How to prove that two PDE's are related?

Question

Say that I have PDE a)

$U_x+U_y=\alpha U$

then I have PDE b)

$U_{xx}+U_{yy}=\beta U$

It is obvious that the first and the second are related by that they are composed of two operators which differ by one degree of differentiation. However, how can I prove that the solutions are related too, and are they?

I was thinking of looking at if the PDEs are hyperbolic, parabolic or elliptic. But I am not sure that is the way to prove it. Also, I am not sure what is the criteria for that two PDEs are "related".

Sorry, I have no more to add, since I am not an expert on PDE theory.

Thanks

Answer 1

$$U_x+U_y=\alpha U \tag 1$$ The solution of Eq.$(1)$ is easy to find thanks to the method of characteristics (or other method) : $$U(x,y)=e^{\alpha x}\Phi(x-y)$$ $\Phi$ is an arbitrary function until some boundary condition be specified.

$U_x=\alpha e^{\alpha x}\Phi+e^{\alpha x}\Phi'$

$U_y=-e^{\alpha x}\Phi'$

$U_{xx}=\alpha^2 e^{\alpha x}\Phi+2\alpha e^{\alpha x}\Phi'+e^{\alpha x}\Phi''$

$U_{yy}=e^{\alpha x}\Phi''$

Putting them into Eq.$(2)$ leads to :

$$U_{xx}+U_{yy}=\beta U \tag 2$$

$\alpha^2 e^{\alpha x}\Phi+2\alpha e^{\alpha x}\Phi'+e^{\alpha x}\Phi''=\beta e^{\alpha x}\Phi$

$$(\alpha^2-\beta) \Phi+2\alpha \Phi'+\Phi''=0\tag 3$$

This is not true in general (any $\alpha,\beta,\Phi$). Thus a relationship between Eqs.$(1)$ and $(2)$ doesn't exist in general.

The solutions of Eqs.$(1)$ and $(2)$ are consistent only in the particular case of solutions of Eq.$(3)$ if :

$\Phi(x-y))=c_1\exp\left((-\alpha+\sqrt{\beta})(x-y) \right)+c_2\exp\left((-\alpha-\sqrt{\beta})(x-y) \right) \tag 4$

In this particular case the common solution of Eqs.$(1)$ and $(2)$ is : $$U(x,y)=c_1e^{\alpha x}\exp\left((-\alpha+\sqrt{\beta})(x-y) \right) + c_2e^{\alpha x}\exp\left((-\alpha-\sqrt{\beta})(x-y) \right) \tag 5$$

Answer 2

Take the four operators given in the OP:

$\frac{\partial^2}{\partial x^2}=A$

$\frac{\partial^2}{\partial y^2}=B$

$\frac{\partial}{\partial x}=C$

$\frac{\partial}{\partial y}=D$

The Da Prato-Grisvard theorem $^1$ states that we can find the sum of two operators A, B, where $A+B$ with natural domain $\mathscr{D}(A+B) = \mathscr{D}(A)\cup\mathscr{D}(B)$. Here, $\mathscr{D}(A)\cup\mathscr{D}(B)$ has similar properties as A and B and one obtains the important feature that the method can be iterated, and hence, complicated operators can be built up from simpler ones. If the operators are non-commuting, matters are, naturally, much more involved. A and B do not commute, and also C and D do not commute, since their order of operation would give different outcomes. However, the Da Prato-Grisvard theorem remains valid if A and B satisfy certain commutator estimates $^2$.

One can find $\mathscr{D}(A)\cup\mathscr{D}(B)$ and see if it admits a $H^\infty$ calcululs (to see what this means, see ref $^3$) and then do the same for the $\mathscr{D}(C)\cup\mathscr{D}(D)$ sum, and if both admit a $H^\infty$ calcululs, one has proven that the two PDEs have some conserved familiarity.

A second option is to use the Dore-Venni theorem$^4$ to find the sum of two non-commuting operators for the case of PDE 1 and PDE 2.

1. G. Da Prato, P. Grisvard. Sommes d’opératours linéaires et équations différentielles opérationelles, J. Math. Pures Appl. 54 (1975), 305–387. MR0442749 (56:1129)

2. Burrage, Kevin, and P. M. Burrage. "General order conditions for stochastic Runge-Kutta methods for both commuting and non-commuting stochastic ordinary differential equation systems." Applied Numerical Mathematics 28.2-4 (1998): 161-177.

https://www.ams.org/journals/tran/2007-359-08/S0002-9947-07-04291-2/S0002-9947-07-04291-2.pdf

3. Hytönen, Tuomas, et al. "The H∞-functional calculus." Analysis in Banach Spaces. Springer, Cham, 2017. 359-478. https://helda.helsinki.fi/bitstream/handle/10138/327524/hytonen_Ch10.pdf?sequence=1

4. S. Monniaux, J. Prüss. A theorem of the Dore-Venni type for non-commuting operators. Transactions Amer. Math. Soc. 349 (1997), 4787–4814. MR1433125 (98b:47005)