A note on the extension of the polar decomposition for the multidimensional Burgers equation

It is shown that the generalizations to more than one space dimension of the pole decomposition for the Burgers equation with finite viscosity and no force are of the form u = -2 viscosity grad log P, where the P's are explicitly known algebraic (or trigonometric) polynomials in the space variables with polynomial (or exponential) dependence on time. Such solutions have polar singularities on complex algebraic varieties.Comment: 3 pages; minor formatting and typos corrected. Submitted to Phys. Rev. E (Rapid Comm.

Entire solutions of hydrodynamical equations with exponential dissipation

We consider a modification of the three-dimensional Navier--Stokes equations and other hydrodynamical evolution equations with space-periodic initial conditions in which the usual Laplacian of the dissipation operator is replaced by an operator whose Fourier symbol grows exponentially as \ue ^{|k|/\kd} at high wavenumbers $|k|$. Using estimates in suitable classes of analytic functions, we show that the solutions with initially finite energy become immediately entire in the space variables and that the Fourier coefficients decay faster than \ue ^{-C(k/\kd) \ln (|k|/\kd)} for any $C<1/(2\ln 2)$. The same result holds for the one-dimensional Burgers equation with exponential dissipation but can be improved: heuristic arguments and very precise simulations, analyzed by the method of asymptotic extrapolation of van der Hoeven, indicate that the leading-order asymptotics is precisely of the above form with $C= C_\star =1/\ln2$. The same behavior with a universal constant $C_\star$ is conjectured for the Navier--Stokes equations with exponential dissipation in any space dimension. This universality prevents the strong growth of intermittency in the far dissipation range which is obtained for ordinary Navier--Stokes turbulence. Possible applications to improved spectral simulations are briefly discussed.Comment: 29 pages, 3 figures, Comm. Math. Phys., in pres

Pole Dynamics And Oscillations For Complex Burgers Equation In The Small Dispersion Limit

A meromorphic solution to Burgers&apos; equation with complex viscosity is analyzed. The equation is linearized via the Cole-Hopf transform which allows for a careful study of the behavior of the singularities of the solution. The asymptotic behavior of the solution as the dispersion coefficient tends to zero is derived. For small dispersion, the time evolution of the poles is found by numerically solving a truncated infinite dimensional Calogero type dynamical system. This system represents a set of compatibility conditions derived from the PDE and a Mittag-Leffler (pole) expansion of the solution. The initial data is provided by high order asymptotic approximations of the poles at the critical time t for the dispersionless solution via the method of steepest descents. The solution is re-constructed using the pole expansion and the location of the poles. The oscillations observed via the singularities are compared to those obtained by a classical stationary phase analysis of the solutio..

Pole Dynamics And Oscillations For Complex Burgers Equation In The Small Dispersion Limit

. A meromorphic solution to Burgers&apos; equation with complex viscosity is analyzed. The equation is linearized via the Cole-Hopf transform which allows for a careful study of the behavior of the singularities of the solution. The asymptotic behavior of the solution as the dispersion coefficient tends to zero is derived. For small dispersion, the time evolution of the poles is found by numerically solving a truncated infinite dimensional Calogero type dynamical system. This system represents a set of compatibility conditions derived from the PDE and a Mittag-Leffler (pole) expansion of the solution. The initial data is provided by high order asymptotic approximations of the poles at the critical time t for the dispersionless solution via the method of steepest descents. The solution is re-constructed using the pole expansion and the location of the poles. The oscillations observed via the singularities are compared to those obtained by a classical stationary phase analysis of the solutio..

Deep learning-based prediction of future myocardial infarction using invasive coronary angiography: a feasibility study.

Angiographic parameters can facilitate the risk stratification of coronary lesions but remain insufficient in the prediction of future myocardial infarction (MI). We compared the ability of humans, angiographic parameters and deep learning (DL) to predict the lesion that would be responsible for a future MI in a population of patients with non-significant CAD at baseline. We retrospectively included patients who underwent invasive coronary angiography (ICA) for MI, in whom a previous angiogram had been performed within 5 years. The ability of human visual assessment, diameter stenosis, area stenosis, quantitative flow ratio (QFR) and DL to predict the future culprit lesion (FCL) was compared. In total, 746 cropped ICA images of FCL and non-culprit lesions (NCL) were analysed. Predictive models for each modality were developed in a training set before validation in a test set. DL exhibited the best predictive performance with an area under the curve of 0.81, compared with diameter stenosis (0.62, p=0.04), area stenosis (0.58, p=0.05) and QFR (0.67, p=0.13). DL exhibited a significant net reclassification improvement (NRI) compared with area stenosis (0.75, p=0.03) and QFR (0.95, p=0.01), and a positive nonsignificant NRI when compared with diameter stenosis. Among all models, DL demonstrated the highest accuracy (0.78) followed by QFR (0.70) and area stenosis (0.68). Predictions based on human visual assessment and diameter stenosis had the lowest accuracy (0.58). In this feasibility study, DL outperformed human visual assessment and established angiographic parameters in the prediction of FCLs. Larger studies are now required to confirm this finding