Potential Theory: Study Caracteristics of Formula

[Raparicio]

Dear friends,

I need to know, in the theory of the potential, what must I do to study the caracteristics of this formulae:

$$0=\vec{v} [\nabla \vec{j}+ \frac{ \partial \vec{D}}{\partial t} ]+ [\nabla \vec{D} \frac {\partial \vec{v}}{\partial t} ]+ \nabla \vec{D} (\vec{v} \nabla ) \vec{v}$$

Thanks you so much

R. Aparicio.

 

[dextercioby]

Dear friends,

I need to know, in the theory of the potential, what must I do to study the caracteristics of this formulae:

$$0=\vec{v} [\nabla \vec{j}+ \frac{ \partial \vec{D}}{\partial t} ]+ [\nabla \vec{D} \frac {\partial \vec{v}}{\partial t} ]+ \nabla \vec{D} (\vec{v} \nabla ) \vec{v}$$

Thanks you so much

R. Aparicio.

I'm not sure what u mean by "characteristics".If u mean the "characteristic curves" or "integral curves",then there is a problem as the equation u posted has 3 vector variables in 3D (probably),which means 9 unknowns.I'm absolutely sure that u couldn't find any integral curves...By the way,maybe if u told us where did u get this eq.and what $\vec{D}$ and $\vec{j}$ mean (i assumed \vec(v) was transport velocity),we could give you some more help...Mathematically speaking it is a quasilinear first order homogeneous PDE.As it's only one eq.and about 9 unknown functions,i'm sure it is part of a system of PDE-s.
So,help us to help you...

Daniel.

 

[Raparicio]

Maxwell

I'm not sure what u mean by "characteristics".If u mean the "characteristic curves" or "integral curves",then there is a problem as the equation u posted has 3 vector variables in 3D (probably),which means 9 unknowns.I'm absolutely sure that u couldn't find any integral curves...By the way,maybe if u told us where did u get this eq.and what $\vec{D}$ and $\vec{j}$ mean (i assumed \vec(v) was transport velocity),we could give you some more help...Mathematically speaking it is a quasilinear first order homogeneous PDE.As it's only one eq.and about 9 unknown functions,i'm sure it is part of a system of PDE-s.
So,help us to help you...

Daniel.

Dear Daniel,

Is an ecuation that I have seen in a book, that is a part of a theory that says that the formula includes the 2 law of Newton. It's a formula that could be a part of the system of formulas from Maxwell. My answer is that if there's a way to resolve it, or if could have information about what really means it.

R.Aparicio.

 

[dextercioby]

Dear Daniel,

Is an ecuation that I have seen in a book, that is a part of a theory that says that the formula includes the 2 law of Newton. It's a formula that could be a part of the system of formulas from Maxwell. My answer is that if there's a way to resolve it, or if could have information about what really means it.

R.Aparicio.

Yes,most of the fluid dynamics equations are derived from the second law of dynamics by Newton.However,a more riguros approach would involve Botzmann's equation.
The only system of equations by Maxwell i know of is the one in electrodynamics.Your equation looks fluid dynamics to me.
I already told u that it is a first order PDE with 9 unknowns and only one equation.It cannot be solved.You need exactly 9 INDEPENDENT equations to solve for each of the components of the 3 vectors.
I cannot give you the signification,since you haven't revealed the context:the book/article/univ.course from where u've gotten it...

Daniel.