Existence and Uniqueness of a Solution to the Cauchy Problem for the Damped Boussinesq Equation

1996 ◽  
Vol 19 (8) ◽  
pp. 639-649 ◽  
Author(s):  
V. Varlamov
Keyword(s):  
Cauchy Problem ◽  
Existence And Uniqueness ◽  
Boussinesq Equation ◽  
The Cauchy Problem ◽  
Uniqueness Of A Solution

scholarly journals A degenerate parabolic–hyperbolic Cauchy problem with a stochastic force

2015 ◽  
Vol 12 (03) ◽  
pp. 501-533 ◽  
Author(s):  
Caroline Bauzet ◽  
Guy Vallet ◽  
Petra Wittbold
Keyword(s):  
Cauchy Problem ◽  
Existence And Uniqueness ◽  
Stochastic Forcing ◽  
Hyperbolic Problem ◽  
Stochastic Force ◽  
The Cauchy Problem ◽  
Degenerate Parabolic ◽  
Entropy Formulation ◽  
Uniqueness Of A Solution ◽  
Nonlinear Degenerate

In this paper, we are interested in the Cauchy problem for a nonlinear degenerate parabolic–hyperbolic problem with multiplicative stochastic forcing. Using an adapted entropy formulation a result of existence and uniqueness of a solution is proven.

scholarly journals Asymptotic behavior of solutions of the damped Boussinesq equation in two space dimensions

1999 ◽  
Vol 22 (1) ◽  
pp. 131-145 ◽  
Author(s):  
Vladimir V. Varlamov
Keyword(s):  
Cauchy Problem ◽  
Existence And Uniqueness ◽  
Boussinesq Equation ◽  
Asymptotic Behavior Of Solutions ◽  
Small Initial Data ◽  
Time Asymptotics ◽  
The Cauchy Problem ◽  
Long Time ◽  
Two Space Dimensions ◽  
Long Time Asymptotics

The Cauchy problem for the damped Boussinesq equation with small initial data is considered in two space dimensions. Existence and uniqueness of its classical solution is proved and the solution is constructed in the form of a series. The major term of its long-time asymptotics is calculated explicitly and a uniform in space estimate of the residual term is given.

scholarly journals Existence, Uniqueness, and Eq–Ulam-Type Stability of Fuzzy Fractional Differential Equation

2021 ◽  
Vol 5 (3) ◽  
pp. 66
Author(s):  
Azmat Ullah Khan Niazi ◽  
Jiawei He ◽  
Ramsha Shafqat ◽  
Bilal Ahmed
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Fractional Differential Equation ◽  
Existence And Uniqueness ◽  
Initial Conditions ◽  
The Cauchy Problem ◽  
Fractional Differential ◽  
Ulam Type Stability ◽  
Fuzzy Fractional Differential Equations ◽  
Fuzzy Fractional Differential Equation

This paper concerns with the existence and uniqueness of the Cauchy problem for a system of fuzzy fractional differential equation with Caputo derivative of order q∈(1,2], 0cD0+qu(t)=λu(t)⊕f(t,u(t))⊕B(t)C(t),t∈[0,T] with initial conditions u(0)=u0,u′(0)=u1. Moreover, by using direct analytic methods, the Eq–Ulam-type results are also presented. In addition, several examples are given which show the applicability of fuzzy fractional differential equations.

scholarly journals A Note on a Meshless Method for Fractional Laplacian at Arbitrary Irregular Meshes

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2843
Author(s):  
Ángel García ◽  
Mihaela Negreanu ◽  
Francisco Ureña ◽  
Antonio M. Vargas
Keyword(s):  
Cauchy Problem ◽  
Porous Medium ◽  
Meshless Method ◽  
Existence And Uniqueness ◽  
Fractional Laplacian ◽  
Porous Medium Equation ◽  
Medium Equation ◽  
The Cauchy Problem ◽  
Irregular Meshes ◽  
Complicated Geometry

The existence and uniqueness of the discrete solutions of a porous medium equation with diffusion are demonstrated. The Cauchy problem contains a fractional Laplacian and it is equivalent to the extension formulation in the sense of trace and harmonic extension operators. By using the generalized finite difference method, we obtain the convergence of the numerical solution to the classical/theoretical solution of the equation for nonnegative initial data sufficiently smooth and bounded. This procedure allows us to use meshes with complicated geometry (more realistic) or with an irregular distribution of nodes (providing more accurate solutions where needed). Some numerical results are presented in arbitrary irregular meshes to illustrate the potential of the method.

scholarly journals Global existence of strong solutions for incompressible hydrodynamic flow of liquid crystals with vacuum

Filomat ◽  
2013 ◽  
Vol 27 (7) ◽  
pp. 1247-1257 ◽  
Author(s):  
Shijin Ding ◽  
Jinrui Huang ◽  
Fengguang Xia
Keyword(s):  
Cauchy Problem ◽  
Liquid Crystals ◽  
Global Existence ◽  
Existence And Uniqueness ◽  
Strong Solutions ◽  
Hydrodynamic Flow ◽  
Three Dimensions ◽  
Small Initial Data ◽  
The Cauchy Problem ◽  
Global Existence And Uniqueness

We consider the Cauchy problem for incompressible hydrodynamic flow of nematic liquid crystals in three dimensions. We prove the global existence and uniqueness of the strong solutions with nonnegative p0 and small initial data.

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