Incompressible Euler Equations and the Effect of Changes at a Distance

Because pressure is determined globally for the incompressible Euler equations, a localized change to the initial velocity will have an immediate effect throughout space. For solutions to be physically meaningful, one would expect such effects to decrease with distance from the localized change, giving the solutions a type of stability. Indeed, this is the case for solutions having spatial decay, as can be easily shown. We consider the more difficult case of solutions lacking spatial decay, and show that such stability still holds, albeit in a somewhat weaker form.Comment: Revised statement of Theorem 1 to include a missing definitio

Vanishing viscosity in the plane for vorticity in borderline spaces of Besov type

AbstractThe existence and uniqueness of solutions to the Euler equations for initial vorticity in BΓ∩Lp0∩Lp1 was proved by Misha Vishik, where BΓ is a borderline Besov space parameterized by the function Γ and 1<p0<2<p1. Vishik established short time existence and uniqueness when Γ(n)=O(logn) and global existence and uniqueness when Γ(n)=O(log12n). For initial vorticity in BΓ∩L2, we establish the vanishing viscosity limit in L2(R2) of solutions of the Navier–Stokes equations to a solution of the Euler equations in the plane, convergence being uniform over short time when Γ(n)=O(logn) and uniform over any finite time when Γ(n)=O(logκn), 0⩽κ<1, and we give a bound on the rate of convergence. This allows us to extend the class of initial vorticities for which both global existence and uniqueness of solutions to the Euler equations can be established to include BΓ∩L2 when Γ(n)=O(logκn) for 0<κ<1

Vanishing viscosity in the plane for vorticity in borderline spaces of Besov type

Abstract The existence and uniqueness of solutions to the Euler equations for initial vorticity in B Γ ∩ L p 0 ∩ L p 1 was proved by Misha Vishik, where B Γ is a borderline Besov space parameterized by the function Γ and 1 &lt; p 0 &lt; 2 &lt; p 1 . Vishik established short time existence and uniqueness when Γ(n) = O(log n) and global existence and uniqueness when Γ(n) = O(log 1 2 n). For initial vorticity in B Γ ∩ L 2 , we establish the vanishing viscosity limit in L 2 (R 2 ) of solutions of the Navier-Stokes equations to a solution of the Euler equations in the plane, convergence being uniform over short time when Γ(n) = O(log n) and uniform over any finite time when Γ(n) = O(log κ n), 0 ≤ κ &lt; 1, and we give a bound on the rate of convergence. This allows us to extend the class of initial vorticities for which both global existence and uniqueness of solutions to the Euler equations can be established to include B Γ ∩ L 2 when Γ(n) = O(log κ n) for 0 &lt; κ &lt; 1

Incompressible fluids with vorticity in Besov spaces

textIn this thesis, we consider soltions to the two-dimensional Euler equations with uniformly continuous initial vorticity in a critical or subcritical Besov space. We use paradifferential calculus to show that the solution will lose an arbitrarily small amount of smoothness over any fixed finite time interval. This result is motivated by a theorem of Bahouri and Chemin which states that the Sobolev exponent of a solution to the two-dimensional Euler equations in a critical or subcritical Sobolev space may decay exponentially with time. To prove our result, one can use methods similar to those used by Bahouri and Chemin for initial vorticity in a Besov space with Besov exponent between 0 and 1; however, we use different methods to prove a result which applies for any Sobolev exponent between 0 and 2. The remainder of this thesis is based on joint work with J. Kelliher. We study the vanishing viscosity limit of solutions of the Navier-Stokes equations to solutions of the Euler equations in the plane assuming initial vorticity is in a variant Besov space introduced by Vishik. Our methods allow us to extend a global in time uniqueness result established by Vishik for the two-dimensional Euler equations in this space.Mathematic