I'm trying to check the well-posedness of the following equation:

$\pmatrix{u\\v}_t$ = $\pmatrix{4/3 & 0 \\ 1 & 0}$$\pmatrix{u\\v}_{xx}$+$\pmatrix{0 & -2/3 \\ 1 & 0}$$\pmatrix{u\\v}_{xy}$

As far I understand, in order to show well-posedness, I have to prove that the energy of the equation is bounded, that is:$$\int a{\lVert{v}\rVert}^2+b{\lVert{v}\rVert}^2 < M$$ where $a$, $b$ and $M$ are constants. Does this need to be proven to hold for all $a$ and $b$ or I can choose their values to my convenience?

Is there a book (or online resource) that contains discussion of this type of problems?

  • 1
    $\begingroup$ Your equation is a bit strange: where did it come from? The time evolution of $v$ is essentially an ODE, and just by counting derivatives you cannot use Cauchy-Kovalevsky as your equation is not hyperbolic, hence bounded energy may not be enough to guarantee well-posedness. But putting that aside: yes, the constants $a,b,M$ need to be chosen. In general such an almost conservation law depends on some algebraic cancellations in the equation and the values of $a,b$ are important to ensure that. $\endgroup$ Jun 19, 2012 at 14:15
  • $\begingroup$ Why can I choose the values of $a$ and $b$? $\endgroup$ Jun 19, 2012 at 14:22
  • 1
    $\begingroup$ Why can you not? Observe that if $\int a\|u\|^2 + b\|v\|^2 \leq M$, then for any $c,d > 0$ you have $$\int c\|u\|^2 + d\|v\|^2 \leq \max(\frac{c}{a},\frac{d}{b}) \int a\|u\|^2 + b\|v\|^2 \leq \max(\frac{c}{a},\frac{d}{b}) M$$ The problem that I was alluding to in my comment above is that for certain $a,b$ the inequality is more easily proven: it is a lot easier to prove the conservation of energy for the linear wave equation when you write the energy as $$ E(u) = \int c^2|u_t|^2 + |u_x|^2 $$ then to directly show that $$\int 5 c^2|u_t|^2 + 2.3 |u_x|^2 $$ is bounded. $\endgroup$ Jun 19, 2012 at 15:20


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