condition which determines whether or not the phenomenon of decay to zero of the total mass of the solution occurs is discovered. Bibliography: 35 titles.
97 MATHEMATICS AND COMPUTING↗
Renormalized entropy solutions of the Cauchy problem for a first-order inhomogeneous quasilinear equation
The concept of a renormalized entropy solution of the Cauchy problem for an inhomogeneous quasilinear equation of the first order is introduced. Existence and uniqueness theorems are proved, together with a comparison principle. Connections with generalized entropy solutions are investigated. Bibliography: 10 titles.
97 MATHEMATICS AND COMPUTING↗
Algorithm for solving the linear Cauchy problem for large systems of ordinary differential equations with the use of parallel computations
An algorithm for solving the linear Cauchy problem for large systems of ordinary differential equations is presented. The algorithm for systems of first-order differential equations is implemented in the EDELWEISS code with the possibility of parallel computations on supercomputers employing the MPI (Message Passing Interface) standard for the data exchange between parallel processes. The solution is represented by a series of orthogonal polynomials on the interval [0, 1]. The algorithm is characterized by simplicity and the possibility to solve nonlinear problems with a correction of the operator in accordance with the solution obtained in the previous iterative process.
74 ATOMIC AND MOLECULAR PHYSICS↗
On the Right Weak Solution of the Cauchy Problem for a Quasilinear Equation of First Order Final Report
Solution of Cauchy problem for quasilinear equation of first order
Hopf, E.↗
On the right weak solution of the Cauchy problem for a quasilinear equation of first order
Right weak solution of Cauchy problem for first order quasi-linear equation
Hopf, E.↗
The Stabilization Rate of Solutions to the Cauchy Problem for Parabolic Equations
We study sufficient conditions on lower-order coefficients of a nondivergence-form parabolic equation that guarantee the power rate of the uniform stabilization of the solution to the Cauchy problem on every compact set K of R{sup N} for any bounded initial function.
97 MATHEMATICS AND COMPUTING↗
On the Classical Fundamental Solutions of the Cauchy Problem for Ultraparabolic Kolmogorov-Type Equations with Two Groups of Spatial Variables
For a Kolmogorov-type ultraparabolic equation with two groups of spatial variables, we establish estimates for the increments of the classical fundamental solution of the Cauchy problem and its derivatives in the spatial variables.
97 MATHEMATICS AND COMPUTING↗
Parameter identification for an abstract Cauchy problem by quasilinearization
A parameter identification problem is considered in the context of a linear abstract Cauchy problem with a parameter-dependent evolution operator. Conditions are investigated under which the gradient of the state with respect to a parameter possesses smoothness properties which lead to local convergence of an estimation algorithm based on quasi-linearization. Numerical results are presented concerning estimation of unknown parameters in delay-differential equations.
Brewer, Dennis W.↗
Parameter identification for an abstract Cauchy problem by quasilinearization
A parameter identification problem is considered in the context of a linear abstract Cauchy problem with a parameter-dependent evolution operator. Conditions are investigated under which the gradient of the state with respect to a parameter possesses smoothness properties which lead to local convergence of an estimation algorithm based on quasi-linearization. Numerical results are presented concerning estimation of unknown parameters in delay-differential equations.
Brewer, Dennis W.↗
On the Cauchy problem for Boltzmann equation modeling a polyatomic gas
In the present article, we consider the Boltzmann equation that models a polyatomic gas by introducing one additional continuous variable, referred to as microscopic internal energy. We establish existence and uniqueness theory in the space homogeneous setting for the full non-linear case, under an extended Grad-type assumption on transition probability rates, which comprises hard potentials for both the relative speed and internal energy with the rate in the interval 0,2, multiplied by an integrable angular part and integrable partition functions. The Cauchy problem is resolved by means of an abstract ordinary differential equation (ODE) theory in Banach spaces for the initial data with finite and strictly positive gas mass and energy, finite momentum, and additionally finite K* polynomial moment, with K* depending on the rate of the transition probability and the structure of a polyatomic molecule or its internal degrees of freedom. Moreover, we prove that polynomially and exponentially weighted Banach space norms associated with the solution are both generated and propagated uniformly in time.
Physics↗