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4. Boltzmann Equation

    1. Propagation of regularity


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      Problem 4.1.

      In the case of Maxwell molecules, can we propagate regularity of initial data for the cutoff Boltzmann equation?

        • Large data global well-posedness for nearly homogeneous non-cutoff Boltzmann

              Wynter (2023) showed global well-posedness for the inhomogeneous (3 + 1)d cutoff Boltzmann equation for large data. In particular, this result required that the collision kernel to be zero for collisions with small relative velocities. With $x \in \mathbb{R}$ and imposing the cutoff, Wynter is able to exploit the Bony functional to get global well-posedness.

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          Problem 4.2.

          [Christopher Henderson] Prove large data global well-posedness for the inhomogeneous (3 +1) non-cutoff Boltzmann equation.

            • Construction of large shocks


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              Problem 4.3.

              [Dominic Wynter] Construct traveling wave solutions to the Boltzmann equation with large amplitude (at least amplitude 1) in dimension 1. Is it possible to observed temperature overshoot, a well-observed phenomena in physics.

                • Caflisch construction for implosion singularity formation

                      In Caflisch (1980), it is shown that any smooth solution of the Euler equations can be used to make a corresponding solution to the Boltzmann equation. Additionally, the formation of implosion singularities is well-studied for compressible fluids.

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                  Problem 4.4.

                  Study the formation of implosion singularities for the Boltzmann equation via the Caflisch construction of solutions

                    • Global well-posedness for 3D incompressible axisymmetric Navier-Stokes equation

                          $$\begin{cases} \partial_t u + u\cdot \nabla u - \Delta u + \nabla p = 0\\ \nabla u = 0\end{cases}$$ In cylindrical coordinates, axisymmetric means that $u = u(r,z)$.

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                      Problem 4.5.

                      [Jiajie Chen] Prove global well-posedness for 3D incompressible axisymmetric Navier-Stokes for large initial data.
                        1. Remark. [Jiajie Chen] It is known that that if the total circulation, given by $$r \times u_\theta \leq |\log(r)|^{-\alpha}$$ for some $\alpha > 1$ and $r < r_0$, then the solution to the incompressible axisymmetric (NS) is globally well-posed. A priori, it is only know that this quantity is conserved.

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