Local existence and blow-up criterion for the Boussinesq equations

1997 ◽  
Vol 127 (5) ◽  
pp. 935-946 ◽  
Author(s):  
Dongho Chae ◽  
Hee-Seok Nam
Keyword(s):  
Global Existence ◽  
External Force ◽  
Existence And Uniqueness ◽  
Blow Up ◽  
Local Existence ◽  
Boussinesq Equations ◽  
Passive Scalar ◽  
Smooth Solutions ◽  
Local Existence And Uniqueness ◽  
Blow Up Criterion

SynopsisIn this paper, we prove local existence and uniqueness of smooth solutions of the Boussinesq equations. We also obtain a blow-up criterion for these smooth solutions. This shows that the maximum norm of the gradient of the passive scalar controls the breakdown of smooth solutions of the Boussinesq equations. As an application of this criterion, we prove global existence of smooth solutions in the case of zero external force.

scholarly journals Local existence and blow-up criterion of Hölder continuous solutions of the Boussinesq equations

1999 ◽  
Vol 155 ◽  
pp. 55-80 ◽  
Author(s):  
Dongho Chae ◽  
Sung-Ki Kim ◽  
Hee-Seok Nam
Keyword(s):  
Existence And Uniqueness ◽  
Blow Up ◽  
Local Existence ◽  
Boussinesq Equations ◽  
Passive Scalar ◽  
Continuous Solutions ◽  
Hölder Continuous ◽  
Local Existence And Uniqueness ◽  
Local Solutions ◽  
Blow Up Criterion

AbstractIn this paper we prove the local existence and uniqueness of C1+γ solutions of the Boussinesq equations with initial data υ0, θ0 ∈ C1+γ, ω0, ∇θ0 ∈ Lq for 0 < γ < 1 and 1 < q < 2. We also obtain a blow-up criterion for this local solutions. More precisely we show that the gradient of the passive scalar θ controls the breakdown of C1+γ solutions of the Boussinesq equations.

scholarly journals The Burgers’ equation with stochastic transport: shock formation, local and global existence of smooth solutions

2019 ◽  
Vol 26 (6) ◽  
Author(s):  
Diego Alonso-Orán ◽  
Aythami Bethencourt de León ◽  
So Takao
Keyword(s):  
Global Existence ◽  
Existence And Uniqueness ◽  
Stochastic Equation ◽  
Burgers Equation ◽  
Blow Up ◽  
Local Existence ◽  
Smooth Solutions ◽  
Local Existence And Uniqueness ◽  
Stochastic Transport ◽  
Global Existence And Uniqueness

Abstract In this work, we examine the solution properties of the Burgers’ equation with stochastic transport. First, we prove results on the formation of shocks in the stochastic equation and then obtain a stochastic Rankine–Hugoniot condition that the shocks satisfy. Next, we establish the local existence and uniqueness of smooth solutions in the inviscid case and construct a blow-up criterion. Finally, in the viscous case, we prove global existence and uniqueness of smooth solutions.

scholarly journals Blow-up criterion for the density dependent inviscid Boussinesq equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Li Li ◽  
Yanping Zhou
Keyword(s):  
Velocity Field ◽  
Energy Method ◽  
Blow Up ◽  
A Priori Estimates ◽  
A Priori ◽  
Boussinesq Equations ◽  
Smooth Solutions ◽  
Density Dependent ◽  
Priori Estimates ◽  
Blow Up Criterion

Abstract In this work, we consider the density-dependent incompressible inviscid Boussinesq equations in $\mathbb{R}^{N}\ (N\geq 2)$ R N ( N ≥ 2 ) . By using the basic energy method, we first give the a priori estimates of smooth solutions and then get a blow-up criterion. This shows that the maximum norm of the gradient velocity field controls the breakdown of smooth solutions of the density-dependent inviscid Boussinesq equations. Our result extends the known blow-up criteria.

scholarly journals On the Well-Posedness of the Boussinesq Equation in the Triebel-Lizorkin-Lorentz Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Zhaoyin Xiang ◽  
Wei Yan
Keyword(s):  
Boussinesq Equation ◽  
Blow Up ◽  
Boussinesq Equations ◽  
Lorentz Spaces ◽  
Well Posedness ◽  
Smooth Solutions ◽  
Blow Up Criterion

We establish the local well-posedness and obtain a blow-up criterion of smooth solutions for the Boussinesq equations in the framework of Triebel-Lizorkin-Lorentz spaces. The main ingredients of our proofs are Littlewood-Paley decomposition and the paradifferential calculus.

scholarly journals On a system of nonlinear pseudoparabolic equations with Robin-Dirichlet boundary conditions

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Le Thi Phuong Ngoc ◽  
Khong Thi Thao Uyen ◽  
Nguyen Huu Nhan ◽  
Nguyen Thanh Long
Keyword(s):  
Weak Solution ◽  
Lower Bound ◽  
Existence And Uniqueness ◽  
Dirichlet Boundary ◽  
Blow Up ◽  
Local Existence ◽  
Initial Energy ◽  
Dirichlet Boundary Conditions ◽  
Local Existence And Uniqueness ◽  
Pseudoparabolic Equations

<p style='text-indent:20px;'>In this paper, we investigate a system of pseudoparabolic equations with Robin-Dirichlet conditions. First, the local existence and uniqueness of a weak solution are established by applying the Faedo-Galerkin method. Next, for suitable initial datum, we obtain the global existence and decay of weak solutions. Finally, using concavity method, we prove blow-up results for solutions when the initial energy is nonnegative or negative, then we establish here the lifespan for the equations via finding the upper bound and the lower bound for the blow-up times.</p>

REMARKS ON THE BLOW-UP CRITERION FOR THE 3-D BOUSSINESQ EQUATIONS

1999 ◽  
Vol 09 (09) ◽  
pp. 1323-1332 ◽  
Author(s):  
NAOYUKI ISHIMURA ◽  
HIROKO MORIMOTO
Keyword(s):  
Blow Up ◽  
Boussinesq Equations ◽  
Maximum Norm ◽  
Smooth Solutions ◽  
Temperature Function ◽  
Blow Up Criterion

We consider the problem of blow-up of smooth solutions for the 3-D Boussinesq equations. Owing to the viscosity, we prove that the maximum norm of the gradient of vorticity controls the breakdown of the solutions; the scalar temperature function is shown to be irrelevant to the breakdown.

Sign in / Sign up

Export Citation Format

Share Document