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Local-in-time existence of strong solutions of the n-dimensional Burgers equation via discretizations

In: The Strength of Nonstandard Analysis

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  • João Paulo Teixeira

    (Instituto Superior Técnico)

Abstract

Consider the equation: $$ u_t = \nu \Delta u - (u \cdot \nabla )u + f{\mathbf{ }}for{\mathbf{ }}x \in [0,1]^n {\mathbf{ }}and{\mathbf{ }}t \in (0,\infty ), $$ together with periodic boundary conditions and initial condition u(t, 0) = g(x). This corresponds a Navier-Stokes problem where the incompressibility condition has been dropped. The major difficulty in existence proofs for this simplified problem is the unbounded advection term, (u · ∇)u. We present a proof of local-in-time existence of a smooth solution based on a discretization by a suitable Euler scheme. It will be shown that this solution exists in an interval [0, T), where T ≤ 1/C, with C depending only on n and the values of the Lipschitz constants of f and u at time 0. The argument given is based directly on local estimates of the solutions of the discretized problem.

Suggested Citation

  • João Paulo Teixeira, 2007. "Local-in-time existence of strong solutions of the n-dimensional Burgers equation via discretizations," Springer Books, in: Imme van den Berg & Vítor Neves (ed.), The Strength of Nonstandard Analysis, chapter 23, pages 349-366, Springer.
    • Handle: RePEc:spr:sprchp:978-3-211-49905-4_23
    DOI: 10.1007/978-3-211-49905-4_23
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