COUPLING EULER AND VLASOV EQUATIONS IN THE CONTEXT OF SPRAYS: THE LOCAL-IN-TIME, CLASSICAL SOLUTIONS

2006 ◽  
Vol 03 (01) ◽  
pp. 1-26 ◽  
Author(s):  
CÉLINE BARANGER ◽  
LAURENT DESVILLETTES
Keyword(s):  
Existence And Uniqueness ◽  
Coupled Model ◽  
Small Time ◽  
Classical Solutions ◽  
Liquid Droplets ◽  
Complex Flows ◽  
Fluid Equation ◽  
Vlasov Equations ◽  
System Coupling ◽  
Well Posed

Sprays are complex flows made of liquid droplets surrounded by a gas. They can be modeled by introducing a system coupling a kinetic equation (for the droplets) of Vlasov type and a (Euler-like) fluid equation for the gas. In this paper, we prove that, for the so-called thin sprays, this coupled model is well-posed, in the sense that existence and uniqueness of classical solutions holds for small time, provided the initial data are sufficiently smooth and their support have suitable properties.

LOCAL SMOOTH SOLUTIONS OF A THIN SPRAY MODEL WITH COLLISIONS

2010 ◽  
Vol 20 (02) ◽  
pp. 191-221 ◽  
Author(s):  
JULIEN MATHIAUD
Keyword(s):  
Boltzmann Equation ◽  
Initial Data ◽  
Euler Equations ◽  
Existence And Uniqueness ◽  
Well Posedness ◽  
Liquid Droplets ◽  
Smooth Solutions ◽  
Perfect Gas ◽  
Complex Flows

Sprays are complex flows made of liquid droplets surrounded by a gas. The aim of this paper is to study the local in time well-posedness of a collisional thin spray model, that is a coupling between Euler equations for a perfect gas and a Vlasov–Boltzmann equation for the droplets. We prove the existence and uniqueness of (local in time) solutions for this problem as soon as initial data are smooth enough.

COEFFICIENT INVERSE PROBLEMS FOR THE PARABOLIC EQUATION WITH GENERAL WEAK DEGENERATION

2021 ◽  
Vol 9 (1) ◽  
pp. 91-106
Author(s):  
N. Huzyk ◽  
O. Brodyak
Keyword(s):  
Parabolic Equation ◽  
Inverse Problems ◽  
Existence And Uniqueness ◽  
Space Variable ◽  
Time Derivative ◽  
Classical Solutions ◽  
Linear Polynomial ◽  
Coefficient Inverse Problems ◽  
Degenerate Parabolic ◽  
Increasing Function

It is investigated the inverse problems for the degenerate parabolic equation. The mi- nor coeffcient of this equation is a linear polynomial with respect to space variable with two unknown time-dependent functions. The degeneration of the equation is caused by the monotone increasing function at the time derivative. It is established conditions of existence and uniqueness of the classical solutions to the named problems in the case of weak degeneration.

EXISTENCE AND UNIQUENESS OF WEAK AND CLASSICAL SOLUTIONS FOR A FOURTH-ORDER SEMILINEAR BOUNDARY VALUE PROBLEM

2019 ◽  
Vol 61 (3) ◽  
pp. 305-319
Author(s):  
CRISTIAN-PAUL DANET
Keyword(s):  
Boundary Value Problem ◽  
Variational Methods ◽  
Fourth Order ◽  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Classical Solutions ◽  
Maximum Principles ◽  
Uniqueness Results

This paper is concerned with the problem of existence and uniqueness of weak and classical solutions for a fourth-order semilinear boundary value problem. The existence and uniqueness for weak solutions follows from standard variational methods, while similar uniqueness results for classical solutions are derived using maximum principles.

scholarly journals A reaction infiltration problem: classical solutions

1997 ◽  
Vol 40 (2) ◽  
pp. 275-291 ◽  
Author(s):  
John Chadam ◽  
Xinfu Chen ◽  
Roberto Gianni ◽  
Riccardo Ricci
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Parabolic Equation ◽  
Elliptic Equation ◽  
Boundary Conditions ◽  
Classical Solution ◽  
Existence And Uniqueness ◽  
Classical Solutions ◽  
Bounded Domains ◽  
Global Classical Solution

In this paper, we consider a reaction infiltration problem consisting of a parabolic equation for the concentration, an elliptic equation for the pressure, and an ordinary differential equation for the porosity. Existence and uniqueness of a global classical solution is proved for bounded domains Ω⊂RN, under suitable boundary conditions.

Exponential Stability Estimate of Fully Nonlinear Aceive Equation by Boundary Control

2011 ◽  
Vol 467-469 ◽  
pp. 1078-1083
Author(s):  
Dian Chen Lu ◽  
Ruo Yu Zhu
Keyword(s):  
Fixed Point ◽  
Exponential Stability ◽  
Dispersion Equation ◽  
Boundary Control ◽  
Existence And Uniqueness ◽  
Stability Estimate ◽  
Banach Fixed Point Theorem ◽  
Operator Semigroups ◽  
Fully Nonlinear ◽  
Well Posed

The well-posed problem for the fully nonlinear Aceive diffusion and dispersion equation on the domain [0, 1] is investigated by using boundary control. The existence and uniqueness of the solutions with the help of the Banach fixed point theorem and the theory of operator semigroups are verified. By using some inequalities and integration by parts, the exponential stability of the fully nonlinear Aceive diffusion and dispersion equation with the designed boundary feedback is also proved.

scholarly journals The periodic quasigeostrophic equations: existence and uniqueness of strong solutions

1982 ◽  
Vol 91 (3-4) ◽  
pp. 185-203 ◽  
Author(s):  
A. F. Bennett ◽  
P. E. Kloeden
Keyword(s):  
Existence And Uniqueness ◽  
Hyperbolic Equations ◽  
Three Dimensional ◽  
Strong Solutions ◽  
Coupled System ◽  
Classical Solutions ◽  
Schauder Estimates ◽  
Initial Vorticity ◽  
Relative Potential ◽  
Quasigeostrophic Equations

SynopsisThe periodic quasigeostrophic equations are a coupled system of a second order elliptic equation for a streamfunction and first order hyperbolic equations for the relative potential vorticity and surface potential temperatures, on a three-dimensional domain which is periodic in both horizontal spatial co-ordinates. Such equations are used in both numerical and theoretical studies in meteorology and oceanography. In this paper Schauder estimates and a Schauder fixed point theorem are used to prove the existence and uniqueness of strong, that is classical, solutions of the periodic quasigeostrophic equations for a finite interval of time, which is inversely proportional to the sum of the norms of the initial vorticity and surface temperatures.

scholarly journals On a global classical solution of a quasilinear hyperbolic equation

1989 ◽  
Vol 12 (1) ◽  
pp. 29-37 ◽  
Author(s):  
Y. Ebihara ◽  
D. C. Pereira
Keyword(s):  
Hyperbolic Equation ◽  
Classical Solution ◽  
Existence And Uniqueness ◽  
Classical Solutions ◽  
Torsional Vibrations ◽  
Global Classical Solution ◽  
Quasilinear Hyperbolic Equation ◽  
Global Classical Solutions

In this paper we establish the existence and uniqueness of global classical solutions for the equation which arises in the study of the extensional vibrations of thin rod, or torsional vibrations of thin rod.

scholarly journals INVERSE PROBLEM FOR THE TIME-FRACTIONAL EULER-BERNOULLI BEAM EQUATION

2021 ◽  
Vol 26 (3) ◽  
pp. 503-518
Author(s):  
Ibrahim Tekin ◽  
He Yang
Keyword(s):  
Existence And Uniqueness ◽  
Numerical Experiments ◽  
Small Time ◽  
Beam Equation ◽  
Time Interval ◽  
Existence And Uniqueness Theorem ◽  
Bernoulli Beam ◽  
Additional Measurement ◽  
Small Time Interval ◽  
Euler Bernoulli Beam

In this paper, the classical Euler-Bernoulli beam equation is considered by utilizing fractional calculus. Such an equation is called the time-fractional EulerBernoulli beam equation. The problem of determining the time-dependent coefficient for the fractional Euler-Bernoulli beam equation with homogeneous boundary conditions and an additional measurement is considered, and the existence and uniqueness theorem of the solution is proved by means of the contraction principle on a sufficiently small time interval. Numerical experiments are also provided to verify the theoretical findings.

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