Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  study coupled nonlinear parabolic equations for a fluid described by a material density ρ and a temperature \gJ, both functions of space and time. In one dimension,  find some stationary solutions corresponding to fixing the temperature on the boundary, with no-escape boundary conditions for the material. For the special case, where the temperature on the boundary is the same at both ends, the linearized equations for small perturbations about a stationary solution at uniform temperature and density are derived; they are subject to boundary conditions, Dirichlet for \gJ and no-flow conditions for the material. The spectrum of the generator L of time evolution, regarded as an operator on L2, is shown to be real, discrete and non-positive, even though L is not self-adjoint. This result is necessary for the stability of the stationary state, but might not be sufficient. The problem lies in the fact that L is not a sectorial operator, since its numerical range is C.
['R. F. Streater']
I am writing a research paper on a idea Stability of a Hot Smoluchowski Fluid
[{"from": "human", "value": "I am writing a research paper on a idea Stability of a Hot Smoluchowski Fluid"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  study coupled nonlinear parabolic equations for a fluid described by a material density \u03c1 and a temperature \\gJ, both functions of space and time. In one dimension,  find some stationary solutions corresponding to fixing the temperature on the boundary, with no-escape boundary conditions for the material. For the special case, where the temperature on the boundary is the same at both ends, the linearized equations for small perturbations about a stationary solution at uniform temperature and density are derived; they are subject to boundary conditions, Dirichlet for \\gJ and no-flow conditions for the material. The spectrum of the generator L of time evolution, regarded as an operator on L2, is shown to be real, discrete and non-positive, even though L is not self-adjoint. This result is necessary for the stability of the stationary state, but might not be sufficient. The problem lies in the fact that L is not a sectorial operator, since its numerical range is C."}]