Uniform well-posedness and inviscid limit for the KdV-Burgers and mKdV-Burgers equations on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"> <mml:mi mathvariant="double-struck">T</mml:mi> </mml:math>

This article investigates the unconditional uniform well-posedness and inviscid limit for the periodic Korteweg-de Vries-Burgers (KdV-B) and modified Korteweg-de Vries-Burgers (mKdV-B) equations: ∂ t u + ∂ x 3 u − ε ∂ x 2 u = ∂ x ( u α ) , u ( 0 ) = u 0 , where α ∈ { 2 , 3 } , ε ∈ ( 0 , 1 ] is the diffusion coefficient, and u : R + × T → R . For the KdV-B equation ( α = 2 ), we establish unconditional uniform global well-posedness in H s ( T ) for s ≥ 0 , uniformly for all ε ∈ [ 0 , 1 ] . This result is achieved via differentiation by parts, without relying on auxiliary function spaces. Furthermore, we prove that as ε → 0 the solutions converge in C ( [ 0 , T ] ; H s ) ( s ≥ 0 ) to those of the KdV equation for any T > 0 . For the mKdV-B equation ( α = 3 ), we establish analogous results in H s ( T ) for s ≥ 1 / 2 by introducing a new method to handle the case s = 1 / 2 .