Abstract In this article, the global existence of a unique strong solution to the 2D Sobolev equation with Burgers’ type nonlinearity is established using weak or weak* compactness type arguments. When the forcing function (f ≠ 0) is in L∞(L2), new a priori bounds are derived, which are valid uniformly in time as t↦∞ and with respect to the dispersion coefficient μ as μ↦0. It is further shown that the solution of the Sobolev equation converges to the solution of the 2D-Burgers’ equation with order O(μ). A finite element method is applied to approximate the solution in the spatial direction and the existence of a global attractor is derived for the semidiscrete scheme. Further, using a priori bounds and an integral operator, optimal error estimates are derived in L∞(L2)-norm, which hold uniformly with respect to μ as μ → 0. Since the constants in the error estimates have exponential growth in time, therefore, under a certain uniqueness condition, the error bounds are derived which are uniformly in time. More importantly, all the above results remain valid as μ tends to zero. Finally, this paper concludes with some numerical examples.

中文翻译：

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具有 Burgers 型非线性的二维 Sobolev 方程的有限元 Galerkin 方法

摘要 在本文中，使用弱或弱*紧致性类型参数建立了具有 Burgers 类型非线性的二维 Sobolev 方程的唯一强解的全局存在性。当强制函数 (f ≠ 0) 在 L∞(L2) 中时，新的先验界限被推导出来，其在时间上一致有效为 t↦∞ 并且相对于色散系数 μ 为 μ↦0。进一步表明，Sobolev 方程的解收敛到 O(μ) 阶 2D-Burgers 方程的解。应用有限元方法在空间方向上逼近解，并推导出半离散方案的全局吸引子的存在性。此外，使用先验界限和积分算子，可以在 L∞(L2)-范数中导出最佳误差估计，该范数相对于 μ 一致地保持为 μ → 0。由于误差估计中的常数随时间呈指数增长，因此，在一定唯一性条件下，推导出在时间上均匀的误差界限。更重要的是，当 μ 趋于零时，所有上述结果仍然有效。最后，本文以一些数值例子作为结论。