Schwartz Function Valued Solutions of the Euler and the Navier-Stokes Equations

Dio, Philipp di

doi:10.21203/rs.3.rs-2462466/v1

2023

DOI: 10.21203/rs.3.rs-2462466/v1

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Schwartz Function Valued Solutions of the Euler and the Navier-Stokes Equations

Philipp di Dio

Abstract: We prove the existence of a solution for the second order system of partial differential equations ∂tf = Δf + g . ∇f + h . f + k by a Montel space version of Arzelà--Ascoli and bound all Schwartz semi-norms. We find that for the Euler and the Navier--Stokes equations the vorticity remains a Schwartz function as long as the classical solution exists. Our approach is not affected by viscosity. It treats the hyperbolic Euler and the parabolic Navier--Stokes equation simultaneously. 2020 MSC: 35Q30, 76D03, 76D05

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Time-Dependent Moments From the Heat Equation and a Transport Equation

Curto

Dio

2022

International Mathematics Research Notices

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We present a new connection between the classical theory of full and truncated moment problems and the theory of partial differential equations, as follows. For the classical heat equation $\partial _t u = {\nu } \Delta u$, with initial data $u_0 \in {\mathcal {S}}(\mathds {R}^n)$, we first compute the moments $s_{\alpha }(t)$ of the unique solution $u \in {\mathcal {S}}(\mathds {R}^n)$. These moments are polynomials in the time variable, of degree comparable to $\alpha $, and with coefficients satisfying a recursive relation. This allows us to define the polynomials for any sequence, and prove that they preserve some of the features of the heat kernel. In the case of moment sequences, the polynomials trace a curve (which we call the heat curve), which remains in the moment cone for positive time, but may wander outside the moment cone for negative time. This provides a description of the boundary points of the moment cone, which are also moment sequences. We also study how the determinacy of a moment sequence behaves along the heat curve. Next, we consider the transport equation $\partial _t u = ax \cdot \nabla u$ and conduct a similar analysis. Along the way we incorporate several illustrating examples. We show that while $\partial _t u = {\nu }\Delta u + ax\cdot \nabla u$ has no explicit solution, the time-dependent moments can be explicitly calculated.

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Time-Dependent Moments From the Heat Equation and a Transport Equation

Curto

Dio

2022

International Mathematics Research Notices

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Schwartz Function Valued Solutions of the Euler and the Navier-Stokes Equations

Cited by 1 publication

References 17 publications

Time-Dependent Moments From the Heat Equation and a Transport Equation

Time-Dependent Moments From the Heat Equation and a Transport Equation

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