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.

scholarly journals The Global Well-Posedness for Large Amplitude Smooth Solutions for 3D Incompressible Navier–Stokes and Euler Equations Based on a Class of Variant Spherical Coordinates

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1195
Author(s):  
Shu Wang ◽  
Yongxin Wang
Keyword(s):  
Initial Data ◽  
Euler Equations ◽  
Large Amplitude ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Navier Stokes ◽  
Well Posedness ◽  
Spherical Coordinates ◽  
Smooth Solutions ◽  
Euler Systems

This paper investigates the globally dynamical stabilizing effects of the geometry of the domain at which the flow locates and of the geometry structure of the solutions with the finite energy to the three-dimensional (3D) incompressible Navier–Stokes (NS) and Euler systems. The global well-posedness for large amplitude smooth solutions to the Cauchy problem for 3D incompressible NS and Euler equations based on a class of variant spherical coordinates is obtained, where smooth initial data is not axi-symmetric with respect to any coordinate axis in Cartesian coordinate system. Furthermore, we establish the existence, uniqueness and exponentially decay rate in time of the global strong solution to the initial boundary value problem for 3D incompressible NS equations for a class of the smooth large initial data and a class of the special bounded domain described by variant spherical coordinates.

GLOBAL CENTERED WAVES AND CONTACT DISCONTINUITIES OF THE AXISYMMETRIC EULER EQUATIONS

2006 ◽  
Vol 03 (01) ◽  
pp. 143-193 ◽  
Author(s):  
PAUL GODIN
Keyword(s):  
Initial Data ◽  
Euler Equations ◽  
Contact Discontinuity ◽  
Existence Results ◽  
Smooth Solutions ◽  
Perfect Gas ◽  
Rotation Invariant ◽  
Contact Discontinuities ◽  
Positive Time ◽  
Two Space Dimensions

Global existence results have been obtained by Serre and Grassin–Serre for smooth solutions to the Euler equations of a perfect gas, provided the initial data belong to suitable spaces, the initial sound speed is small, and the initial velocity forces particles to spread out. We work in two space dimensions and start with initial data which are rotation invariant around 0 and of the type considered by Serre and Grassin–Serre. We then consider slightly perturbed initial data which are also rotation invariant around 0 and jump across a given circle centered at 0, in such a way that there is a solution with these perturbed initial data which presents two centered waves (in radial coordinates) and one contact discontinuity for small positive time. We show that this solution is global in positive time and keeps the same structure.

scholarly journals Local well-posedness of the Boltzmann equation with polynomially decaying initial data

2020 ◽  
Vol 13 (4) ◽  
pp. 837-867 ◽  
Author(s):  
Christopher Henderson ◽  
◽  
Stanley Snelson ◽  
Andrei Tarfulea ◽  
◽  
...  
Keyword(s):  
Boltzmann Equation ◽  
Initial Data ◽  
Well Posedness ◽  
The Boltzmann Equation

Well-posedness and limit behaviors for a stochastic higher order modified Camassa–Holm equation

2016 ◽  
Vol 16 (06) ◽  
pp. 1650019
Author(s):  
Lin Lin ◽  
Guangying Lv ◽  
Wei Yan
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Existence And Uniqueness ◽  
Local Existence ◽  
Higher Order ◽  
Well Posedness ◽  
Local Existence And Uniqueness ◽  
Holm Equation ◽  
The Cauchy Problem

This paper is devoted to the Cauchy problem for a stochastic higher order modified-Camassa–Holm equation [Formula: see text] The local existence and uniqueness with initial data [Formula: see text], [Formula: see text] and [Formula: see text], is established. The limit behaviors of the solution are examined as [Formula: see text].

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.

scholarly journals Optimal L p – L q -Type Decay Rates of Solutions to the Three-Dimensional Nonisentropic Compressible Euler Equations with Relaxation

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Rong Shen ◽  
Yong Wang
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Euler Equations ◽  
Existence And Uniqueness ◽  
Three Dimensional ◽  
Decay Rates ◽  
Compressible Euler Equations ◽  
Compressible Euler ◽  
Formula Type ◽  
Math Phys

In this paper, we consider the three-dimensional Cauchy problem of the nonisentropic compressible Euler equations with relaxation. Following the method of Wu et al. (2021, Adv. Math. Phys. Art. ID 5512285, pp. 1–13), we show the existence and uniqueness of the global small H k k ⩾ 3 solution only under the condition of smallness of the H 3 norm of the initial data. Moreover, we use a pure energy method with a time-weighted argument to prove the optimal L p – L q 1 ⩽ p ⩽ 2 , 2 ⩽ q ⩽ ∞ -type decay rates of the solution and its higher-order derivatives.

scholarly journals On the well-posedness of the anisotropically-reduced two-dimensional Kuramoto-Sivashinsky Equation

2022 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
David Massatt
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Existence And Uniqueness ◽  
Well Posedness ◽  
Two Dimensional ◽  
Periodic Domain ◽  
Uniqueness Of Solutions ◽  
Global Existence And Uniqueness ◽  
Sivashinsky Equation

<p style='text-indent:20px;'>We address the global existence and uniqueness of solutions for the anisotropically reduced 2D Kuramoto-Sivashinsky equations in a periodic domain with initial data <inline-formula><tex-math id="M1">\begin{document}$ u_{01} \in L^2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ u_{02} \in H^{-1 + \eta} $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M3">\begin{document}$ \eta &gt; 0 $\end{document}</tex-math></inline-formula>.</p>

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