New dissipated energy for nonnegative weak solution of unstable thin-film equations

الأصل · EN

The fluid thin film equation hₜ = - (hⁿ hxxx)ₓ - a₁(hᵐ hₓ)ₓ is known to conserve mass ∫h dx, and in the case of a₁ ≤ 0, to dissipate entropy ∫h³/² ⁻ ⁿdx (see [8]) and the L²-norm of the gradient ∫hₓ²dx (see [3]). For the special case of a₁ = 0 a new dissipated quantity ∫ hαhₓ²dx was recently discovered for positive classical solutions by Laugesen (see [15]). We extend it in two ways. First, we prove that Laugesen's functional dissipates strong nonnegative generalized solutions. Second, we prove the full α-energy ∫(1/2 hα hₓ² - a₁hα⁺ ᵐ ⁻ ⁿ ⁺ ²(α+ m - n + 1)(α+ m - n + 2)) dx dissipation for strong nonnegative generalized solutions in the case of the unstable porous media perturbation a₁> 0 and the critical exponent m = n+2.