Blow-up or No Blow-up? the Role of Convection in 3D Incompressible Navier-Stokes Equations
|
|
- Emery George
- 5 years ago
- Views:
Transcription
1 Blow-up or No Blow-up? the Role of Convection in 3D Incompressible Navier-Stokes Equations Thomas Y. Hou Applied and Comput. Mathematics, Caltech Joint work with Zhen Lei; Congming Li, Ruo Li, and Guo Luo PDE Conference in honor of Blake Temple s 60th birthday May 2, 2011
2 Motivation The question of whether the 3D incompressible Navier-Stokes equations can develop a finite time singularity from smooth initial data is one of the seven Clay Millennium Problems. u t + (u )u = p + ν u, u = 0, (1) with initial condition u(x, 0) = u 0. Define vorticity ω = u, then ω is governed by ω t + (u )ω = u ω + ν ω. (2) Note u is formally of the same order as ω. Thus the vortex stretching term u ω ω 2. So far, most regularity analysis uses energy estimates and treats the nonlinear terms as a small perturbation to the diffusion term. The global regularity can be obtained only for small data.
3 A brief review Global existence for small data (Leray, Ladyzhenskaya, Kato, etc). If u 0 L p (p 3) or u 0 L 2 u 0 L 2 is small, then the 3D Navier-Stokes equations have a globally smooth solution. Non-blowup criteria due to G. Prodi 59, J. Serrin 63. A weak solution u of the 3D Navier-Stokes equations is smooth on [0, T ] R 3 provided that u L q t Lp x ([0,T ] R 3 ) < for some p, q satisfying 3 p + 2 q 1 with 3 < p and 2 q <. The critical case of p = 3, q = was proved by L. Escauriaza, G. Seregin, and V. Sverak in Partial regularity theory (Caffarelli-Kohn-Nirenberg 82, F. Lin 98) For any suitable weak solution of the 3D Navier-Stokes equations on an open set in space-time, the one-dimensional Hausdorff measure of the associated singular set is zero.
4 Review continud Beale-Kato-Majda non-blow-up criterion, u ceases to be classical at T if and only if T 0 ω (t) dt =. Geometry regularity of direction field of ω: Constantin, Fefferman and Majda (1996). Let ω = ω ξ. The solution of 3D Euler is smooth solution up to T if (1) u L (Ω t) is bounded; and (2) t 0 ξ 2 L (Ω τ ) dτ is uniformly bounded for all t < T. Localized non-blow-up criteria, Deng-Hou-Yu, Denote by L(t) the arclength of a vortex line segment L t around the maximum vorticity. If (i) max Lt ( u ξ + u n ) C U (T t) A with A < 1, (ii) C L (T t) B L(t) C 0 / max Lt ( κ, ξ ) with B < 1 A, then the 3D Euler remains smooth up to T. For the axisymmetric Navier-Stokes equations, Chen-Strain-Tsai-Yau [07] and Koch-Nadirashvili-Seregin-Sverak [07] recently proved that if u(x, t) C t 1/2 where C is allowed to be large, then the velocity field u is regular at time zero.
5 The role of convection and 3D Euler Equations Due to the incompressibility condition, the convection term does not contribute to the energy norm of velocity or the L p -norm of ω. ω t + (u )ω = u ω + ν ω. The convection term and the vortex stretching term can be reformulated as a commutator: ω t + (u )ω (ω )u = ν ω. When we consider the two terms together, we preserve the Lagrangian structure of the solution of the 3D Euler equations: ω(x (α, t), t) = X α (α, t)ω 0(α), det( X (α, t)) 1 α where X (α, t) is the flow map: X t = u(x, t), X (α, 0) = α. Convection tends to severely deform and flatten the support of maximum vorticity. Such deformation tends to weaken the nonlinearity of vortex stretching dynamicallty.
6 Computation of Hou and Li, J. Nonlinear Science, 2006 Figure: Two slightly perturbed antiparallel vortex tubes at t=0 and t=6
7 Figure: The local 3D vortex structures and vortex lines around the maximum vorticity at t = 17 with resolution
8 Dynamic depletion of vortex stretching 35 ξ u ω 30 c 1 ω log( ω ) c 2 ω Figure: Study of the vortex stretching term in time, resolution The fact ξ u ω c 1 ω log ω plus ω = ξ u ω implies ω bounded by doubly exponential. D Dt
9 Log log plot of maximum vorticity in time Figure: The plot of log log ω vs time, resolution
10 The Stabilizing Effect of Convection, joint with C. Li Consider the 3D axi-symmetric incompressible Navier-Stokes equations ( ut θ + u r ur θ + u z uz θ = ν 2 1 ) r 2 u θ 1 r ur u θ, (3) ( ωt θ + u r ωr θ + u z ωz θ = ν 2 1 ) r 2 ω θ + 1 ( (u θ ) 2) r + 1 z r ur ω θ, (4) ( 2 1 r 2 ) ψ θ = ω θ, (5) where u θ, ω θ and ψ θ are the angular components of the velocity, vorticity and stream function respectively, and u r = (ψ θ ) z u z = 1 r (rψθ ) r. Note that equations (3)-(5) completely determine the evolution of the 3D axisymmetric Navier-Stokes equations.
11 Reformulation of axisymmetric Navier-Stokes equations In [CPAM 08], Hou and Li introduced the following new variables: u 1 = u θ /r, ω 1 = ω θ /r, ψ 1 = ψ θ /r, (6) and derived the following equivalent system that governs the dynamics of u 1, ω 1 and ψ 1 as follows: t u 1 + u r r u 1 + u z z u 1 = ν ( r r ) r + z 2 u1 + 2u 1 ψ 1z, ) z, t ω 1 + u r r ω 1 + u z z ω 1 = ν ( r r ) r + z 2 ω1 + ( u1 2 ( r r ) r + z 2 ψ1 = ω 1, where u r = rψ 1z, u z = 2ψ 1 + rψ 1r. Liu and Wang [SINUM07] showed that if u is a smooth velocity field, then u θ, ω θ and ψ θ must satisfy: u θ r=0 = ω θ r=0 = ψ θ r=0 = 0. Thus u 1, ψ 1 and ω 1 are well defiend. (7)
12 Stabilizing effect of convection through an exact 1D model for the 3D Navier-Stokes equations In [Hou-Li, CPAM, 61 (2008), no. 5, ], we derived an exact 1D model along the z-axis for the Navier-Stokes equations: (u 1 ) t + 2ψ 1 (u 1 ) z = ν(u 1 ) zz + 2 (ψ 1 ) z u 1, (8) (ω 1 ) t + 2ψ 1 (ω 1 ) z = ν(ω 1 ) zz + ( u1 2 ) z, (9) (ψ 1 ) zz = ω 1. (10) Let ũ = u 1, ṽ = (ψ 1 ) z, and ψ = ψ 1. The above system becomes (ũ) t + 2 ψ(ũ) z = ν(ũ) zz 2ṽũ, (11) (ṽ) t + 2 ψ(ṽ) z = ν(ṽ) zz + (ũ) 2 (ṽ) 2 + c(t), (12) where ṽ = ( ψ) z, ṽ z = ω, and c(t) is an integration constant to enforce the mean of ṽ equal to zero.
13 The 1D model is exact! A surprising result is that the above 1D model is exact. Theorem 2. Let u 1, ψ 1 and ω 1 be the solution of the 1D model (8)-(10) and define u θ (r, z, t) = ru 1 (z, t), ω θ (r, z, t) = rω 1 (z, t), ψ θ (r, z, t) = rψ 1 (z, t). Then (u θ (r, z, t), ω θ (r, z, t), ψ θ (r, z, t)) is an exact solution of the 3D Navier-Stokes equations. Theorem 2 tells us that the 1D model (8)-(10) preserves some essential nonlinear structure of the 3D axisymmetric Navier-Stokes equations.
14 Energy method does not work for the 1D model! A standard energy estimate for the 1D model would give we get 1 d 2 dt 1 d 2 dt ũ 2 dz = 3 ṽ 2 dz = (ũ) 2 ṽdz ν ũ 2 ṽdz ũ 2 z dz, (ṽ) 3 dz ν 1 0 ṽ 2 z dz. One can obtain essentially the same result for the corresponding reaction-diffusion model by dropping convection and c(t): (ũ) t = ν(ũ) zz 2ṽũ, (ṽ) t = ν(ṽ) zz + (ũ) 2 (ṽ) 2, (13) which admits finite time blowup solutions. It is not clear how to control the nonlinear vortex stretching like terms by the diffusion terms, unless we assume T 0 ṽ L dt <, t T.
15 Global Well-Posedness of the full 1D Model Theorem 3. Assume that ũ(z, 0) and ṽ(z, 0) are in C m [0, 1] with m 1 and periodic with period 1. Then the solution (ũ, ṽ) of the 1D model will be in C m [0, 1] for all times and for ν 0. Proof. The key is to obtain a priori pointwise estimate for the Lyapunov function ũ 2 z + ṽ 2 z. Differentiating the ũ and ṽ-equations w.r.t z, we get (ũ z ) t + 2 ψ(ũ z ) z 2ṽũ z = 2ṽũ z 2ũṽ z + ν(ũ z ) zz, (ṽ z ) t + 2 ψ(ṽ z ) z 2ṽṽ z = 2ũũ z 2ṽṽ z + ν(ṽ z ) zz. Note that the convection term contributes to stability by cancelling one of the nonlinear terms on the right hand side. This gives (ũ z ) t + 2 ψ(ũ z ) z = 2ũṽ z + ν(ũ z ) zz, (14) (ṽ z ) t + 2 ψ(ṽ z ) z = 2ũũ z + ν(ṽ z ) zz. (15)
16 Multiplying (14) by 2ũ z and (15) by 2ṽ z, we have (ũz 2 ) t + 2 ψ(ũ z 2 ) z = 4ũũ z ṽ z + 2νũ z (ũ z ) zz, (16) (ṽz 2 ) t + 2 ψ(ṽ z 2 ) z = 4ũũ z ṽ z + 2νṽ z (ṽ z ) zz. (17) Now, we add (16) to (17). Surprisingly, the nonlinear vortex stretching-like terms cancel each other exactly. We get (ũ2 z + ṽz 2 ) + 2 ψ ( ũ 2 t z + ṽz 2 ) = 2ν (ũ z z(ũ z ) zz + ṽ z (ṽ z ) zz ). Moreover we can rewrite the diffusion term in the following form: (ũ2 z + ṽz 2 ) + 2 ψ ( ũ 2 t z + ṽz 2 ) = ν ( ũ 2 z z + ṽz 2 ) 2ν [ (ũ zz zz ) 2 + (ṽ zz ) 2]. Thus, (ũz 2 + ṽz 2 ) satisfies a maximum principle for all ν 0: ũz 2 + ṽz 2 L (ũ 0 ) 2 z + (ṽ 0 ) 2 z L.
17 Construction of a family of globally smooth solutions Theorem 4. Let φ(r) be a smooth cut-off function and u 1, ω 1 and ψ 1 be the solution of the 1D model. Define u θ (r, z, t) = ru 1 (z, t)φ(r) + ũ(r, z, t), ω θ (r, z, t) = rω 1 (z, t)φ(r) + ω(r, z, t), ψ θ (r, z, t) = rψ 1 (z, t)φ(r) + ψ(r, z, t). Then there exists a family of globally smooth functioons ũ, ω and ψ such that u θ, ω θ and ψ θ are globally smooth solutions of the 3D Navier-Stokes equations with finite energy.
18 A New 3D Model for NSE, [Hou-Lei, CPAM, 09] Recall the reformulated 3D Navier-Stokes equations: t u 1 + u r r u 1 + u z z u 1 = ν ( r r ) r + z 2 u1 + 2u 1 ψ 1z, t ω 1 + u r r ω 1 + u z z ω 1 = ν ( r r ) r + z 2 ω1 + ( ) u1 2 z, ( r r ) r + z 2 ψ1 = ω 1, (18) where u r = rψ 1z, u z = 2ψ 1 + rψ 1r. Our 3D model is derived by simply dropping the convective term from (18): t u 1 = ν ( r r ) r + z 2 u1 + 2u 1 ψ 1z, t ω 1 = ν ( r r ) r + z 2 ω1 + (u1 2) z, ( (19) r r ) r + z 2 ψ1 = ω 1. Note that (19) is already a closed system, and u 1 = u θ /r characterizes the axial vorticity near r = 0.
19 Properties of the 3D Model [Hou-Lei, CPAM, 09] This 3D model shares many important properties with the axisymmetric Navier-Stokes equations. First of all, one can define an incompressible velocity field in the model equations (19). u(t, x) = u r (t, r, z)e r + u θ (t, r, z)e θ + u z (t, r, z)e z, (20) u θ = ru 1, u r = rψ 1z, u z = 2ψ 1 + rψ 1r, (21) where x = (x 1, x 2, z), r = x1 2 + x 2 2. It is easy to check that u = r u r + z u z + ur r = 0, (22) which is the same as the Navier-Stokes equations.
20 Properties of the 3D Model continued Our model enjoys the following properties ([Hou-Lei, CPAM-09]): Theorem 5. Energy identity. The strong solution of (19) satisfies 1 d ( u Dψ 1 2) ( Du1 r 3 drdz + ν D 2 ψ 1 2) r 3 drdz = 0, 2 dt which has been proved to be equivalent to that of the Navier-Stokes equations. Here D is the first order derivative operator defined in R 5. Theorem 6. A non-blowup criterion of Beale-Kato-Majda type. A smooth solution (u 1, ω 1, ψ 1 ) of the model (19) for 0 t < T blows up at time t = T if and only if T where u is defined in (20)-(21). 0 u BMO(R 3 )dt =,
21 Properties of the 3D Model continued Theorem 7. A non-blowup criterion of Prodi-Serrin type. A weak solution (u 1, ω 1, ψ 1 ) of the model (19) is smooth on [0, T ] R 3 provided that u θ L q t Lp x ([0,T ] R 3 ) < (23) for some p, q satisfying 3 p + 2 q 1 with 3 < p and 2 q <. Theorem 8. An analog of Caffarelli-Kohn-Nirenberg partial regularity result [Hou-Lei, CMP-09]. For any suitable weak solution of the 3D model equations (19) on an open set in space-time, the one-dimensional Hausdorff measure of the associated singular set is zero.
22 Potential singularity formation of the 3D model It is interesting to study the invicsid model. t u 1 = 2u 1 ψ 1z, t ω 1 = (u1 2 ) z, ( r r ) r + z 2 ψ1 = ω 1. (24) If we let v = log(u1 2 ), then we can further reduce the 3D model to the following nonlocal nonlinear wave equation: v tt = 4 ( ( ) 1 e v ) zz, (25) where = ( 2 r + 3 r r + 2 z ), and e v r 3 drdz C 0. Note that ( ) 1 is a positive smoothing operator. This is a nonlinear nonlocal wave equation along the z-direction.
23 Numerical evidence for a potential finite time singularity Innitial condition we consider in our numerical computations is given by u 1 (z, r, 0) = (1 + sin(4πz))(r 2 1) 20 (r 2 1.2) 30, 0 r 1, ψ 1 (z, r, 0) = 0, ω 1 (z, r, 0) = 0. A second order finite difference discretization is used in space, and the classical fourth order Runge-Kutta method is used to discretize in time. We use the following coordinate transformation along the r-direction to achieve the adaptivity: r = f (α) α 0.9 sin(πα)/π. We use an effective resolution up to for the 3D problem.
24 A 3D view of u 1 at t = 0.02.
25 A 3D view of u 1 at t =
26 Asymptotic blowup rate: u 1 T = and C = C (T t), with 10 x u (blue) vs C/(T t) (red) in time, T= , C= , N=4096, ν=0.001 u, N= Fitted solution u Time
27 Local alignment of u 1 and ψ 1z at t = Recall (u 1 ) t = 2u 1 ψ 1z + ν u 1, (ω 1 ) t = (u 2 1 ) z + ν ω u 1 vs (ψ 1 ) z at r=0, t=0.02, N z =4096, N r =400, t= , ν=0.001 u (ψ 1 ) z z
28 Stabilizing effect of convection. To study the stabilizing effect of convection, we add the convection term back to the 3D model and solve the Navier-Stokes equations using the solution of the 3D model at t = 0.02 as the initial condition Maximum u 1 in time, N z =2048, N r =1024, t= , ν=0.001, full NSE u Time
29 Contours of initial data for u 1. Contour of u 1 at t=0.02, N z =4096, N r =400, t= , ν=0.001, 3D model r z
30 Contours of u 1 at t = 0.021, solution of full NSE. Contour of u 1 at t=0.021, N z =2048, N r =1024, t= , ν=0.001, full NSE r z
31 Contours of u 1 at t = , solution of full NSE. Contour of u 1 at t=0.0235, N z =2048, N r =1024, t= , ν=0.001, full NSE r z
32 Recent theortical progress for the 3D model Theorem 9. [Hou-Shi-Wang, 09]. Consider the 3D inviscid model { u t = 2uψ z, ω t = (u 2 ) z, ( ) x 2 + y 2 + z 2 (26) ψ = ω, 0 x, y, z 1, with boundary condition ψ = 0 at x = 0, 1, y = 0, 1, z = 1, and (α ψ z + ψ) z=0 = 0 for some 0 < α < 1. If the initial conditions, u 0 and ψ 0, are smooth, satisfying u 0 = 0 at z = 0, 1, and log(u 0 )φ(x, y, z)dxdydz 0, (ψ 0 ) z φ(x, y, z)dxdydz > 0, [0,1] 3 [0,1] 3 where φ(x, y, z) = sin(x) sin(y) cosh(α(1 z)), then the 3D inviscid model must develop a finite time singularity. Moreover, if the ω-equation is viscous and ω satisfies the same boundary condition as ψ, then the 3D model with partial viscosity must develop a finite time singularity.
33 Global regularity for a class of initial-boundary data Let v = ψ z, we can rewrite the 3D model as follows: { u t = 2uv v t = (u 2 ) zz, (x, z) Ω = [0, δ] 3. (27) The initial and boundary conditions are given as follows: v Ω = 4, v(x, y, z, 0) = v 0 (x, y, z), u(x, y, z, 0) = u 0 (x, y, z). (28) Theorem 10. Assume u 2 0, v 0 H m (Ω) for m 2 and v 0 4 for x Ω, then the solution of the 3D model remains smooth for all times as long as the following holds max (4C m, 1) ( v 0 H m + C m u 2 0 H m) δ < 1, (29) where C m is a Sobolev interpolation constant.
34 Recent progress on potential blowup of full NSE, joint with Dr. Guo Luo Surprisingly, a recent study of the full 3D NSE and Euler equations reveals that for certain axi-symmetric initial data with swirl, the combined effect of convection and vortex stretching may have the potential to drive the NSE/Euler to a finite time blow-up. Our preliminary numerical study shows that the solution of NSE could grow rapidly in time with a strong, apparently quadratic nonlinearity, between u 1 and ψ 1z, u 1t + u r (u 1 ) r + u z (u 1 ) z = 2u 1 ψ 1z + ν u 1, which could potentially lead to a finite time blow-up. The dynamic evolution of u 1 and w 1 on the (r, z)-plane can be visualized in the following contour plots.
35 Contour plots of u 1 at t = 0.02 Figure: Contour plot of u 1 at t=0.02.
36 Contour plots of u 1 at t = 0.03 Figure: Contour plot of u 1 at t=0.03.
37 Contour plots of u 1 at t = Figure: Contour plot of u 1 at t=0.036.
38 Contour plots of u 1 at t = Figure: Contour plot of u 1 at t=0.043.
39 Contour plots of w1 at t = 0.02 Figure: Contour plot of w1 at t=0.02. T. Y. Hou, Applied Mathematics, Caltech On Global Regularity of 3D NSE
40 Contour plots of w1 at t = 0.03 Figure: Contour plot of w1 at t=0.03. T. Y. Hou, Applied Mathematics, Caltech On Global Regularity of 3D NSE
41 Contour plots of w 1 at t = Figure: Contour plot of w 1 at t=0.036.
42 Contour plots of w 1 at t = Figure: Contour plot of w 1 at t=0.043.
43 Velocity field and streamlines at t = The solution can also be visualized in 3D. Figure: Velocity field (u r, u θ, u z) and streamlines at t =
44 Vorticity iso-surface at t = Figure: Iso-surface of vorticity ω (with ω 1 max ω ) at t = Note the nontrivial geometry of the iso-surface.
45 Vortex lines at t = Figure: Vortex lines at t = Note the high curvature of the vortex lines near the maximum vorticity region.
46 Log log plot of the maximum of u 1 and w 1 The following figure shows the rapid, super-double-exponential growth of the maximum of u 1 and w 1 in time. 1.9 log(log(max u1 )) as a function of t log(log(max u1 )) t Figure: Log log plot of maximum u 1 as a function of time.
47 Log log plot of the maximum of u 1 and w 1 (Cont d) 2.6 log(log(max w1 )) as a function of t log(log(max w1 )) t Figure: Log log plot of maximum w 1 as a function of time. To the best of our knowledge, this type of strong nonlinearity has not been observed before.
48 The quadratic nonlinearity, u 1t + u u 1 = 2u 1 ψ 1z We observe that the nonlinear u 1 -stretching term, 2ψ 1z, is comparable with u 1 at the maximum of u 1. This quadratic nonlinearity is the driving force behind the rapid growth of u 1 and w Nonlinear stretching of u1 a function of t 2 phi1 z max u t Figure: Ratio of the u 1-stretching term, 2ψ 1, to u 1 at the maximum of u 1.
49 Understanding the strong nonlinearity To understand how this quadratic nonlinearity arises, we plot the cross sections of the solution along the r- and z-coordinate lines that pass through the maximum of u T=0.043, r cross section through max u1, Nr=2048 u1 phi1 z w1 (rescaled) phi1 (rescaled) r Figure: Cross section of the solution at t = along the r-line that passes through the maximum of u 1.
50 Understanding the strong nonlinearity (Cont d) T=0.043, z cross section through max u1, Nz=8192 u1 phi1 z w1 (rescaled) phi1 (rescaled) z Figure: Cross section of the solution at t = along the z-line that passes through the maximum of u 1.
51 Understanding the strong nonlinearity (Cont d) We observe from the above figures how u 1 and ψ 1z are closely aligned, both in r- and in z-directions. More interestingly, since the radial velocity u r = rψ 1z is negative near the maximum of u 1. Recall that the u 1 -equation is as follows: u 1t + u r (u 1 ) r + u z (u 1 ) z = 2u 1 ψ 1z + ν u 1, Thus, the r-convection term u r (u 1 ) r creates a compressive like force along the radial direction that introduces a sharp front for u 1 and pushes the peak of u 1 towards the axis r = 0. Recall that CKN s partial regulairty implies that the solution of the axisymmetric NSE can only blow up at r = 0, if there is a blowup at all.
52 Approaching the axis r = 0 To see how fast the solution is approaching r = 0, we plot the location (r 1, z 1 ) of the maximum of u 1 as a function of time. 3.5 x 10 3 r1 2 as a function of t, where r1 is the r coordinate of maxloc u r t Figure: Plot of r 2 1 as a function of time, where (r 1, z 1) is the point at which u 1 attains its maximum.
53 Approaching the axis r = 0 (Cont d) 3.5 x 10 3 z1 2 as a function of t, where z1 is the z coordinate of maxloc u z t Figure: Plot of z 2 1 as a function of time, where (r 1, z 1) is the point at which u 1 attains its maximum.
54 Asymptotic blowup rate Assuming that the solution blows up at a finite time T with a blowup rate O(T t) α, we can obtain a crude estimate of T and α using a line fitting procedure. 9 x /max u1 as a function of t /max u t Figure: Plot of 1/ max u 1 as a function of time.
55 Asymptotic blowup rate (Cont d) 1/max w1 1/2 as a function of t /max w1 1/ t Figure: Plot of 1/ max w 1 1/2 as a function of time.
56 Asymptotic blowup rate (Cont d) 0.4 1/max phi1 2 as a function of t /phi t Figure: Plot of 1/ max ψ 1 2 as a function of time. The estimated singularity time is T
57 A few remarks on the potentially singular solution This preliminary study shows that the solution of the axisymmetric NSE may develop a finite time singularity, with the blowup rate u 1 = O(T t) 1, w 1 = O(T t) 2, ψ 1 = O(T t) 1/2, if the nonlinear alignment between u 1 and ψ 1,z is maintained for a sufficiently long period of time. We are carrying out more careful numerical studies to confirm whether a finite-time blowup can occur for NSE (and also for Euler). An important question is whether the profile near the potentially singular region remains dynamically stable. If this is the case, we can not rule out the possibility of a finite time blow-up.
58 Concluding Remarks Our study shows that convection could play a dual role in the dynamic stability or finite time singularity formation of 3D incompressible flow. One the one hand, we construct an exact 1D model, in which we show that convection plays an essential role to deplete the nonlinear vortex stretching, and the 1D model has global regularity. We also construct a new 3D model by neglecting the convection term. This 3D model shares almost all properties of the Navier-Stokes equations, but could develop finite time singularities. On the other hand, our recent study suggests that convection may not always stablize the nonlinear vortex stretching. For initial data with certain symmetry, the combined effect of convection and vortex stretching has the potential to lead to a finite time singularity. It remains to understand whether the nearly singular behavior we observed numerically is dynamically stable. If so, it may suggest a possible way to construct a singular solution analytically.
59 References T. Y. Hou and R. Li, Dynamic Depletion of Vortex Stretching and Non-Blowup of the 3-D Incompressible Euler Equations, J. Nonlinear Science, 16 (2006), T. Y. Hou and C. Li, Dynamic Stability of the 3D Axisymmetric Navier-Stokes Equations with Swirl, CPAM, 61 (2008), T. Y. Hou and Z. Lei, On the Stabilizing Effect of Convection in 3D Incompressible Flows, CPAM, 62, pp , T. Y. Hou, Z. Shi, and S. Wang, On Singularity Formation of a 3D Model for Incompressible Navier-Stokes Equations, 2009, arxiv: v1 [math.ap]. T. Y. Hou, Blow-up or no blowup? A unified computational and analytical approach to 3D incompressible Euler and Navier-Stokes equations, Acta Numerica, pp , T. Y. Hou and G. Luo, A class of potentially singular solutions for 3D incompressible Euler and Navier-Stokes equations, in preparation, 2011.
60 Happy 60th Birthday, Blake
61 Growth rate in velocity Here we present the loglog plot of maximum velocity in the r, θ and z coordinates log(log(max u i )) as a function of t log(log(max u θ )) log(log(max u r )) log(log(max u z )) t Figure: Plot of max u as a function of time.
62 Growth rate in vorticity Here we present the loglog plot of maximum vorticity in the r, θ and z coordinates log(log(max w i )) as a function of t log(log(max w θ )) log(log(max w r )) log(log(max w z )) t Figure: Plot of max ω as a function of time.
The Role of Convection and Nearly Singular Behavior of the 3D Navier-Stokes Equations
The Role of Convection and Nearly Singular Behavior of the 3D Navier-Stokes Equations Thomas Y. Hou Applied and Comput. Mathematics, Caltech PDE Conference in honor of Blake Temple, University of Michigan
More informationBlow-up or No Blow-up? Fluid Dynamic Perspective of the Clay Millennium Problem
Blow-up or No Blow-up? Fluid Dynamic Perspective of the Clay Millennium Problem Thomas Y. Hou Applied and Comput. Mathematics, Caltech Research was supported by NSF Theodore Y. Wu Lecture in Aerospace,
More informationThe interplay between theory and computation in the study of 3D Euler equations
The interplay between theory and computation in the study of 3D Euler equations Thomas Y. Hou Applied and Computational Mathematics California Institute of Technology Joint work with Guo Luo and Pengfei
More informationPotentially Singular Solutions of the 3D Axisymmetric Euler Equations
Potentially Singular Solutions of the 3D Axisymmetric Euler Equations Thomas Y. Hou Applied and Computational Mathematics California Institute of Technology Joint work with Guo Luo IPAM Workshop on Mathematical
More informationIncompressible Navier-Stokes Equations in R 3
Incompressible Navier-Stokes Equations in R 3 Zhen Lei ( ) School of Mathematical Sciences Fudan University Incompressible Navier-Stokes Equations in R 3 p. 1/5 Contents Fundamental Work by Jean Leray
More informationBlow-up or no blow-up? A unified computational and analytic approach to 3D incompressible Euler and Navier Stokes equations
Acta Numerica (29), pp. 277 346 c Cambridge University Press, 29 doi: 1.117/S962492964218 Printed in the United Kingdom Blow-up or no blow-up? A unified computational and analytic approach to 3D incompressible
More informationThe enigma of the equations of fluid motion: A survey of existence and regularity results
The enigma of the equations of fluid motion: A survey of existence and regularity results RTG summer school: Analysis, PDEs and Mathematical Physics The University of Texas at Austin Lecture 1 1 The review
More informationOn Finite Time Singularity and Global Regularity of an Axisymmetric Model for the 3D Euler Equations
On Finite Time Singularity and Global Regularity of an Axisymmetric Model for the 3D Euler Equations arxiv:23.298v [math.ap] 4 Mar 22 Thomas Y. Hou Zhen Lei Shu Wang Chen Zou March 5, 22 Abstract We investigate
More informationThe Dual Role of Convection in 3D. Navier-Stokes equations.
1 The Dual Role of Convection in 3D Navier-Stokes Equations Thomas Y. Hou a, Zuoqiang Shi b, Shu Wang c Abstract We investigate the dual role of convection on the large time behavior of the 3D incompressible
More informationApplying Moser s Iteration to the 3D Axially Symmetric Navier Stokes Equations (ASNSE)
Applying Moser s Iteration to the 3D Axially Symmetric Navier Stokes Equations (ASNSE) Advisor: Qi Zhang Department of Mathematics University of California, Riverside November 4, 2012 / Graduate Student
More informationof some mathematical models in physics
Regularity, singularity and well-posedness of some mathematical models in physics Saleh Tanveer (The Ohio State University) Background Mathematical models involves simplifying assumptions where "small"
More informationGlobal regularity of a modified Navier-Stokes equation
Global regularity of a modified Navier-Stokes equation Tobias Grafke, Rainer Grauer and Thomas C. Sideris Institut für Theoretische Physik I, Ruhr-Universität Bochum, Germany Department of Mathematics,
More informationOn Global Well-Posedness of the Lagrangian Averaged Euler Equations
On Global Well-Posedness of the Lagrangian Averaged Euler Equations Thomas Y. Hou Congming Li Abstract We study the global well-posedness of the Lagrangian averaged Euler equations in three dimensions.
More informationTopics in Fluid Dynamics: Classical physics and recent mathematics
Topics in Fluid Dynamics: Classical physics and recent mathematics Toan T. Nguyen 1,2 Penn State University Graduate Student Seminar @ PSU Jan 18th, 2018 1 Homepage: http://math.psu.edu/nguyen 2 Math blog:
More informationarxiv: v2 [math.ap] 6 Sep 2007
ON THE REGULARITY OF WEAK SOLUTIONS OF THE 3D NAVIER-STOKES EQUATIONS IN B 1, arxiv:0708.3067v2 [math.ap] 6 Sep 2007 A. CHESKIDOV AND R. SHVYDKOY ABSTRACT. We show that if a Leray-Hopf solution u to the
More informationVorticity and Dynamics
Vorticity and Dynamics In Navier-Stokes equation Nonlinear term ω u the Lamb vector is related to the nonlinear term u 2 (u ) u = + ω u 2 Sort of Coriolis force in a rotation frame Viscous term ν u = ν
More informationarxiv: v2 [physics.flu-dyn] 8 Dec 2013
POTENTIALLY SINGULAR SOLUTIONS OF THE 3D INCOMPRESSIBLE EULER EQUATIONS GUO LUO AND THOMAS Y. HOU arxiv:131.497v2 [physics.flu-dyn] 8 Dec 213 December 1, 213 Abstract Whether the 3D incompressible Euler
More informationThe 3D Euler and 2D surface quasi-geostrophic equations
The 3D Euler and 2D surface quasi-geostrophic equations organized by Peter Constantin, Diego Cordoba, and Jiahong Wu Workshop Summary This workshop focused on the fundamental problem of whether classical
More informationOn Smoothness of Suitable Weak Solutions to the Navier-Stokes Equations
On Smoothness of Suitable Weak Solutions to the Navier-Stokes Equations G. Seregin, V. Šverák Dedicated to Vsevolod Alexeevich Solonnikov Abstract We prove two sufficient conditions for local regularity
More informationA LOWER BOUND ON BLOWUP RATES FOR THE 3D INCOMPRESSIBLE EULER EQUATION AND A SINGLE EXPONENTIAL BEALE-KATO-MAJDA ESTIMATE. 1.
A LOWER BOUND ON BLOWUP RATES FOR THE 3D INCOMPRESSIBLE EULER EQUATION AND A SINGLE EXPONENTIAL BEALE-KATO-MAJDA ESTIMATE THOMAS CHEN AND NATAŠA PAVLOVIĆ Abstract. We prove a Beale-Kato-Majda criterion
More informationDIRECTION OF VORTICITY AND A REFINED BLOW-UP CRITERION FOR THE NAVIER-STOKES EQUATIONS WITH FRACTIONAL LAPLACIAN
DIRECTION OF VORTICITY AND A REFINED BLOW-UP CRITERION FOR THE NAVIER-STOKES EQUATIONS WITH FRACTIONAL LAPLACIAN KENGO NAKAI Abstract. We give a refined blow-up criterion for solutions of the D Navier-
More informationc 2014 Society for Industrial and Applied Mathematics
MULTISCALE MODEL. SIMUL. Vol. 12, No. 4, pp. 1722 1776 c 214 Society for Industrial and Applied Mathematics TOWARD THE FINITE-TIME BLOWUP OF THE 3D AXISYMMETRIC EULER EQUATIONS: A NUMERICAL INVESTIGATION
More informationRecent progress on the vanishing viscosity limit in the presence of a boundary
Recent progress on the vanishing viscosity limit in the presence of a boundary JIM KELLIHER 1 1 University of California Riverside The 7th Pacific RIM Conference on Mathematics Seoul, South Korea 1 July
More informationLevel Set Dynamics and the Non-blowup of the 2D Quasi-geostrophic Equation
Level Set Dynamics and the Non-blowup of the 2D Quasi-geostrophic Equation J. Deng, T. Y. Hou, R. Li, X. Yu May 31, 2006 Abstract In this article we apply the technique proposed in Deng-Hou-Yu [7] to study
More informationu t + u u = p (1.1) u = 0 (1.2)
METHODS AND APPLICATIONS OF ANALYSIS. c 2005 International Press Vol. 12, No. 4, pp. 427 440, December 2005 004 A LEVEL SET FORMULATION FOR THE 3D INCOMPRESSIBLE EULER EQUATIONS JIAN DENG, THOMAS Y. HOU,
More informationGlobal well-posedness of the primitive equations of oceanic and atmospheric dynamics
Global well-posedness of the primitive equations of oceanic and atmospheric dynamics Jinkai Li Department of Mathematics The Chinese University of Hong Kong Dynamics of Small Scales in Fluids ICERM, Feb
More informationOn the well-posedness of the Prandtl boundary layer equation
On the well-posedness of the Prandtl boundary layer equation Vlad Vicol Department of Mathematics, The University of Chicago Incompressible Fluids, Turbulence and Mixing In honor of Peter Constantin s
More informationarxiv: v1 [math.ap] 16 May 2007
arxiv:0705.446v1 [math.ap] 16 May 007 Regularity criterion for 3D Navier-Stokes equations in terms of the direction of the velocity Alexis Vasseur October 3, 018 Abstract In this short note, we give a
More informationSpatial Profiles in the Singular Solutions of the 3D Euler Equations and Simplified Models
Spatial Profiles in the Singular Solutions of the 3D Euler Equations and Simplified Models Thesis by Pengfei Liu In Partial Fulfillment of the Requirements for the degree of Doctor of Philosophy CALIFORNIA
More informationRemarks on the blow-up criterion of the 3D Euler equations
Remarks on the blow-up criterion of the 3D Euler equations Dongho Chae Department of Mathematics Sungkyunkwan University Suwon 44-746, Korea e-mail : chae@skku.edu Abstract In this note we prove that the
More informationNonuniqueness of weak solutions to the Navier-Stokes equation
Nonuniqueness of weak solutions to the Navier-Stokes equation Tristan Buckmaster (joint work with Vlad Vicol) Princeton University November 29, 2017 Tristan Buckmaster (Princeton University) Nonuniqueness
More informationCitation 数理解析研究所講究録 (2016), 1995:
Swirling flow of the axi-symmetric Titlenear a saddle point and no-slip bou New Developments for Elucidating In Author(s) 野津, 裕史 ; 米田, 剛 Citation 数理解析研究所講究録 (2016), 1995: 9-13 Issue Date 2016-04 URL http://hdl.handle.net/2433/224715
More informationThe Euler Equations and Non-Local Conservative Riccati Equations
The Euler Equations and Non-Local Conservative Riccati Equations Peter Constantin Department of Mathematics The University of Chicago November 8, 999 The purpose of this brief note is to present an infinite
More informationTOWARD THE FINITE-TIME BLOWUP OF THE 3D AXISYMMETRIC EULER EQUATIONS: A NUMERICAL INVESTIGATION
TOWARD THE FINITE-TIME BLOWUP OF THE 3D AXISYMMETRIC EULER EQUATIONS: A NUMERICAL INVESTIGATION GUO LUO AND THOMAS Y. HOU Abstract. Whether the 3D incompressible Euler equations can develop a singularity
More informationAnisotropic partial regularity criteria for the Navier-Stokes equations
Anisotropic partial regularity criteria for the Navier-Stokes equations Walter Rusin Department of Mathematics Mathflows 205 Porquerolles September 7, 205 The question of regularity of the weak solutions
More informationWeek 6 Notes, Math 865, Tanveer
Week 6 Notes, Math 865, Tanveer. Energy Methods for Euler and Navier-Stokes Equation We will consider this week basic energy estimates. These are estimates on the L 2 spatial norms of the solution u(x,
More informationFINITE TIME BLOW-UP FOR A DYADIC MODEL OF THE EULER EQUATIONS
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 357, Number 2, Pages 695 708 S 0002-9947(04)03532-9 Article electronically published on March 12, 2004 FINITE TIME BLOW-UP FOR A DYADIC MODEL OF
More informationBlow up of solutions for a 1D transport equation with nonlocal velocity and supercritical dissipation
Blow up of solutions for a 1D transport equation with nonlocal velocity and supercritical dissipation Dong Li a,1 a School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, NJ 854,
More informationON THE REGULARITY OF WEAK SOLUTIONS OF THE 3D NAVIER-STOKES EQUATIONS IN B 1
ON THE REGULARITY OF WEAK SOLUTIONS OF THE 3D NAVIER-STOKES EQUATIONS IN B 1, A. CHESKIDOV AND R. SHVYDKOY ABSTRACT. We show that if a Leray-Hopf solution u to the 3D Navier- Stokes equation belongs to
More informationBorel Summability in PDE initial value problems
Borel Summability in PDE initial value problems Saleh Tanveer (Ohio State University) Collaborator Ovidiu Costin & Guo Luo Research supported in part by Institute for Math Sciences (IC), EPSRC & NSF. Main
More informationFundamentals of Fluid Dynamics: Ideal Flow Theory & Basic Aerodynamics
Fundamentals of Fluid Dynamics: Ideal Flow Theory & Basic Aerodynamics Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI (after: D.J. ACHESON s Elementary Fluid Dynamics ) bluebox.ippt.pan.pl/
More informationNonlinear Analysis. A regularity criterion for the 3D magneto-micropolar fluid equations in Triebel Lizorkin spaces
Nonlinear Analysis 74 (11) 5 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na A regularity criterion for the 3D magneto-micropolar fluid equations
More informationF11AE1 1. C = ρν r r. r u z r
F11AE1 1 Question 1 20 Marks) Consider an infinite horizontal pipe with circular cross-section of radius a, whose centre line is aligned along the z-axis; see Figure 1. Assume no-slip boundary conditions
More informationV (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t)
IV. DIFFERENTIAL RELATIONS FOR A FLUID PARTICLE This chapter presents the development and application of the basic differential equations of fluid motion. Simplifications in the general equations and common
More informationAvailable online at J. Math. Comput. Sci. 4 (2014), No. 3, ISSN:
Available online at http://scik.org J. Math. Comput. Sci. 4 (2014), No. 3, 587-593 ISSN: 1927-5307 A SMALLNESS REGULARITY CRITERION FOR THE 3D NAVIER-STOKES EQUATIONS IN THE LARGEST CLASS ZUJIN ZHANG School
More informationGlobal well-posedness and decay for the viscous surface wave problem without surface tension
Global well-posedness and decay for the viscous surface wave problem without surface tension Ian Tice (joint work with Yan Guo) Université Paris-Est Créteil Laboratoire d Analyse et de Mathématiques Appliquées
More informationVortex stretching in incompressible and compressible fluids
Vortex stretching in incompressible and compressible fluids Esteban G. Tabak, Fluid Dynamics II, Spring 00 1 Introduction The primitive form of the incompressible Euler equations is given by ( ) du P =
More informationUNIVERSITY OF EAST ANGLIA
UNIVERSITY OF EAST ANGLIA School of Mathematics May/June UG Examination 2007 2008 FLUIDS DYNAMICS WITH ADVANCED TOPICS Time allowed: 3 hours Attempt question ONE and FOUR other questions. Candidates must
More informationarxiv: v1 [math.ap] 3 Nov 2017
BLOW UP OF A HYPERBOLIC SQG MODEL HANG YANG arxiv:7.255v [math.ap] 3 Nov 27 Abstract. This paper studies of a variation of the hyperbolic blow up scenario suggested by Hou and Luo s recent numerical simulation
More information3 Generation and diffusion of vorticity
Version date: March 22, 21 1 3 Generation and diffusion of vorticity 3.1 The vorticity equation We start from Navier Stokes: u t + u u = 1 ρ p + ν 2 u 1) where we have not included a term describing a
More information0 = p. 2 x + 2 w. z +ν w
Solution (Elliptical pipe flow (a Using the Navier Stokes equations in three dimensional cartesian coordinates, given that u =, v = and w = w(x,y only, and assuming no body force, we are left with = p
More informationOSGOOD TYPE REGULARITY CRITERION FOR THE 3D NEWTON-BOUSSINESQ EQUATION
Electronic Journal of Differential Equations, Vol. 013 (013), No. 3, pp. 1 6. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu OSGOOD TYPE REGULARITY
More informationJournal of Differential Equations
J. Differential Equations 48 (1) 6 74 Contents lists available at ScienceDirect Journal of Differential Equations www.elsevier.com/locate/jde Two regularity criteria for the D MHD equations Chongsheng
More informationA regularity criterion for the 3D NSE in a local version of the space of functions of bounded mean oscillations
Ann. I. H. Poincaré AN 27 (2010) 773 778 www.elsevier.com/locate/anihpc A regularity criterion for the 3D NSE in a local version of the space of functions of bounded mean oscillations Zoran Grujić a,,
More informationBoundary conditions and estimates for the linearized Navier-Stokes equations on staggered grids
Boundary conditions and estimates for the linearized Navier-Stokes equations on staggered grids Wendy Kress and Jonas Nilsson Department of Scientific Computing Uppsala University Box S-75 4 Uppsala Sweden
More informationThis note presents an infinite-dimensional family of exact solutions of the incompressible three-dimensional Euler equations
IMRN International Mathematics Research Notices 2000, No. 9 The Euler Equations and Nonlocal Conservative Riccati Equations Peter Constantin This note presents an infinite-dimensional family of exact solutions
More informationPartial regularity for fully nonlinear PDE
Partial regularity for fully nonlinear PDE Luis Silvestre University of Chicago Joint work with Scott Armstrong and Charles Smart Outline Introduction Intro Review of fully nonlinear elliptic PDE Our result
More informationThe Navier Stokes Equations for Incompressible Flows: Solution Properties at Potential Blow Up Times
The Navier Stokes Equations for Incompressible Flows: Solution Properties at Potential Blow Up Times Jens Lorenz Department of Mathematics and Statistics, UNM, Albuquerque, NM 873 Paulo Zingano Dept. De
More informationOn the Boundary Partial Regularity for the incompressible Navier-Stokes Equations
On the for the incompressible Navier-Stokes Equations Jitao Liu Federal University Rio de Janeiro Joint work with Wendong Wang and Zhouping Xin Rio, Brazil, May 30 2014 Outline Introduction 1 Introduction
More informationA New Regularity Criterion for the 3D Navier-Stokes Equations via Two Entries of the Velocity Gradient
Acta Appl Math (014) 19:175 181 DOI 10.1007/s10440-013-9834-3 A New Regularity Criterion for the 3D Navier-Stokes Euations via Two Entries of the Velocity Gradient Tensor Zujin Zhang Dingxing Zhong Lin
More informationOn the limiting behaviour of regularizations of the Euler Equations with vortex sheet initial data
On the limiting behaviour of regularizations of the Euler Equations with vortex sheet initial data Monika Nitsche Department of Mathematics and Statistics University of New Mexico Collaborators: Darryl
More informationCHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION
CHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION 7.1 THE NAVIER-STOKES EQUATIONS Under the assumption of a Newtonian stress-rate-of-strain constitutive equation and a linear, thermally conductive medium,
More informationOn Some Variational Optimization Problems in Classical Fluids and Superfluids
On Some Variational Optimization Problems in Classical Fluids and Superfluids Bartosz Protas Department of Mathematics & Statistics McMaster University, Hamilton, Ontario, Canada URL: http://www.math.mcmaster.ca/bprotas
More informationREGULARITY OF GENERALIZED NAVEIR-STOKES EQUATIONS IN TERMS OF DIRECTION OF THE VELOCITY
Electronic Journal of Differential Equations, Vol. 00(00), No. 05, pp. 5. ISSN: 07-669. UR: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu REGUARITY OF GENERAIZED NAVEIR-STOKES
More informationGetting started: CFD notation
PDE of p-th order Getting started: CFD notation f ( u,x, t, u x 1,..., u x n, u, 2 u x 1 x 2,..., p u p ) = 0 scalar unknowns u = u(x, t), x R n, t R, n = 1,2,3 vector unknowns v = v(x, t), v R m, m =
More informationMULTISCALE ANALYSIS IN LAGRANGIAN FORMULATION FOR THE 2-D INCOMPRESSIBLE EULER EQUATION. Thomas Y. Hou. Danping Yang. Hongyu Ran
DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS Volume 13, Number 5, December 2005 pp. 1153 1186 MULTISCALE ANALYSIS IN LAGRANGIAN FORMULATION FOR THE 2-D INCOMPRESSIBLE EULER
More informationarxiv:math/ v1 [math.ap] 6 Mar 2007
arxiv:math/73144v1 [math.ap] 6 Mar 27 ON THE GLOBAL EXISTENCE FOR THE AXISYMMETRIC EULER EQUATIONS HAMMADI ABIDI, TAOUFIK HMIDI, AND SAHBI KERAANI Abstract. This paper deals with the global well-posedness
More informationFrequency Localized Regularity Criteria for the 3D Navier Stokes Equations. Z. Bradshaw & Z. Grujić. Archive for Rational Mechanics and Analysis
Frequency Localized Regularity Criteria for the 3D Navier Stokes Equations Z. Bradshaw & Z. Gruić Archive for Rational Mechanics and Analysis ISSN 0003-9527 Arch Rational Mech Anal DOI 10.1007/s00205-016-1069-9
More informationTRANSIENT FLOW AROUND A VORTEX RING BY A VORTEX METHOD
Proceedings of The Second International Conference on Vortex Methods, September 26-28, 21, Istanbul, Turkey TRANSIENT FLOW AROUND A VORTEX RING BY A VORTEX METHOD Teruhiko Kida* Department of Energy Systems
More informationProperties at potential blow-up times for the incompressible Navier-Stokes equations
Properties at potential blow-up times for the incompressible Navier-Stokes equations Jens Lorenz Department of Mathematics and Statistics University of New Mexico Albuquerque, NM 87131, USA Paulo R. Zingano
More informationProbing Fundamental Bounds in Hydrodynamics Using Variational Optimization Methods
Probing Fundamental Bounds in Hydrodynamics Using Variational Optimization Methods Bartosz Protas and Diego Ayala Department of Mathematics & Statistics McMaster University, Hamilton, Ontario, Canada URL:
More informationRegularity and Decay Estimates of the Navier-Stokes Equations
Regularity and Decay Estimates of the Navier-Stokes Equations Hantaek Bae Ulsan National Institute of Science and Technology (UNIST), Korea Recent Advances in Hydrodynamics, 216.6.9 Joint work with Eitan
More informationRecent results for the 3D Quasi-Geostrophic Equation
Recent results for the 3D Quasi-Geostrophic Equation Alexis F. Vasseur Joint work with Marjolaine Puel (U. of Nice, France) and Matt Novack (UT Austin) The University of Texas at Austin Transport Phenomena
More informationMaximum Rate of Growth of Enstrophy in Solutions of the Fractional Burgers Equation
Maximum Rate of Growth of Enstrophy in Solutions of the Fractional Burgers Equation arxiv:161.9578v [physics.flu-dyn] 15 Jan 18 Dongfang Yun and Bartosz Protas Department of Mathematics and Statistics,
More informationREGULARITY CRITERIA FOR WEAK SOLUTIONS TO 3D INCOMPRESSIBLE MHD EQUATIONS WITH HALL TERM
Electronic Journal of Differential Equations, Vol. 2018 (2018), No. 10, pp. 1 12. ISSN: 1072-6691. UL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EGULAITY CITEIA FO WEAK SOLUTIONS TO D INCOMPESSIBLE
More informationRegularity Theory a Fourth Order PDE with Delta Right Hand Side
Regularity Theory a Fourth Order PDE with Delta Right Hand Side Graham Hobbs Applied PDEs Seminar, 29th October 2013 Contents Problem and Weak Formulation Example - The Biharmonic Problem Regularity Theory
More informationFluid Dynamics Exercises and questions for the course
Fluid Dynamics Exercises and questions for the course January 15, 2014 A two dimensional flow field characterised by the following velocity components in polar coordinates is called a free vortex: u r
More informationPartial regularity for suitable weak solutions to Navier-Stokes equations
Partial regularity for suitable weak solutions to Navier-Stokes equations Yanqing Wang Capital Normal University Joint work with: Quansen Jiu, Gang Wu Contents 1 What is the partial regularity? 2 Review
More informationThe 2D Magnetohydrodynamic Equations with Partial Dissipation. Oklahoma State University
The 2D Magnetohydrodynamic Equations with Partial Dissipation Jiahong Wu Oklahoma State University IPAM Workshop Mathematical Analysis of Turbulence IPAM, UCLA, September 29-October 3, 2014 1 / 112 Outline
More informationVortex stretching and anisotropic diffusion in the 3D NSE
University of Virginia SIAM APDE 2013 ; Lake Buena Vista, December 7 3D Navier-Stokes equations (NSE) describing a flow of 3D incompressible viscous fluid read u t + (u )u = p + ν u, supplemented with
More informationREGULARITY AND EXISTENCE OF GLOBAL SOLUTIONS TO THE ERICKSEN-LESLIE SYSTEM IN R 2
REGULARITY AND EXISTENCE OF GLOBAL SOLUTIONS TO THE ERICKSEN-LESLIE SYSTEM IN R JINRUI HUANG, FANGHUA LIN, AND CHANGYOU WANG Abstract. In this paper, we first establish the regularity theorem for suitable
More informationGLOBAL REGULARITY OF LOGARITHMICALLY SUPERCRITICAL 3-D LAMHD-ALPHA SYSTEM WITH ZERO DIFFUSION
GLOBAL REGULARITY OF LOGARITHMICALLY SUPERCRITICAL 3-D LAMHD-ALPHA SYSTEM WITH ZERO DIFFUSION KAZUO YAMAZAKI Abstract. We study the three-dimensional Lagrangian-averaged magnetohydrodynamicsalpha system
More informationQuasi-two Dimensional Hydrodynamics and Interaction of Vortex Tubes
Quasi-two Dimensional Hydrodynamics and Interaction of Vortex Tubes Vladimir Zakharov 1 Introduction This paper is long overdue. The most of the results presented here were obtained in 1986-87. Just a
More information5.1 Fluid momentum equation Hydrostatics Archimedes theorem The vorticity equation... 42
Chapter 5 Euler s equation Contents 5.1 Fluid momentum equation........................ 39 5. Hydrostatics................................ 40 5.3 Archimedes theorem........................... 41 5.4 The
More informationASYMPTOTIC STRUCTURE FOR SOLUTIONS OF THE NAVIER STOKES EQUATIONS. Tian Ma. Shouhong Wang
DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS Volume 11, Number 1, July 004 pp. 189 04 ASYMPTOTIC STRUCTURE FOR SOLUTIONS OF THE NAVIER STOKES EQUATIONS Tian Ma Department of
More informationRelative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system
Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system Institute of Mathematics, Academy of Sciences of the Czech Republic, Prague joint work
More informationA new regularity criterion for weak solutions to the Navier-Stokes equations
A new regularity criterion for weak solutions to the Navier-Stokes equations Yong Zhou Department of Mathematics, East China Normal University Shanghai 6, CHINA yzhou@math.ecnu.edu.cn Proposed running
More informationMultiscale Computation of Isotropic Homogeneous Turbulent Flow
Multiscale Computation of Isotropic Homogeneous Turbulent Flow Tom Hou, Danping Yang, and Hongyu Ran Abstract. In this article we perform a systematic multi-scale analysis and computation for incompressible
More informationSufficient conditions on Liouville type theorems for the 3D steady Navier-Stokes equations
arxiv:1805.07v1 [math.ap] 6 May 018 Sufficient conditions on Liouville type theorems for the D steady Navier-Stokes euations G. Seregin, W. Wang May 8, 018 Abstract Our aim is to prove Liouville type theorems
More informationOn the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals
On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals Fanghua Lin Changyou Wang Dedicated to Professor Roger Temam on the occasion of his 7th birthday Abstract
More informationEuler Equations of Incompressible Ideal Fluids
Euler Equations of Incompressible Ideal Fluids Claude BARDOS, and Edriss S. TITI February 27, 27 Abstract This article is a survey concerning the state-of-the-art mathematical theory of the Euler equations
More informationPAPER 331 HYDRODYNAMIC STABILITY
MATHEMATICAL TRIPOS Part III Thursday, 6 May, 016 1:30 pm to 4:30 pm PAPER 331 HYDRODYNAMIC STABILITY Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal
More informationMultiscale Computation of Isotropic Homogeneous Turbulent Flow
Multiscale Computation of Isotropic Homogeneous Turbulent Flow Tom Hou, Danping Yang, and Hongyu Ran Abstract. In this article we perform a systematic multi-scale analysis and computation for incompressible
More informationarxiv: v1 [math.ap] 21 Dec 2016
arxiv:1612.07051v1 [math.ap] 21 Dec 2016 On the extension to slip boundary conditions of a Bae and Choe regularity criterion for the Navier-Stokes equations. The half-space case. H. Beirão da Veiga, Department
More informationTOPICS IN MATHEMATICAL AND COMPUTATIONAL FLUID DYNAMICS
TOPICS IN MATHEMATICAL AND COMPUTATIONAL FLUID DYNAMICS Lecture Notes by Prof. Dr. Siddhartha Mishra and Dr. Franziska Weber October 4, 216 1 Contents 1 Fundamental Equations of Fluid Dynamics 4 1.1 Eulerian
More informationα 2 )(k x ũ(t, η) + k z W (t, η)
1 3D Poiseuille Flow Over the next two lectures we will be going over the stabilization of the 3-D Poiseuille flow. For those of you who havent had fluids, that means we have a channel of fluid that is
More informationAn Eulerian-Lagrangian Approach for Incompressible Fluids: Local Theory
An Eulerian-Lagrangian Approach for Incompressible Fluids: Local Theory Peter Constantin Department of Mathematics The University of Chicago December 22, 2000 Abstract We study a formulation of the incompressible
More informationWeek 2 Notes, Math 865, Tanveer
Week 2 Notes, Math 865, Tanveer 1. Incompressible constant density equations in different forms Recall we derived the Navier-Stokes equation for incompressible constant density, i.e. homogeneous flows:
More informationRelation between Distributional and Leray-Hopf Solutions to the Navier-Stokes Equations
Relation between Distributional and Leray-Hopf Solutions to the Navier-Stokes Equations Giovanni P. Galdi Department of Mechanical Engineering & Materials Science and Department of Mathematics University
More informationChapter 9: Differential Analysis
9-1 Introduction 9-2 Conservation of Mass 9-3 The Stream Function 9-4 Conservation of Linear Momentum 9-5 Navier Stokes Equation 9-6 Differential Analysis Problems Recall 9-1 Introduction (1) Chap 5: Control
More information