Weak existence and uniqueness for linear PDE system

I am interested in proving existence/uniqueness to: find $u(x,t)$, $v(x,t)$ such that $$u_t - a_1u_{xx} - a_2u_x - a_3u -a_4v = f$$ $$v_t - a_5u_{xx} - a_6u_x - a_7u - a_8v_{xx} - a_9v_x - a_{10}v = g$$ holds. The $a_i$ are functions of $(x,t)$. I don't mind which space the coefficients and solutions lie in, so some Sobolev time-dependent space is fine. (I'd prefer Hölder functions but it seems (according to what I asked on Mathoverflow; see http://mathoverflow.net/questions/104507/linear-coupled-parabolic-pde-system-with-holder-continuous-coefficients) this is impossible, so something weaker is fine).

Is there already literature where this is treated? Does anyone know how I prove existence to this system? A lot of the literature I come across look at something similar but the notation they use is a bit confusing as they never define it, so am unsure whether this falls into their schemes.

Thanks

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Could you post a link to your question on MathOverflow, so that the answers here will not duplicate what you already found out? –  user31373 Aug 18 '12 at 3:18
@LeonidKovalev Sure. –  Bloop Aug 18 '12 at 9:11
You need some kind of ellipticity assumption on coefficients; otherwise even the stationary case ($u,v$ independent on $t$) may have nonunique solutions. –  user31373 Aug 19 '12 at 22:27
@LVK Thanks for the bounty! –  Bloop Aug 21 '12 at 16:03

The apparent difficulty here is that the term $u_{xx}$ appears in the second equation, so standard results do not seem to apply directly. However, a simple trick can reduce it to a standard parabolic system.

To explain the main idea, let us take the model system $$\begin{split} u_t&=u_{xx}+f,\\ v_t&=v_{xx}+u_{xx}+g. \end{split}$$ We see that the system is well-posed by just solving the first equation for $u$ and plugging it into the second equation, in which the term $u_{xx}$ would be treated as a source term. Perhaps this method can be extended to treat the original system, but it seems to be cumbersome. So we try to develop a method that does not involve solving one equation and plugging into the other. The idea is as follows. We introduce the new variable $w=u_x$, which satisfies $$w_t = w_{xx}+f_x.$$ This equation is derived simply by differentiating the equation of $u$ w.r.t. $x$. Now the equation for $v$ reads $$v_t = v_{xx} + w_x +g,$$ which does not involve $w_{xx}$, so standard results can be applied to the $(v,w)$ system.

We need to take care of one more obstacle in order to apply the above trick to the original system. Let me illustrate it by the model problem $$\begin{split} u_t&=\alpha u_{xx}+\beta u+\gamma v,\\ v_t&=\delta v_{xx}+\varepsilon u_{xx}+\zeta u + \eta v, \end{split}$$ where $\alpha,\beta,\gamma,\delta,\varepsilon,\zeta$, and $\eta$ are sufficiently regular functions. When we differentiate the first equation in $x$, we quickly realize that we cannot write the resulting equation in terms of $w\equiv u_x$, because of the term $\beta_xu$. The plot thickens when we see that there is $\zeta u$ in the second equation as well. The variable $u$ seems to be unavoidable. A way out would be to simply consider the equation for $u$ in addition to the two equations for $v$ and $w$. This means that we consider the following system $$\begin{split} u_t&=\alpha u_{xx}+\beta u+\gamma v,\\ v_t&=\delta v_{xx}+\varepsilon w_{x}+\zeta u + \eta v,\\ w_t&=\alpha w_{xx}+\alpha_xw_x + \beta w+\beta_x u+\gamma v_x+\gamma_x v, \end{split}$$ for which standard results are available. The initial condition for $w$ must of course be $u_x$, and since the third equation is derived from the first equation, one can show that the equations preserve the constraint $w-u_x=0$, so our 3-equation system produces the same solutions as the original 2-equation system.

Now let us discuss the boundary conditions. The question is what would be the boundary condition for $w$? Clearly, everything is fine for the periodic boundary condition, and for the whole space case. The Neumann boundary condition for $u$ is ok too, since this becomes a Dirichlet condition for $w$. In general (for instance if you have the Dirichlet condition for $u$), we need to impose the boundary condition $w-u_x=0$ at the boundary.

I don't recall any literature that explicitly treat systems that I called "standard", but the results can easily be constructed. Roughly speaking we just have to go through the proofs for single equations, and replace the word "equations" with "system". One relatively clean approach would be to combine elliptic theory with semigroup theory. Introductory references in this regard are Friedman's PDE, Wloka's PDE, and Taylor's 3 volume PDE. More specialized ones are Eidel'man's Parabolic systems, and Krylov's two books, one treating elliptic and parabolic equations in Sobolev spaces, and the other treating the same equations in Hölder spaces. Books on elliptic systems are MacLean's Strongly elliptic systems and boundary integral equations, and Chen and Wu's Second order elliptic equations and elliptic systems. A canonical reference on semigroup theory is Pazy's Semigroups of linear operators and applications to PDE.

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Thanks for the answer, I really appreciate it. You write that "standard results are available" for the 3-equation system. Can you please give me a reference to these? I guess it's in Holder space? PS: you're missing a subscript $x$ on $v$ in the PDE for $w$. –  Bloop Aug 21 '12 at 18:02
@Bloop: Thanks for spotting the typo. I added some references. –  timur Aug 22 '12 at 2:30
Is it easy to explain why the presence of the Laplacian term $u_{xx}$ in the second equation causes so many problems? –  Court Aug 24 '12 at 13:51