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I often come across the term dissipative (partial) differential equation in mathematical articles, especially in the context of hypocoercivity and entropy methods. I now have an intuitive idea of ​​what that means. For example, the heat equation or the damped wave equation are considered dissipative. But is there a rigorous mathematical definition of it or an easy characterization? I can well imagine that there will be different ones depending on the author. (I am not asking about the physical intuition behind it.) Is it possible to give a characterization for an evolution equation (Cauchy-Problem) $$ \partial_t u + A(u) = f(t)$$ through the operator $A$? And is there a connection to the concept of dissipative operators in semigroup theory (Lumer-Phillips-Theorem)? Would be grateful for good references too.

EDIT: I already tried to read "Entropy Methods for Diffusive Partial Differential Equations" by Ansgar Jüngel and "Hypocoercivity" by Cédric Villani, where the term "dissipative equation" is used a lot. But there was no adapted definition for this framework. I just found the script "Dissipative Partial Differential Equations and Dynamical Systems" by C.E.Wayne who tries to give a definition but rather in context of dynamical systems.

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    $\begingroup$ I somewhat doubt that there can be a satisfying answer; this rather seems to be a question of the type "ask three mathematicians, get eleven different answers". For instance, even the claim that the heat equation is dissipative doesn't seem so clear: if you consider the heat equation on the whole space or on a domain with Neumann boundary conditions, no energy is lost. Whether one wants to call this "dissipative" is up to interpretation. $\endgroup$
    – Jochen Glueck
    Commented Aug 5, 2023 at 22:15
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    $\begingroup$ Anyway, I upvoted since I have non-zero hope that someone has a more useful (and more comprehensive) answer than "it depends on who you ask" - and if so, I'd be really interested in reading it. $\endgroup$
    – Jochen Glueck
    Commented Aug 5, 2023 at 22:17
  • $\begingroup$ Spectrum of A (as in the given equation above) lies on the right-half plane plus the imaginary axis. If A has that property, then solution of the equation will be bounded for well-behaved forcing $f$, via use of variation of constants formula. These definitions regularly pop up in control theory literature. $\endgroup$
    – Piyush Grover
    Commented Aug 5, 2023 at 22:43
  • $\begingroup$ @PiyushGrover: I'm not sure I follow your comment. Spectrum in the closed right half-plane does not imply boundedness of the solutions, even if the forcing term is $0$. In finite dimension this requires, in addition, that the eigenvalues on the imaginary axis are semi-simple. In infinite dimensions, one gets even less boundedness information from the location of the spectrum alone. Generally speaking, the concept "dissipativity" seems to be closer related to the numerical range of $A$ than to the spectrum of $A$. $\endgroup$
    – Jochen Glueck
    Commented Aug 5, 2023 at 23:46
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    $\begingroup$ @Jochen Glueck: Concerning the heat equation on the whole space or with no flux boundary condition, I agree in the physical sense. But doesn't that depend on what you define as energy/entropy? If I think of the $L^2-$Energy, it would decrease in time and it would make sense to call it "dissipative". $\endgroup$
    – kumquat
    Commented Aug 6, 2023 at 12:55

2 Answers 2

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Q: Is there a rigorous mathematical definition or an easy characterization of a dissipative PDE?

Dissipative Operators and Hyperbolic Systems of Partial Differential Equations by R.S. Phillips gives a characterization for a hyperbolic PDE:

The initial value problem $\partial y/\partial t=Ly$ with $L$ a linear differential operator in a Hilbert space $H$ with inner product $\langle y,z\rangle$ is dissipative if $$\frac{d}{dt}\langle y,y\rangle=\langle Ly,y\rangle+\langle y,Ly\rangle\leq 0\;\text{for all}\; y\in H.$$

The quantity $\langle y,y\rangle$ can be identified with the energy of the system.

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  • $\begingroup$ ... and thus includes most hyperbolic systems, as well as the Schroedinger equation (RHS being 0)? $\endgroup$
    – Willie Wong
    Commented Sep 18, 2023 at 2:46
  • $\begingroup$ If the RHS is 0, Phillips calls the operator "conservative" (see definition 1.1.2); the Schrödinger equation is then both conservative and dissipative in his terminology. $\endgroup$
    – Carlo Beenakker
    Commented Sep 18, 2023 at 6:11
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In some sense, the so-called "entropy" does not have a unified definition in the mathematics community and sometimes the notion of "entropy" is identified (loosely speaking) as certain convex Lyapunov functional which assists us in the study of the problem of "convergence to equilibrium". I personally highly recommend the (relatively condense) monograph titled Entropy Methods for Diffusive Partial Differential Equations written by Ansgar Jüngel for such topics. It contains many essential techniques in the study of dissipative PDEs and also have extensive pointers to related (cutting edge) research articles.

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