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1. Blowup and global well-posedness for the wave kinetic equation

Wave kinetic equations (WKE) describe the evolution of the energy spectrum $\mathbb{E}[|\hat{u}(k,t)|^2]$ of a system of weakly nonlinear interacting waves. An example of such an equation is the homogeneous 4-WKE equation, $$ \partial_tf = \int_{\mathbb{R}^{3d}}\delta(\Sigma)\delta(\Omega)ff_1f_2f_3\left(\frac{1}{f} + \frac{1}{f_1} - \frac{1}{f_2} - \frac{1}{f_3}\right)dv_1dv_2dv_3 $$ where $\Sigma = k + k_1 - k_2 - k_3$ and $\Omega = \omega(k) + \omega(k_1) - \omega(k_2) - \omega(k_3)$ and $\omega$ is the dispersion relation of the dispersive equation from which the WKE is derived.

    1. Continuation after blowup


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      Problem 1.1.

      If solutions of the wave kinetic equation blowup in $L^\infty$, is there continuation of these solutions after the blowup?

        • Blowup of solutions to a wave kinetic equation from smooth initial data


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          Problem 1.2.

            • Monotone quantities

                  The entropy for the wave kinetic equation is given by $$\int \log(f).$$ In the cases of other kinetic equations like the Landau and Boltzmann equations, other monotone quantities have been found like the Fisher information.

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              Problem 1.3.

              Can we find any monotone hydrodynamic quantities for a wave kinetic equation?

                • Conditions for global well-posedness


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                  Problem 1.4.

                  Can we find conditions on the initial data so that the homogeneous wave kinetic equation is globally well-posed?

                      Cite this as: AimPL: Integro-differential equations in many-particle interacting systems, available at http://aimpl.org/manyparticle.