In the problem of a fluid interace problem I have two equations in the 2D Fluid Interface problem of the velocity of a fluid and vorticity in the weak formulation. With an energy bound of the system obtained by Gronwall's Lemma such as:
$\begin{equation}E(t)\leq -1/C log(exp(-C E(0)-Ct^2)\end{equation}$
For a positive constant $C$.
My question is, in which conditions we will have the local existence and uniqueness. The existence follows from the weak formulation? Because if we have a PDEs system in the weak formulation there exist a Time $T$ such that exists at least one weak solution? Shouldn't I use Picard theorem and Lipschitz condition of the PDE operator in order to prove the existence of solutions?
If we obtain an energy bound and we consider the difference of two solutions $\omega(x,t)=z_1(x,t)-z_2(x,t)$ and both satisfies that they have the same energy at the beggining having a priori estimates of the energy implies uniqueness?.
Thanks in advance.
In general you cannot deduce existence from a priori estimates and must use some limiting argument by approximate solutions (common in the case of evolution equations) and then extract a convergent sequence of functions converging to a weak solution of your PDE. Usually one uses Grönwall's inequality to provide an upper bound which is used to make some argument about the convergence of an approximate solution to your weak solution. For example the Galerkin method is often applied to parabolic (linear and nonlinear) equations by projecting the PDE onto a finite subspace of some Hilbert space you expect solutions to live in, generating a sequence of ODEs which hopefully weakly converges to a solution of the PDE. The existence of the ODEs follows from standard ODE existence and uniqueness.
As for uniqueness, yes, if you have an a priori estimate on the difference of two solutions you can get uniqueness. Note however, that for any nonlinear problem, it is not in general true that the difference of two solutions solves the PDE. You often need to derive new a priori estimates for the difference.
For example, consider $u_t - u_{xx} = F(u)$ with initial data $u_0$. Suppose you have obtained some energy estimate for $||u||_{H^s}$ using Grönwall of the form $$ ||u||_{H^s} \lesssim ||u_0||_{H^s}G(t) $$ Where we suppose that $G(t)$ is monotone increasing and does not blow up on $[0,T]$. Then suping in time gives us $$ ||u||_{L^\infty(0,T;H^s)} \lesssim ||u_0||_{H^s}G(T) $$ so that when $u_0=0$ we get the zero solution (if such a solution exists). However if $u,v$ are two solutions and $w:=u-v$, then $w$ satisfies $w_t-w_{xx} =F(u)-F(v)\ne F(u-v) = F(w)$ (in general) with intial data $w_0=0$. Therefore our energy estimates do not apply to the PDE for $w$ and we must make new estimates for the $w$ PDE. I.e. it may not be true that we can bound the solution by some multiple of the initial data. This is a common problem in fluids because generally the equations are nonlinear.
Majda and Bertozzi's "Vorticity and Incompressible Flow" has a good exposition of uniqueness via a priori estimates in Chapter 3 where they prove local well posedness of Euler and Navier-Stokes.