1. I have started proving the existence of weak solution through faedo-galerkin method in a finite dimensional subspace.
It evolved into a Matrix IVP with vector of polynomial entries on right hand side.
2. Used weak compactness theorem (Banach–Alaoglu theorem ) to find a weak limit.
3. After deriving some a-priori estimates for that weak limit as well as for finite dimensional sequences in some Bochner spaces (space-time Lebesgue) it has been shown that the time derivative is in the required Dual space.
4. A lemma on Gelfand triplet is used to show that the weak solution is continuous in time.
5. Using the above continuity argument for the sub sequence and its weak limit, the limit has been passed inside the operators of the finite dimensional formulation (Faedo-Galerkin) of the problem. Hence the Existence.
6. To prove the uniqueness of the solution Lipschitz(L) continuity of polynomial being used to show the the norm of the difference of two different solution is bounded by a negative energy, provided L, epsilon and lambda satisfy a constraint.
7. Continuous dependence on the data was proved using Lipschitz constant and with fixed time interval.
8. Showing the total Energy dissipation needs some more grey activity in my brain. (Even,Bertozzi et al couldn't establish it though it is evident from computations)
9. Inner-product of time derivative and bi-harmonic term has been proved to be equal to the time derivative of L2 inner product of the the Laplacians.
10. The numerical scheme used was Convex-Concave splitting in time and Fourier spectral form in space. Due to periodic boundary conditions, it becomes an algebraic expressions resulting direct evaluation.
11. Time complexity has been analysed and is found to be of the order of ann fft algorithm.
It evolved into a Matrix IVP with vector of polynomial entries on right hand side.
2. Used weak compactness theorem (Banach–Alaoglu theorem ) to find a weak limit.
3. After deriving some a-priori estimates for that weak limit as well as for finite dimensional sequences in some Bochner spaces (space-time Lebesgue) it has been shown that the time derivative is in the required Dual space.
4. A lemma on Gelfand triplet is used to show that the weak solution is continuous in time.
5. Using the above continuity argument for the sub sequence and its weak limit, the limit has been passed inside the operators of the finite dimensional formulation (Faedo-Galerkin) of the problem. Hence the Existence.
6. To prove the uniqueness of the solution Lipschitz(L) continuity of polynomial being used to show the the norm of the difference of two different solution is bounded by a negative energy, provided L, epsilon and lambda satisfy a constraint.
7. Continuous dependence on the data was proved using Lipschitz constant and with fixed time interval.
8. Showing the total Energy dissipation needs some more grey activity in my brain. (Even,
9. Inner-product of time derivative and bi-harmonic term has been proved to be equal to the time derivative of L2 inner product of the the Laplacians.
10. The numerical scheme used was Convex-Concave splitting in time and Fourier spectral form in space. Due to periodic boundary conditions, it becomes an algebraic expressions resulting direct evaluation.
11. Time complexity has been analysed and is found to be of the order of ann fft algorithm.
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