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Topics of Regularity in PDE workshop

On some unsolved mathematical problems for viscous fluids O.A. LADYZHENSKAYA St.Petersburg Steklov Mathematical Institute, Russian Academy of Sciences, RUSSIA

The main part of this paper is based on the talk given by me in 1994 in USA at the conference devoted to the nonlinear problems in PDE. It was supposed to publish a special volume of Proceedings of that conference but this was not carried out and my talk was not published. As the most of problems I mentioned in my talk are open up to now I present them in this paper again. I completed my list by some new interesting unsolved problems and excluded the problems which were solved during these seven years. I also excluded topics which was not related with hydrodynamics. We consider some boundary and initial boundary value problems for the Navier-Stokes equations - the most accepted model for viscous incompressible fluids. Consider first the case of bounded domains Ω ⊂ Rn , n = 2, 3. The equation has the form  L(v) ≡ vt − ν∆v + vk vxk = f − ∇p, (1) in Ω × R+ . div v = 0. In the case of the first initial boundary value problem we add to (1) the following conditions (2)

v ∂Ω×R+ = a ∂Ω×R+ ,

(3)

v t=0 = ϕ.

Here x ∈ Ω, t ∈ R+ ≡ [0, +∞), v : Ω × R+ → Rn is the velocity field, vk are (cartesian) components of v, p : Ω × R+ → R is the pressure, ν is a positive number (the coefficient of viscosity), vt and vxk are partial derivatives of v with respect to t and xk . The functions f : Ω × R+ → Rn , a : ∂Ω × R+ → Rn and ϕ : Ω → Rn are known and v, p have to be found.

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O. A. Ladyzhenskaya

In the corresponding stationary problem we seek solutions v : Ω → Rn , p : Ω → R of the system (4)

−ν∆v + vk vxk = f − ∇p, div v = 0.

(5)

v ∂Ω = a ∂Ω .

It is easy to see that for solvability of (4), (5) (and (1), (2), (3) too) the field a ∂Ω has to satisfy the condition Z (6) a · n ds = 0, ∂Ω

where n(x) x∈∂Ω is an exterior unit normal to ∂Ω. In [1] (see also [2] – [4]) we have proved global solvability of the problem (4), (5) (i.e., we did not put any restrictions on the values of ν > 0, or the sizes or shape of Ω, as well as on the values of the norms of f and a) if the equality (6) is fulfilled for all connected components Sk of ∂Ω, i.e. if Z a · n ds = 0, k = 1, . . . , m, ∂Ω = ∪m (7) k=1 Sk . Sk

First we prove (under very moderate hypothesis on the smoothness of all data) the existence of a generalized solution v ∈ W 1,2 (Ω; Rn ) (at least one). Then we show that the smoothness in reality depends only of the smoothness of data and the dependence has a local character. The proof of existence is based on an a priori estimate of the norm W 1,2 of all possible (generalized) solutions v λ of the problems (8)

−ν∆v + λvk vxk = f − ∇p, λ ∈ [0, 1], div v = 0, v ∂Ω = a ∂Ω .

In [1] and [2] there is one method of estimation and in [4] another. In both of these methods we use the fact that the field a ∂Ω can be extended on all Ω as a field a = rot b, b ∈ W22 (Ω; Rn ) (in case of n = 2 b is taken in the form: b1 = b2 ≡ 0, b3 = ψ(x1 , x2 ) and a1 = ψx2 , a2 = −ψx1 ). For such extension a ∂Ω has to satisfy (7). Let us describe the third method (related to the first one) in which we do not need the representation a = rot b but we do use the restriction (7).

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Topics of Regularity in PDE workshop

Denote by a(x), x ∈ Ω, any solenoidal field on Ω from W 1,2 (Ω; Rn ) equal to a|∂Ω on ∂ΩR(so that div a = 0). Suppose that there is no common majorant for kvxλ k ≡ ( Ω |∇v λ |2 dx)1/2 . Then there is a sequence {v λm }∞ m=1 for which λm → λ0 ∈ [0, 1] and Am ≡ kvxλm k → 0. Without loss of generality we can λm converge weakly in W 1,2 (Ω; Rn ) and strongly in suppose that v m ≡ A−1 m v 2 n L (Ω; R ) to some w. It is shown, as in [4], that w does not depend on the continuation of a ∂Ω on Ω and satisfies the following relations: Z ν − λ0

(9)

wk wxk a dx = 0, Ω

(10)

div w = 0,

w|∂Ω = 0,

w ∈ W 1,2 (Ω; Rn ).

From (9) and (10) it follows wk wxk = ∇q,

(11) Z (12)

Z

ν = λ0

∇q · a ds = λ0

wk wxk a dx = λ0 Ω

Z



q(a · n) ds Ω

and q S = qk = const,

(13)

k

If m = 1 then

k = 1, . . . , m.

Z

Z q(a · n)ds = λ0 q1

λ0 ∂Ω

a · nds = 0 ∂Ω

which contradicts (12). We have the same contradiction in the case (7) with any m > 1. For m > 1 the homogeneous system (10), (11) can have infinitely many nontrivial solutions, with different values qk on different Sk . It is easy to see that for the circular domain (14)

Ω = {x = (x1 , x2 ) : 0 < r1 < |x| < r2 < ∞} ⊂ R2

any vector field (15)

w1 = ϕ(r) sin θ,

w2 = −ϕ(r) cos θ,

29

O. A. Ladyzhenskaya where r = |x|, θ = arctg xx21 , satisfies (10) and (11) with Z r 1 dφ(ρ) 2 (16) q(r) = − dρ + const dρ r1 ρ if only φ(r) is a smooth function and d φ(r) ≡ φ0 (r) = 0, dr

(17) For such w R

q(a · n)ds = q2

a · nds + q1

|x|=r2

∂Ω

= (q2 − q1 )

R

for r = rk ,

R

R

a · nds =

|x|=r1

a · nds = −

|x|=r2

k = 1, 2.

Rr2 1 r1

(φ(ρ))2 dρ ρ

R

a · nds.

|x|=r2

Our calculations show ”how bad” the stationary problem is for the Euler equation, even if Ω ⊂ R2 and for incompressible case. For the Navier-Stokes system the following problem is open: Problem 1. Is there a common majorant for the norms in W 1,2 (Ω; Rn ) of all possible solutions v λ ∈ W 1,2 (Ω; Rn ) of the problem (8) if ∂Ω = ∪m k=1 Sk , m > 1, and a satisfies (6) but not (7)? If the problem has positive answer then the global solvability of the problem (4), (5) follows from our previous considerations. At first it is reasonable to study the problem (4) in domains (14) with x x2  1 a(x) = α , , α∈R |x|2 |x|2 and arbitrary f (x). The method to obtain the global unique solvability of (4), (5) which was introduced by me in [1] — [4] can be applied also to the case of unbounded domains Ω including domains having noncompact boundaries. For the external problem in case of a domain with compact boundary and condition at infinity (18)

v(x) → a0

as |x| → ∞,

this methods gives the existence of at least one solution v which is a sum of some solenoidal field A(x) satisfying the conditions (5) and (18), and a field u

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Topics of Regularity in PDE workshop

belonging to the Hilbert space H(Ω) (the later is a closure with respect to the norm of Dirichlet integral of the set J0∞ (Ω) of infinitely smooth solenoidal vector fields having a compact support in Ω. The norm in H(Ω) is given by kukH(Ω) = kux kL2 (Ω) ). It turns out that the elements of H(Ω) have a different behaviour in two- and three dimensional case. For Ω ⊂ R3 an element u ∈ H(Ω) tends to zero as |x| → ∞ (in an appropriate sense). But in two dimensional situation u can tend to some nonzero constant. By this diference I explain the known Stokes phenomena which is the following problem: for the Stokes system −∆u + ∇p = f,

div u = 0,

u|∂Ω = a,

in case of an exterior domain Ω the restriction (18) must be put if Ω ⊂ R3 and must not be put if Ω ⊂ R2 . In the last case the value of a0 is determined by Ω and the data f and a ∂Ω . For the Navier-Stokes system in case of Ω ⊂ R3 the restriction (18) must be assumed as well as in the case of the Stokes system. But in two dimensional case the matter is unclear. This is our second problem: Problem 2. Does the system (4), (5) in case of an exterior domain Ω ⊂ R2 with a compavt boundary have at least one solution satisfying (18) with a prescribed value of a0 ? Some questions concerning nonstationary dynamics are connected with Problem 1. We have proved the global unique solvability of the problem (1), (2) (3) for n = 2 (in a different phase spaces) under some reasonablly moderate hypotheses on the data ([2] — [7]). The solutions are continuous in t. Moreover, for t > 0 they have better properties than for t = 0 (if only f , a ∂Ω and ∂Ω are good enough). In this case, the problem (1), (2), (3) has the property of instant smoothing. If a ∂Ω×R+ in (2) is not identically zero we reduce the problem (1), (2), (3) to a new system for the function u(x, t) = v(x, t) − a(x, t) and p(x, t) taking as a(x, t) a smooth solenoidal extension of a ∂Ω×R+ on Q = Ω × R+ . This new system has the form (19)

L0 (u) = L(u) + ak uxk + uk axk = f 0 − ∇p, div u = 0,

31

O. A. Ladyzhenskaya where f 0 = f − at + ν∆a − ak axk , (20)

u ∂Ω = 0,

(21)

u|t=0 = ϕ − a ≡ ϕ0 .

The lower order terms ak uxk + uk axk do not destroy the global unique solv1,2 ability of the problem (1), (2), (3) if, for example, a ∈ L∞ (Ω, R2 )). loc (R+ ; W Suppose that f |Q , a|∂Ω×R+ and the extension a|Q do not depend on t. Then we have proved the existence of a compact minimal global B- attractor M for the problem (1), (2), (3) if a|∂Ω satisfies the restrictions (7) (see [7], [8]). We used these restrictions to take a special extension of a|∂Ω constructed in the study of stationary problem (4), (5). For this a|Ω the solution u of (19), (20), (21) satisfy the inequality ν ν 1 1d ku(t)k2 + kux (t)k2 ≤ (f 0 , u(t)) ≤ kf 0 k(−1) kux (t)k ≤ kux (t)k2 + kf 0 k2(−1) 2 dt 2 4 ν and therefore (22)

1d ν 1 ku(t)k2 + kux (t)k2 ≤ kf 0 k2(−1) . 2 dt 4 ν

Here k · k is the norm in L2 (Ω, Rn ) and k · k(−1) is the norm in the Hilbert space H −1 which is dual to the Hilbert space H 1 = {u ∈ W01,2 (Ω, Rn ), div u = 0} relative to H 0 — a subspace of L2 (Ω; Rn ). (We deal here with the case n = 2, but the definition of spaces are the same for n = 2 and for n = 3). Space H 0 is the orthogonal supplement in L2 (Ω; Rn ) to the subspace G(Ω) = {∇q : q ∈ W 1,2 (Ω; R)}. An equivalent definition of H 0 is the following: H 0 = Closure of J0∞ (Ω) in L2 (Ω; Rn ), where the set J0∞ (Ω) was defined above (see [2] — [5] for details).

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Topics of Regularity in PDE workshop

In the phase space H 0 the problem (1) with a ∈ W 1,2 (Ω; R2 ), f ∈ H −1 has a unique solution u(t, ϕ) for any ϕ ∈ H 0 . The solution operators Ut : ϕ → u(t, ϕ) form a continuous semigroup {Ut , t ∈ R+ , H 0 } of the first class (see [7] — [10]). This semigroup has the same properties which we had established in [7] for the case a ≡ 0. The inequality (22) guaranties the existence of a Babsorbing ball BR (0) ⊂ H 0 . This means that for any bounded set B ⊂ H 0 the sets Ut (B) belongs to BR (0) for all t ≥ t1 (B). The minimal global Battractor M0 for {Ut , R+ , H 0 } lies in BR (0) and has many nice properties (for example, it has finite fractal dimension). Due to this the problem (1), (2), (3) has a minimal global B- attractor M = M0 + {a}. The following interesting problem is open: Problem 3. Suppose that n = 2, f ∈ L2 (Ω; R2 ), a|∂Ω has a solenoidal extension on Ω which belongs to W 1,2 (Ω, R2 ) (so a satisfies (6)), and ∂Ω = ∪m k=1 Sk , m > 1. Is it possible to prove that there is a bounded B- attracting set B 0 ⊂ H 0 + {a} for the problem (1), (2), (3)? Remark 1. According to one of the theorems given in [10] (see also [8], [9]) about the abstract semigroups of the class 1 it is sufficient to prove the existence of a ball B 0 such that any solution v(t, ϕ) of (1), (2), (3) with ϕ ∈ H 0 + {a} tends to B 0 as t → ∞. Remark 2. If the problem 2 (or the Remark 1) has positive solution then the problem (1), (2), (3) has a compact minimal global B- attractor M. Remark 3. It is reasonable in the beginning to consider the problem (1), (2), (3) in the circular domains (14) and prove that sup kv(t, ϕ)k < ∞. t∈R+

Let us formulate the next open problem on the behavior of the solutions of some magnitodynamical systems when t → ∞. Problem 4. To find a bounded domain Ω in R3 and a solenoidal field v = v(x), x ∈ Ω such that the problem

(23)

Ht + γ rot rot H + rot(H × v) = 0, γ = const > 0, div H = 0, H · n|∂Ω = 0, (rot H)τ |∂Ω = 0, H|t=0 = ψ

33

O. A. Ladyzhenskaya

has solutions H(t, ψ) : Ω → R3 which do not vanish as t → ∞. Here wτ |∂Ω = w − (w · n)n|∂Ω is a projection of w on the tangent plane to ∂Ω. It is desirable that v satisfies the boundary condition v · n|∂Ω = 0.

Problem 4 is known as a ”Dynamo problem” on the existence of nonvanishing magnetic fields H(t, ψ) : Ω → Rn . It is a ”linearization” of the following more general (Problem 5) for the magneto-dynamical system

(24)

vt − ν∆v − v × rot v + H × rot H = −∇(p + 21 |v|2 ) + f, Ht + γ rot rot H + rot(H × v) = rot j, div v = 0, div H = 0, v|∂Ω = a, a · n|∂Ω = 0, Hn |∂Ω = (rot H)τ |∂Ω = 0, v|t=0 = ϕ, H|t=0 = ψ.

Here ν = const > 0, γ = const > 0 and Ω is a bounded domain in Rn . For the two-dimensional case we proved the global unique solvability of (24) under some moderate hypotheses on all data ([11], [12]). The solutions (v(t, ϕ, ψ), H(t, ϕ, ψ)) depends continuously on t (in the chosen phase-space, for example in X = H 0 × H 0 ) and even more: they are better for t > 0 than arbitrary element of X (so we have instant smoothing). If f , j and a do not depend on t then the solution operators Wt : (ϕ, ψ) → (v(t, ϕ, ψ), H(t, ϕ, ψ)) form a continuos semigroup {Wt , t ∈ R+ , X}. It is bounded, continuous, has a bounded B- absorbing set and belongs to class 1. Therefore it has a compact minimal global H- attractor M = M(a, f, j, Ω) ⊂ X with some nice properties. However, the following problem is open: Problem 5. To find Ω, f and a (with the properties: div a = 0 and a·n|∂Ω = 0) such that the attractor M = M(a, f, 0; Ω) for (24) with j ≡ 0 does not replace entirely in the subspace H 0 × {0} of X = H 0 × H 0 . In the study of Problem 4 we considered the domains Ω = {x = (r, θ, ϕ) : 0 < r1 < r < r2 < ∞, θ ∈ [0, 2π], z ∈ R} ⊂ R3 ,

34

Topics of Regularity in PDE workshop

where r, θ, z are the cilindrical coordinates of x, and the velocity fields v(x), x ∈ Ω, having the cilindrical components vr = rc , vθ = 0, vz = 0. Corresponding magnetic fields H have the form H = (0, Hθ = Hθ (r, t), 0) ∈ R3 . We have found some intervals for the parametrs r1 , r2 , c in which the spectral problem corresponding to (23) has nonvanishing solutions H(t, ψ). But these cases do not correspond to a natural experiments: our Ω is unbounded and v · n 6≡ 0. It is interesting to find more ”physical” situations. Let us pass to the nonstationary three dimensional problems for viscous incompressible fluids. The principal open question for the Navier-Stokes equations is: Do the Navier-Stokes equations give a deterministic description of the dynamical processes in viscous incompressible fluids? In other words we do not know if there is a phase space in which the problem (1), (2), (3) is globally unique solvable (I shall not remind the reader of cases for which we know the unique solvability of (1), (2), (3). They are described in [2], [4], and [14]). There is the famous result of E. Hopf on the existence of at least one weak solution to the problem (1), (2), (3) (we call them Hopf’s solutions) [13]. After that there were many attemps to prove uniqueness in the class of Hopf’s solutions or to show that these solutions in reality have some additional smoothness which permits one to prove their uniqueness. My opinion was, and is now, that the class of Hopfs solutions is too large — in it we lose uniqueness. In support of this statement I have constructed two Hopf’s solutions v 0 and v 00 of the same system (1), (3) which satisfy the same boundary condition (see [15]). But I have done this not for boundary condition (2) but for another condition and for domains Ωt , t ∈ R+ changing in time. So it is desirable to solve the following problem: Problem 6. Construct two different Hopf ’s solutions of the problem (1), (2), (3) with the same f , a, ϕ and fixed Ω. Remark 3. The initial boundary value problem taken in [15] is correct in some classes which are slightly more narrow than the class of Hopf’s solutions.

35

O. A. Ladyzhenskaya

These classes are described in [15] and in the second Russian edition [4] (Ch. VI). The other even more interesting problem is the following: Problem 7. To show that some solutions v of the problem (1), (2), (3) n = 3, with smooth f , a, ϕ and ∂Ω can lose smothness at finite times t1 , t2 , . . . < ∞. In particular, the Dirichlet integral kvx (t, ϕ)k can become ∞ as t → tk . In the middle of the sixties I suggested some modifications of the NavierStokes equations (MNS eqs) and proved for them (for n = 2 and n = 3) the global unique solvability of the problem (2), (3) and the global solvability of the problem (5) (see [16], [17], [4]). I supposed for simplicity that a|∂Ω = 0. But the cases of nonzero a|∂Ω can be considered using the same reduction to the zero boundary condition which I applied to the Navier-Stokes equations (and have described above), if only a|∂Ω satisfy the conditions (7). It is desirable to study the problems: Problem 10 . Is there a common majorant in W 1,2+2µ (Ω, Rn ) (n = 2 or 3) for all stationary solutions v λ ∈ W 1,2+2µ (Ω, Rn ), λ ∈ [0, 1], to the first boundary value problem for MNS eqs if ∂Ω is multiconnected and a|∂Ω satisfies (6) but not (7)? Remark 4. Parameter λ comes in MNS eqs in the term λvk vxk . Problem 20 . Suppose that in the first boundary value problem for MNS eqs, the forces f and a|∂Ω do not depend on t and m > 1. Suppose further that a|∂Ω satisfies (6) (but not (7)). Is the norm kv(t, ϕ)k of solution v(t, ϕ) bounded on the semi-axis t ∈ R+ for any ϕ ∈ X = H 0 + {a}? Is there a bounded Babsorbing set for the semigroup of solution operators {Vt , t ∈ R+ , X} of the problem. Let us note that there are results on the existence of minimal global Battractors M for some classes of MNS eqs ([18] — [20]). In [19] — [24] we considered MNS eqs of the form (25)

vt − div

∂D(ε(v)) + vk vxk = f − ∇p, ∂ε

where εij (v) = 12 (vixj + vjxi ) are components of the symmetric rate tensor ε(v) = (εij (v)), i, j = 1, . . . , n, D : Mns → R+ is a dissipative potential,

36

Topics of Regularity in PDE workshop

and Mns denotes the space of symmetric n × n- matrices. The velocity field v : Ω × R+ ⊂ Rn → Rn and the pressure p : Ω × R+ → R have to be found. For D(ε) = ν|ε|2 , ν = const > 0, the system (25) is the Navier-Stokes system. We have studied the system (25) for arbitrary convex D : Mns → R+ satisfying the hypothesis: D(ε) ≥ ν1 |ε|2 + ν2 |ε|2+2µ ,

D(0) = 0,

with ν1 > 0, ν2 ≥ 0, µ ≥ 0 for n = 2, ν1 ≥ 0, ν2 > 0, µ ≥ 41 for n = 3, and

∂D(ε) ≤ ν3 (1 + |ε|) + ν2 ν4 |ε|1+2µ ∂ε

with ν3 , ν4 > 0. In my papers and in papers of other mathematicians generalized solutions of (25) were studied. These solutions generally speaking had as much all weak derivatives involved into the system. For nonstationary problems in these classes the uniqueness theorem takes place. The regularity (which means better smoothness providing the data are smooth) of this generalized solutions was not studied for a long time. But for the last ten years there is a remarkable progress in this direction, see [22] — [27]. Here I would like to formulate the following unsolved problem: Problem 8. Show that in tree dimensional case the system of type (25) can have nonsmooth solutions even if the data of the problem are infinitely smooth (in both stationary and nonstationary cases). To the contrast, in two dimensional case regularity of solutions to (25) was obtained under some restrictions on parameters of the problem (see [22] — [24]) and references there). It would be interesting to improve these restrictions.

References [1] O.A. Ladyzhenskaya, The investigation of the Navier-Stokes equations in the case of stationary incompressible fluids, Uspehi Math. Nauk, Moscow, 13: 4, 1958, pp. 219-220; 14: 3, 1959, pp. 75-97.

O. A. Ladyzhenskaya

37

[2] O.A. Ladyzhenskaya, Mathematical problems of viscous incompressible fluids, Moscow, Fiz. Mat. Giz, 1961. [3] O.A. Ladyzhenskaya, The mathematical theory of viscous incompressible fluids, second edition, Gordon and Beach, New York-London, 1969. (English translation of [2]). [4] O.A. Ladyzhenskaya, Mathematical problems of viscous incompressible fluids, Moscow, ”Nauka”, 1970 (the second Russian edition). [5] O.A. Ladyzhenskaya, The solvability “in large” of the boundary value problem for the Navier-Stokes equations in case of two space variables, Doklady Akad. Nauk SSSR, 123: 3, 1958, pp. 427-429. [6] O.A. Ladyzhenskaya, Solution “in large” of the nonstationary boundary value problem for the Navier-Stokes system with two space variables, Comm. Pure Appl. math., 12: 3, 1959, pp. 427-433. [7] O.A. Ladyzhenskaya, On the dynamical system generated by the NavierStokes equations, Zapiski Nauchnyh Seminarov LOMI, Leningrad, v. 27, 1972, pp. 91 -115 (English translation in J. of Sovet Math., 34, 1975). [8] O.A. Ladyzhenskaya, On finding the minimal global B- attractors for the Navier-Stokes equations and other PDE, Uspehi Math. Nauk, 42: 6, 1987, pp. 25-60. [9] O.A. Ladyzhenskaya, On finding the minimal global B- attractors for semigroup generated by initial boundary value problem to nonlinear dissipative PDE, Preprint LOMI, E-3-1987, pp. 1-54. [10] O.A. Ladyzhenskaya, Attractors for semi-groups and evolution equations, Lezioni Lincei, Roma 1988, Cambrige University Press, 1991. [11] O.A. Ladyzhenskaya and V.A. Solonnikov, The solvability of some nonstationary problems of magneto-hydrodinamics for viscous incompressible fluids, Trudy Math. Inst. of steklov, 59, 1960, pp. 115-173. [12] O.A. Ladyzhenskaya, Attractors in problems of magneto-hydrodynamics, Topological Fluid Mechanics, Proceedings of IUTAM Symposium, Cambrige, August 13-18, 1989, Cambrige University Press, 1990, pp. 782-786. ¨ [13] E. Hopf, Uber die Anfangsewertaufgabe f¨ ur die hydrodynamische Grundgleichungen, Math. Nachrichten, 4, 1950-51, pp. 213-231. [14] A.A. Kiselev and O.A. Ladyzhenskaya, On existence and uniqueness of a solution to the nonstationary problem for viscous incompressible fluids, Izvestia Akad. Nauk SSSR, ser. Math., 21, 1957, pp. 655-680. [15] O.A. Ladyzhenskaya, An example of non-uniqueness in the Hopfs class of weak solutions to the Navier-Stokes equations, Izvestia Akad. Nauk SSSR, ser. Math., 33, 1969, pp. 240-247.

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Topics of Regularity in PDE workshop

[16] O.A. Ladyzhenskaya, On nonlinear problems in continuous mechanics, Proceedings of International Mathematical Congress, 1966, Moscow, 1968, pp. 560-572. [17] O.A. Ladyzhenskaya, On some modifications of the Navier-stokes equations for large gradients of velocity. Zapiski Nauchnyh Seminarov LOMI, v. 7, 1968, pp. 126-154. [18] O.A. Ladyzhenskaya, Limiting regimes for the modified Navier-Stokes equations in three-dimensional space, Zapiski Nauchnyh Seminarov LOMI, v. 84, 1979, pp. 131-146. [19] O.A. Ladyzhenskaya, Attractors for the modifications of three-dimensional Navier-Stokes equations, Philosoph. Trans. of Royal Society of London, Ser. A, v. 346, no. 1679, 1994, 173-190. [20] O.A. Ladyzhenskaya and G.A. Seregin, On semi-groups generated by initialboundary value problems for system describing two-dimensional visco-plastic flows, pp. 1-28. [21] Q. Du and M. Gunzburger, Analysis of a Ladyzhenskaya model for incompressible viscous flow, J. Math. Anal. and Appl., v. 155, no. 1, 1991, pp. 21-45. [22] O.A. Ladyzhenskaya and G.A. Seregin, On smoothness of solutions to the system describing the dynamics of the generalized Newtonian fluids, Izvestia of RAN, ser. Math., v. 62 (1998), no.1, pp. 59-222. [23] O.A. Ladyzhenskaya and G.A. Seregin, On regularity of solutions to the twodimensional dynamical equations for fluids with nonlinear viscosity, Zapiski Nauchn. Seminarov POMI, v. 259 (1999), pp. 145-166. [24] O.A. Ladyzhenskaya and G.A. Seregin, On disjointness of solutions to the MNS equations, Amer. Math. Soc. Translations, v. 189 (1999), no. 2, pp. 159-179. [25] J. Malek, J. Necas, M. Rokyta, M. Ruzicka, Weak and measure-valued solutions to evolution partial differential equations, Applied Math. and Math. Computations, vol. 13, Chapman and Hall, 1996. [26] G.A. Seregin, Partial regularity of solutions to the Modified Navier-Stokes equations, Zapiski Nauchn. Seminarov POMI, v. 259 (1999), pp. 238-253. [27] G.A. Seregin, Interior regularity for solutions to the Modified Navier-Stokes equations, Journal of Math. Fluid Mechanics, v. 1 (1999), no.3, pp.235-281.

On some unsolved mathematical problems for ...

problems I mentioned in my talk are open up to now I present them in this ... data) the existence of a generalized solution v ∈ W1,2(Ω;Rn) (at least one). Then we ...

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Solved and unsolved problems in Number Theory - Daniel Shanks.pdf. Solved and unsolved problems in Number Theory - Daniel Shanks.pdf. Open. Extract.

some puzzles and problems
ing theories of religion, correspondingly, is that “Our usual approaches to the study of religion…(are) largely unusable and inadequate”. As read- ers, we are merely left with puzzles: how did Gill 'see' what is invisible? From whence his convi

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Regional Mathematical Olympiad-2000. Problems and Solutions. 1. Let AC be a line segment in the plane and B a point between A and C. Construct isosceles.

On some conjectures on VOAs
Feb 1, 2013 - generated by fermionic fields bA in BC(g)1,− and cA in BC(g)0,− for A = 1 .... given a symplectic vector space V over C, denote by SB(V ) a VOA ...

Some thoughts on hypercomputation q
18 However it is more convenient for us to give an alternative definition for the ... q N.C.A. da Costa is supported in part by CNPq, Philosophy Section. ..... 247 practical situations) with finite, even if large, specific instances of the halting pr

on some new methodologies for pattern recognition ...
delity between the vector quantizers using SOM code books and surface ..... However, for real-life problems with large data sets the construction of a dendrogram is ... clustering techniques from data mining perspective can be found in [29].

Problems on TORSION
6.73. G = 9. × 10 N/mm. 3. 2. Developed By : Shaikh sir's Reliance Academy, Near Malabar Hotel, Station road, Kolhapur. Contact :[email protected] ...

On some sufficient conditions for distributed QoS ...
May 23, 2008 - other, we study sufficient conditions for determining whether a given set of minimum bandwidth .... In order to keep this paper as self-contained and accessible as possible, we recall along the way ... an optimal schedule has a rationa

Some reflections on incentives for publication: the case ...
Phone: 972-883 6402; Fax: 972-883 6297; E-mail: [email protected] ... in Faria (2002a) the impact of business and political networks can drive the research of ... which is an independent, low-cost way to measure and evaluate .... from a for a top do

On Some Sufficient Conditions for Distributed Quality-of ... - IEEE Xplore
that of an optimal, centralized algorithm. Keywords-distributed algorithms; quality-of-service (QoS); conflict graph; wireless networks; interference models; frac-.

Some Descriptional Complexity Problems in Finite ...
ing developments in theoretical computer science for more than two decades .... Degree of ambiguity is also an intensively investigated con- cept in automata ... us the following classes of nondeterministic finite automata. We denote by NFA ...

PDF Intriguing Mathematical Problems - Oswald Jacoby ...
... A Common Sense Approach to Web Application Design - Robert Hoekman Jr. ... puzzles and oddities of all kinds, compiled by one of the world's best card ...

Challenging Mathematical Problems with Elementary Solutions Vol 2 ...
Challenging Mathematical Problems with Elementary Solutions Vol 2 (Dover) - Yaglom & Yaglom.pdf. Challenging Mathematical Problems with Elementary ...

Challenging Mathematical Problems with Elementary Solutions Vol 2 ...
Dept. of Mathematics,. The University of ... Page 3 of 223. Challenging Mathematical Problems with Elementary Solutions Vol 2 (Dover) - Yaglom & Yaglom.pdf.

Selected-Problems-Of-The-Vietnamese-Mathematical-Olympiad-1962 ...
Whoops! There was a problem previewing this document. Retrying... Download. Connect more apps... Try one of the apps below to open or edit this item. Selected-Problems-Of-The-Vietnamese-Mathematical-Olympiad-1962-2009.pdf. Selected-Problems-Of-The-Vi

Challenging Mathematical Problems with Elementary Solutions Vol 1 ...
Challenging Mathematical Problems with Elementary Solutions Vol 1 (Dover) - Yaglom & Yaglom.pdf. Challenging Mathematical Problems with Elementary ...

On Some Remarkable Concurrences
Nov 18, 2002 - Y -axis of the rectangular coordinate system in Π, respectively. .... of Pure Mathematics and Computer Algebra, Krijgslaan 281-S22, B-.

Some rough notes on GravitoElectroMagnetism.
I found the GEM equations interesting, and explored the surface of them slightly. Here are some notes, mostly as a reference for myself ... looking at the GEM equations mostly generates questions, especially since I don't have the GR background to un