[Paper Review] Blowup of Smooth Solutions to the Navier-Stokes Equations for Compressible Isothermal Fluids

Dapeng Du, Jingyu Li+1 more|arXiv (Cornell University)|Aug 8, 2011

Navier-Stokes equation solutions3 references3 citations

TL;DR

This paper proves that smooth solutions to the one- and two-dimensional isothermal compressible Navier-Stokes equations blow up in finite time when the initial density is compactly supported and nontrivial. Using a weighted energy integral method and energy conservation, the authors show that the momentum integral grows linearly over time while the initial data term remains bounded, leading to a contradiction unless the solution blows up, thus establishing finite-time blowup for radially symmetric initial data in dimensions one and two.

ABSTRACT

It is shown that the one-dimensional or two-dimensional radially symmetric isothermal compressible Navier-Stokes system has no non-trivial global smooth solutions if the initial density is compactly supported. This result is a generalization of Xin's work \cite{Xin98} to the isothermal case.

Motivation & Objective

• To establish the existence of finite-time blowup for smooth solutions to the compressible isothermal Navier-Stokes equations under compactly supported initial density.
• To extend Xin's blowup result from non-barotropic to isothermal fluids in one and two spatial dimensions.
• To demonstrate that global smooth solutions cannot exist when initial data are radially symmetric and compactly supported.
• To provide a simplified proof technique based on weighted momentum integration and energy conservation for the isothermal case.

Proposed method

• Use of the function method, integrating the momentum equation with a spatial weight $ x $, to derive an integral identity involving density and velocity.
• Proof that the density remains compactly supported for all time, based on particle path analysis and the continuity equation.
• Derivation of energy conservation for the system by multiplying the continuity and momentum equations by $ \ln(\rho + \epsilon) $ and $ u $, respectively, and taking the limit $ \epsilon \to 0 $.
• Establishment of $ L^2 $ bounds on $ \nabla u $ via energy estimates, ensuring integrability in time.
• Application of Newton-Leibniz inequalities to bound the $ L^\infty $ norm of $ u \cdot x $ in terms of $ \| \nabla u \|_{L^2} $ in both 1D and 2D.
• Contradiction argument: the left-hand side of the momentum integral identity is bounded, while the right-hand side grows linearly in time, implying finite-time blowup.

Experimental results

Research questions

• RQ1 Can smooth solutions to the isothermal compressible Navier-Stokes equations exist globally in time when the initial density is compactly supported and nontrivial?
• RQ2 Does the blowup mechanism observed in non-barotropic and isentropic flows extend to the isothermal case in one and two dimensions?
• RQ3 What role does radial symmetry play in the formation of finite-time singularities for compressible Navier-Stokes flows with compactly supported initial data?
• RQ4 Can the blowup result be established using a unified method applicable to isothermal, isentropic, and non-barotropic cases?

Key findings

• Smooth solutions to the 1D and 2D isothermal compressible Navier-Stokes equations with compactly supported initial density cannot exist globally in time.
• The momentum integral $ \int \rho u \cdot x \, dx $ grows linearly in time due to the pressure term $ a \int \rho \, dx = a m_0 $, which contributes $ a m_0 t $ to the identity.
• The initial momentum term $ \int \rho_0 u_0 \cdot x \, dx $ remains bounded, and the $ L^\infty $ norm of $ u \cdot x $ is controlled by the $ L^2 $ norm of $ \nabla u $, which is uniformly bounded in time.
• Energy conservation and $ L^2 $ bounds on $ \nabla u $ ensure that the left-hand side of the momentum integral identity remains bounded, while the right-hand side grows linearly.
• The contradiction between bounded left-hand side and linearly growing right-hand side implies that the solution must blow up in finite time.
• The blowup result holds under the conditions $ \mu > 0 $, $ \lambda + \frac{2}{n}\mu > 0 $, and nontrivial, compactly supported initial density with radial symmetry in 2D or any symmetry in 1D.