scholarly journals Solutions and Drift Homogenization for a Class of Viscous Lake Equations in

2021 ◽  
Author(s):  
Maoting Tong
Keyword(s):  
Continuous Function ◽  
Strong Solution ◽  
Initial Value Problem ◽  
Smooth Function ◽  
Three Dimensional ◽  
Usual Sense ◽  
Bounded Operators ◽  
Initial Value ◽  
Lake Equations ◽  
Local Strong Solution

In this paper we study solutions and drift homogenization for a class of viscous lake equations by using the method of semigroups of bounded operators. Suppose that the initial value i.e.,for some Hölder continuous function onwith smooth function value satisfying and Then the initial value problem (2) for viscous lake equations has a unique smooth local strong solution. Using this result we study the drift homogenization for three-dimensional stationary Stokes equation in the usual sense

The initial-value problem for a fourth-order dispersive closed curve flow on the 2-sphere

2017 ◽  
Vol 147 (6) ◽  
pp. 1243-1277 ◽  
Author(s):  
Eiji Onodera
Keyword(s):  
Fourth Order ◽  
Energy Method ◽  
Initial Value Problem ◽  
Local Existence ◽  
Three Dimensional ◽  
Closed Curve ◽  
Initial Value ◽  
Curve Flow ◽  
Local Existence And Uniqueness ◽  
Loss Of Derivatives

A closed curve flow on the 2-sphere evolved by a fourth-order nonlinear dispersive partial differential equation on the one-dimensional flat torus is studied. The governing equation arises in the field of physics in relation to the continuum limit of the Heisenberg spin chain systems or three-dimensional motion of the isolated vortex filament. The main result of the paper gives the local existence and uniqueness of a solution to the initial-value problem by overcoming loss of derivatives in the classical energy method and the absence of the local smoothing effect. The proof is based on the delicate analysis of the lower-order terms to find out the loss of derivatives and on the gauged energy method to eliminate the obstruction.

scholarly journals Rapidly decreasing behaviour of solutions in nonlinear 3-D-thermoelasticity

1991 ◽  
Vol 43 (1) ◽  
pp. 89-99 ◽  
Author(s):  
Song Jiang
Keyword(s):  
Boundary Value Problem ◽  
Sobolev Spaces ◽  
Initial Value Problem ◽  
Weighted Sobolev Spaces ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Exterior Domains ◽  
Initial Value ◽  
Initial Boundary Value

In this paper we study the asymptotic behaviour, as |x| → ∞, of solutions to the initial value problem in nonlinear three-dimensional thermoelasticity in some weighted Sobolev spaces. We show that under some conditions, solutions decrease fast for each t as x tends to infinity. We also consider the possible extension of the method presented in this paper to the initial boundary value problem in exterior domains.

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