Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2019
The Navier-Stokes equations for viscous, incompressible fluids are studied in the three-dimension... more The Navier-Stokes equations for viscous, incompressible fluids are studied in the three-dimensional periodic domains, with the body force having an asymptotic expansion, when time goes to infinity, in terms of power-decaying functions in a Sobolev-Gevrey space. Any Leray-Hopf weak solution is proved to have an asymptotic expansion of the same type in the same space, which is uniquely determined by the force, and independent of the individual solutions. In case the expansion is convergent, we show that the next asymptotic approximation for the solution must be an exponential decay. Furthermore, the convergence of the expansion and the range of its coefficients, as the force varies are investigated.
We give necessary and sufficient conditions for the existence of weak solutions to the model equa... more We give necessary and sufficient conditions for the existence of weak solutions to the model equation −∆pu = σ u q on R n , in the case 0 < q < p − 1, where σ ≥ 0 is an arbitrary locally integrable function, or measure, and ∆pu = div(∇u|∇u| p−2) is the p-Laplacian. Sharp global pointwise estimates and regularity properties of solutions are obtained as well. As a consequence, we characterize the solvability of the equation −∆pv = b |∇v| p v + σ on R n , where b > 0. These results are new even in the classical case p = 2. Our approach is based on the use of special nonlinear potentials of Wolff type adapted for "sublinear" problems, and related integral inequalities. It allows us to treat simultaneously several problems of this type, such as equations with general quasilinear operators div A(x, ∇u), fractional Laplacians (−∆) α , or fully nonlinear k-Hessian operators.
Calculus of Variations and Partial Differential Equations, 2014
We study finite energy solutions to quasilinear elliptic equations of the type −∆pu = σ u q in R ... more We study finite energy solutions to quasilinear elliptic equations of the type −∆pu = σ u q in R n , where ∆p is the p-Laplacian, p > 1, and σ is a nonnegative function (or measure) on R n , in the case 0 < q < p − 1 (below the "natural growth" rate q = p − 1). We give an explicit necessary and sufficient condition on σ which ensures that there exists a solution u in the homogeneous Sobolev space L 1,p 0 (R n), and prove its uniqueness. Among our main tools are integral inequalities closely associated with this problem, and Wolff potential estimates used to obtain sharp bounds of solutions. More general quasilinear equations with the A-Laplacian divA(x, ∇•) in place of ∆p are considered as well.
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2019
The Navier-Stokes equations for viscous, incompressible fluids are studied in the three-dimension... more The Navier-Stokes equations for viscous, incompressible fluids are studied in the three-dimensional periodic domains, with the body force having an asymptotic expansion, when time goes to infinity, in terms of power-decaying functions in a Sobolev-Gevrey space. Any Leray-Hopf weak solution is proved to have an asymptotic expansion of the same type in the same space, which is uniquely determined by the force, and independent of the individual solutions. In case the expansion is convergent, we show that the next asymptotic approximation for the solution must be an exponential decay. Furthermore, the convergence of the expansion and the range of its coefficients, as the force varies are investigated.
We give necessary and sufficient conditions for the existence of weak solutions to the model equa... more We give necessary and sufficient conditions for the existence of weak solutions to the model equation −∆pu = σ u q on R n , in the case 0 < q < p − 1, where σ ≥ 0 is an arbitrary locally integrable function, or measure, and ∆pu = div(∇u|∇u| p−2) is the p-Laplacian. Sharp global pointwise estimates and regularity properties of solutions are obtained as well. As a consequence, we characterize the solvability of the equation −∆pv = b |∇v| p v + σ on R n , where b > 0. These results are new even in the classical case p = 2. Our approach is based on the use of special nonlinear potentials of Wolff type adapted for "sublinear" problems, and related integral inequalities. It allows us to treat simultaneously several problems of this type, such as equations with general quasilinear operators div A(x, ∇u), fractional Laplacians (−∆) α , or fully nonlinear k-Hessian operators.
Calculus of Variations and Partial Differential Equations, 2014
We study finite energy solutions to quasilinear elliptic equations of the type −∆pu = σ u q in R ... more We study finite energy solutions to quasilinear elliptic equations of the type −∆pu = σ u q in R n , where ∆p is the p-Laplacian, p > 1, and σ is a nonnegative function (or measure) on R n , in the case 0 < q < p − 1 (below the "natural growth" rate q = p − 1). We give an explicit necessary and sufficient condition on σ which ensures that there exists a solution u in the homogeneous Sobolev space L 1,p 0 (R n), and prove its uniqueness. Among our main tools are integral inequalities closely associated with this problem, and Wolff potential estimates used to obtain sharp bounds of solutions. More general quasilinear equations with the A-Laplacian divA(x, ∇•) in place of ∆p are considered as well.
Uploads
Papers by Dat Cao