Well-posedness of a complex PDE.

[Alone]

I asked my question at math.stackexchange with no reply as of yet, here's my question:
https://math.stackexchange.com/questions/2448845/well-posedness-of-a-complex-pde

Hope I could have some assistance here.

[EDIT by moderator: Added copy of question here.]

I have the following PDE:

$$u_t= \imath \bigg[ A_1 u_{xx}+A_2 u_{yy} \bigg] + B_1u_x+ B_2u_y+Cu+F(x,y,t)$$
$$u(t=0,x,y)=f(x,y);f(x,y)=f(x+2\pi,y+2\pi)$$

Where $u\in \mathbb{C}^N$, i.e a complex vector with $N$ entries, $A_j,B_j,C$ are complex $N\times N$ matrices, and $A_j$ are Hermitian matrices, i.e $(\bar{A_j}^T)=A_j$ where the bar is complex conjugate and $T$ is transpose.
All the matrices are constant matrices.
I want to find a nontrivial condition for when this PDE is well-posed and give an energy estimate.

So I started by calculating :$\partial_t(\langle u,u \rangle = \langle u_t, u \rangle + \langle u,u_t \rangle$; I am not sure how to proceed from here.

Any tips?

Thanks.

 

[Euge]

Hi Alan,

What conditions does $F$ satisfy? Also, you need to know what (Banach) space your iniitial data belongs to.

 

[Alone]

Hi @Euge , $F\in C^{\infty}(x,y,t)$, $f\in C^{\infty}(x,y)$, as in $f,F$ are smooth functions in their corresponding variables.

The definition of well posedness is that there exists a unique $2\pi$-periodic solution, $u$ and that there exist constants $A,B$ which don't depend on $f,F$ s.t:

$$\| u(\cdot , t) \| \le Be^{At}(\| f\|+\max_{ 0\le \tau \le t} \| F(\cdot , \tau)\|)$$

 

[Euge]

Re: Well-Posedness Of A Complex Pde.

Hi @Euge , $F\in C^{\infty}(x,y,t)$, $f\in C^{\infty}(x,y)$, as in $f,F$ are smooth functions in their corresponding variables.

The definition of well posedness is that there exists a unique $2\pi$-periodic solution, $u$ and that there exist constants $A,B$ which don't depend on $f,F$ s.t:

$$\| u(\cdot , t) \| \le Be^{At}(\| f\|+\max_{ 0\le \tau \le t} \| F(\cdot , \tau)\|)$$

That's a bit better. What is $\iota$? What is spatial domain, and how are the norms defined? Writing this down will provide better clarity.

When dealing with general pde like this, it helps to work with a simplified equations where one or more of the constants equal 0. Look for symmetries. If your domain is unbounded, work in the Schwarz class. These tips can help in finding meaningful energy or Strichartz estimates. Just be careful with the $F$-term -- the situation may be vastly different with it than without it, like how the nonlinear Schrödinger equation with potential is a different animal from the corresponding equation without potential.

 

[Alone]

Re: Well-Posedness Of A Complex Pde.

That's a bit better. What is $\iota$? What is spatial domain, and how are the norms defined? Writing this down will provide better clarity.

When dealing with general pde like this, it helps to work with a simplified equations where one or more of the constants equal 0. Look for symmetries. If your domain is unbounded, work in the Schwarz class. These tips can help in finding meaningful energy or Strichartz estimates. Just be careful with the $F$-term -- the situation may be vastly different with it than without it, like how the nonlinear Schrödinger equation with potential is a different animal from the corresponding equation without potential.

It's not iota, but \imath as in $i=\sqrt{-1}$.

The spatial domain is some compact square $[0,1]\times [0,1]$.
The norms are defined as follows:
$u=(u_1,\ldots,u_N)$, then:
$$\| u \|^2_{L^2} = \sum_{j=1}^N \|u_j\|^2_{L^2}$$

We're using the L^2 norms component wise.
As I have written in my first post I am trying to use this relation: $\partial_t \langle u , u \rangle = \langle u_t,u\rangle +\langle u , u_t \rangle$, and then plugging in PDE instead of $u_t$, but I don't see how to proceed from there.

 

[Euge]

Re: Well-Posedness Of A Complex Pde.

Take the spatial Fourier transform on both sides of the PDE to obtain a form $\partial_t\widehat{u} = \Lambda \widehat{u} + \widehat{F}$, $\widehat{u}(\xi,0) = \widehat{f}(\xi)$. The $\Lambda$ will be something like $\sqrt{-1}[-4\pi^2\xi_1^2A_1 - 4\pi^2\xi_2^2A_2 + 2\pi \xi_1B_1 + 2\pi \xi_2B_2 + C]$, for $(\xi_1,\xi_2)$ in the frequency domain. So $u(\xi,t) = e^{\Lambda t}\widehat{f}(\xi) + \int_0^t e^{(t-s)\Lambda}\widehat{F}(\xi,s)\, ds$. Using the triangle and Minkowski inequalities, one obtains
$$\|u(t)\|_2 \lesssim e^{\lambda t}\|\widehat{f}\|_2 + \int_0^t e^{(t-s)\lambda}\|\widehat{F}(\cdot,s)\|_2\, ds \lesssim e^{\lambda t}(\|\widehat{f}\|_2 + \max_{0 \le s \le 1} \|\widehat{F}(\cdot,s)\|_2$$ where $\lambda = \|\Lambda\|$.

By Plancherel’s theorem, $\|\widehat{f}\|_2 = \|f\|_2$ and $\|\widehat{F}(\cdot,s)\|_2 = \|F(\cdot,s)\|_2$. Thus
$$\|u\|_2 \lesssim e^{\lambda t}(\|f\|_2 + \max_{0\le s \le 1} \|F(\cdot,s)\|_2)$$

To get existence and uniqueness, one usually starts by writing the corresponding integral equation and setting up a Banach contraction scheme, so that Banach’s fixed point theorem can be applied.