Global asymptotics toward the rarefaction wave for solutions of viscous 𝑝-system with boundary effect

Source abstract

The initial-boundary value problem on the half-line R + ⋅ = ( 0 , ∞ ) {R_ + } \cdot = \left ( 0, \infty \right ) for a system of barotropic viscous flow v t − u x = 0 , u t + p ( v ) x = μ ( u x v ) x {v_t} - {u_x} = 0, {u_t} + p{\left ( v \right )_x} = \mu {\left ( {\frac {{{u_x}}}{v}} \right )_x} is investigated, where the pressure p ( v ) = v − γ ( γ ≥ 1 ) p\left ( v \right ) = {v^{ - \gamma }}\left ( \gamma \ge 1 \right ) for the specific volume v > 0 v > 0 . Note that the boundary value at x = 0 x = 0 is given only for the velocity u u , say u − {u_ - } and that the initial data ( v 0 , u 0 ) ( x ) \left ( {v_0}, {u_0} \right )\left ( x \right ) have the constant states ( v + , u + ) \left ( {v_ + }, {u_ + } \right ) at x = + ∞ x = + \infty with v 0 ( x ) > 0 , v + > 0 {v_0}\left ( x \right ) > 0, {v_ + } > 0 . If u − > u + {u_ - } > {u_ + } , then there is a unique v − {v_ - } such that ( v + , u + ) ∈ R 2 ( v − , u − ) \left ( {v_ + }, {u_ + } \right ) \in {R_2}\left ( {v_ - }, {u_ - } \right ) (the 2-rarefaction curve) and hence there exists the 2-rarefaction wave ( v 2 R , u 2 R ) ( x / t ) \left ( v_2^R, u_2^R \right )\left ( x/t \right ) connecting ( v − , u − ) \left ( {v_ - }, {u_ - } \right ) with ( v + , u + ) \left ( {v_ + }, {u_ + } \right ) . Our assertion is that, if u − > u + {u_ - } > {u_ + } , then there exists a global solution ( v , u ) ( t , x ) \left ( v, u \right )\left ( t, x \right ) in C 0 ( [ 0 , ∞ ) ; H 1 ( R + ) ) {C^0}\left ( \left [ 0, \infty \right );{H^1}\left ( {R_ + } \right ) \right ) , which tends to the 2-rarefaction wave ( v 2 R , u 2 R ) ( x / t ) | x ≥ 0 \left ( v_2^R, u_2^R \right )\left ( x/t \right )\left | {_{x \ge 0}} \right . as t → ∞ t \to \infty in the maximum norm, with no smallness condition on | u + − u − | \left | {{u_ + } - {u_ - }} \right | and ‖ ( v 0 − v + , u 0 − u + ) ‖ H 1 {\left \| {\left ( {{v_0} - {v_ + }, {u_0} - {u_ + }} \right )} \right \|_{{H^1}}} , nor restriction on γ ( ≥ 1 ) \gamma \left ( \ge 1 \right ) . A similar result to the corresponding Cauchy problem is also obtained. The proofs are given by an elementary L 2 {L^{2}} -energy method.