id author title date pages extension mime words sentence flesch summary cache txt ejde-1971 Kim, Tujin Non-steady magneto-hydrodynamics-heat system with joule and buoyancy effects under mixed boundary conditions 2025 42 .pdf application/pdf 17906 699 83 (4.38) EJDE-2025/119 NON-STEADY MAGNETOHYDRODYNAMICS-HEAT SYSTEMS 19 Taking into account (4.37), (4.38) and applying the inequality |a + b|p ≤ 2p(|a|p + |b|p), p ∈ (1,∞), we have I2 ≡ 1 ∥u∥L6(0,T ;V) ∣∣∣ ∫ T 0 [ ek1t⟨curl(ŵ + v0)× (ŵ + v0), u⟩ ] dt ∣∣∣ ≤ c ∥u∥L6(0,T ;V) ∫ T 0 ∥ curl(ŵ + v0)∥L2∥(ŵ + v0)∥1/2L2 ∥(ŵ + v0)∥1/2V ∥u∥L6 dt ≤ ∥ŵ + v0∥1/2C(0,T ;L2) ( c ∥u∥L6(0,T ;V) ∥ŵ + v0∥3/2L9/5(0,T ;V) ∥u∥L6(0,T ;V) ) ≤ ∥ŵ + v0∥C([0,T ];HV) + c∥(ŵ + v0)∥3L9/5(0.T ;V) ≤ c∥ŵ′∥1/2 L6/5(0,T ;V∗) ∥ŵ∥1/2L6(0,T ;V) + ∥v0∥+ c∥ŵ + v0∥3L6(0,T ; Let us estimate∣∣∣ ∫ T 0 ek1t⟨(ŵ + v0)θ0,∇θ̂⟩ dt ∣∣∣ = ∣∣∣ ∫ T 0 [ ek1t⟨ŵθ0,∇θ̂⟩+ ek1t⟨v0θ0,∇θ̂⟩ ] dt ∣∣∣. First, we have∣∣∣ ∫ T 0 ek1t⟨ŵθ0,∇θ̂⟩ dt ∣∣∣ ≤ ∫ T 0 ek1t∥ŵ∥L3∥θ0∥L6∥∇θ̂∥ dt ≤ κ0 12 ∥θ̂∥2 L2(0,T ;W 1,2 ΓD ) + c′Te4k1T ε ∥θ0∥4W 1,2 + ε 6 ∥ŵ∥6L6(0,T ;V). cache/ejde-1971.pdf txt/ejde-1971.txt