A generalised Ladyzhenskaya inequality and a coupled parabolic-elliptic problem
|
|
- Stephen Farmer
- 6 years ago
- Views:
Transcription
1 A generalised Ladyzhenskaya inequality and a coupled parabolic-elliptic problem Dave McCormick joint work with James Robinson and José Rodrigo Mathematics and Statistics Centre for Doctoral Training University of Warwick February 26th, 2013
2 A coupled parabolic-elliptic MHD system We consider the following modified system of equations for magnetohydrodynamics on a bounded domain Ω R 2 : u + p = (B )B tb ε B + (u )B = (B )u, with u = B = 0 and Dirichlet boundary conditions. This is like the standard MHD system, but with the terms tu + (u )u removed. Theorem Given u 0, B 0 L 2 (Ω) with u 0 = B 0 = 0, for any T > 0 there exists a unique weak solution (u, B) with u L (0, T; L 2, ) L 2 (0, T; H 1 ) and B L (0, T; L 2 ) L 2 (0, T; H 1 ). We prove this using both a generalisation of Ladyzhenskaya s inequality, and some elliptic regularity theory for L 1 forcing.
3 Weak solutions of the Navier Stokes equations Consider the Navier Stokes equations on a domain Ω R n, n = 2 or 3: tu + (u )u u + p = 0, with u = 0, and Dirichlet boundary conditions. Theorem (Leray, 1934, and Hopf, 1951) Given u 0 L 2 (Ω) with u 0 = 0, there exists a weak solution u of the Navier Stokes equations satisfying u L (0, T; L 2 ) L 2 (0, T; H 1 ). Moreover, if n = 2, this weak solution is unique. The same is true if Ω = R n, or if Ω = [0, 1] n with periodic boundary conditions.
4 Weak solutions of NSE: existence Let u m be the mth Galerkin approximation: i.e., the solution of tu m + P m [(u m )u m ] u m + p m = 0. Taking the L 2 inner product with u m, we get Fact tu m, u m + (u m )u m, u m u m, u m + p m, u m = 0. }{{} =0 If u = 0, and u, v, w = 0 on Ω, then (u )v, w = (u )w, v. Hence 1 d 2 dt um 2 + u m 2 = 0, so integrating in time yields u m (t) 2 + t 0 u m (s) 2 ds = u m (0) 2 u 0 2. Hence u m are uniformly bounded in L (0, T; L 2 ) L 2 (0, T; H 1 ).
5 Ladyzhenskaya s inequality To get uniform bounds on tu m = u m P m [(u m )u m ], one uses: Ladyzhenskaya s inequality in 2D (1958) u L 4 c u 1/2 L 2 u 1/2 L 2. Ladyzhenskaya s inequality yields a priori bounds on the nonlinear term (u m )u m : (u m )u m φ = (u m )φ u m u m 2 L 4 φ L 2, so (u m )u m H 1 u m 2 L 4 c um L 2 u L 2, and thus (u m )u m L 2 (0, T; H 1 ), and hence tu m L 2 (0, T; H 1 ). Theorem (Aubin, 1963, and Lions, 1969) If u m L 2 (0, T; H 1 ) and tu m L 2 (0, T; H 1 ) uniformly, then a subsequence u m k u L 2 (0, T; L 2 ) (strongly).
6 Magnetic relaxation (Moffatt, 1985) Aim to construct stationary solutions of the Euler equations with non-trivial topology: (u )u + p = 0. Consider the MHD equations with zero magnetic resistivity u t u + (u )u + p = (B )B B t + (u )B = (B )u and assume that smooth solutions exist for all t 0 (open even in 2D). Energy equation 1 d ( u 2 + B 2) + u 2 = 0. 2 dt So B decreases while u = 0. Since the magnetic helicity H = A B is preserved, where B = A and A = 0, B is bounded below: ( 2 c B 4 B 2 A 2 A B) = H 2. So u(t) 0 as t (Nuñez, 2007) and B(t) B with p = (B )B.
7 Magnetic relaxation (Moffatt, 2009) The dynamics are arbitrary, so consider instead the simpler model The new energy equation is u + p = (B )B B t + (u )B = (B )u. 1 d 2 dt B 2 + u 2 = 0. So u 2 0 = u(t) 0 and B(t) B as t. Open: does u(t) 0 as t in this case? To address existence of solutions we make two simplifications: we consider 2D, and regularise the B equation: u + p = (B )B B t ε B + (u )B = (B )u.
8 A priori estimates We consider the 2D system u + p = (B )B u = 0 B t ε B + (u )B = (B )u B = 0. Toy version of 3D Navier Stokes, which in vorticity form (ω = u) is ω t ν ω + (u )ω = (ω )u. Take inner product with u in the first equation, with B in the second equation and add: We get: u 2 = (B )B, u = (B )u, B 1 d 2 dt B 2 + ε B 2 = (B )u, B 1 d 2 dt B 2 + ε B 2 + u 2 = 0. B L (0, T; L 2 ) L 2 (0, T; H 1 ), u L 2 (0, T; H 1 ).
9 A priori estimates What can we say about u? We know that B L (0, T; L 2 ). Note that [(u )v] i = u j j v i = j (u j v i ) =: (u v), since v = 0. We can write the equation for u as Elliptic regularity works for p > 1: u + p = (B )B = (B B). }{{} L 1 u + p = f, f L p = u W 2,p (e.g. Temam, 1979). If we could take p = 1 then we would expect, for RHS f with f L 1, to get u W 1,1 L 2. In fact for RHS f with f L 1 we get u L 2,, where L 2, is the weak-l 2 space.
10 L p, : weak L p spaces For f : R n R define d f (α) = µ{x : f(x) > α}. Note that f p L R p = f(x) p f(x) p α p d f (α). n {x: f(x) >α} For 1 p < set } f L p, = inf {C : d f (α) Cp = sup{γd f (γ) 1/p : γ > 0}. The space L p, (R n ) consists of all those f such that f L p, <. L p L p, x n/p L p, (R n ) but / L p (R n ). if f L p, (R n ) then d f (α) f p L p, α p. α p
11 L p, : weak L p spaces Just as with strong L p spaces, we can interpolate between weak L p spaces: Weak L p interpolation Take p < r < q. If f L p, L q, then f L r and f L r c p,r,q f p(q r)/r(q p) L p, f q(r p)/r(q p) L q,. Recall Young s inequality for convolutions: if 1 p, q, r and 1 p + 1 = 1 q + 1 r then E f L p E L q f L r. There is also a weak form, which requires stronger conditions on p, q, r: Weak form of Young s inequality for convolutions If 1 r < and 1 < p, q <, and = then p q r E f L p, E L q, f L r.
12 Elliptic regularity in L 1 Fundamental solution of Stokes operator on R 2 is E ij (x) = δ ij log x + x ix j x 2, i.e. solution of u + p = f is u = E f. Solution of u + p = f is u = E ( f) = ( E) f. Note that Thus E L 2, and so x k k E ij = δ ij x + δ ikx j + δ jk x i x ix j x k 1 2 x 2 x 4 x. f L 1 = u = E f L 2,. If we consider the problem in a bounded domain we have the same regularity. We replace the fundamental solution E by the Dirichlet Green s function G satisfying G = δ(x y) G Ω = 0. Mitrea & Mitrea (2011) showed that in this case we still have G L 2,. So on our bounded domain, u L (0, T; L 2, ).
13 Estimates on time derivatives: t B L 2 (0, T; H 1 )? Take v H 1 with v H 1 = 1. Then tb, v = ε B (u )B + (B )u, v ε B v + 2 u L 4 B L 4 v L 2. so tb H 1 ε B + 2 u L 4 B L 4. Standard 2D Ladyzhenskaya inequality gives B L 4 c B 1/2 B 1/2 ; but we only have uniform bounds on u in L 2,. If f L 4 c f 1/2 L 2, f 1/2 then which would yield tb H 1 ε B + c u 1/2 L 2, B 1/2 u 1/2 B 1/2 tb L 2 (0, T; H 1 ).
14 Generalised Ladyzhenskaya inequality In 2D, f 2 L 4 c f L 2 f L 2. Proof: (i) write f 2 = 2 f j f dx j and integrate (f 2 ) 2. (ii) use the Sobolev embedding Ḣ 1/2 L 4 and interpolation in Ḣ s : f L 4 c f Ḣ1/2 c f 1/2 L 2 f 1/2 Ḣ 1. In fact, using the theory of interpolation spaces: Since Ḣ 1 BMO in 2D, this yields f L 4 c f 1/2 L 2, f 1/2 BMO. f L 4 c f 1/2 L 2, f 1/2 Ḣ 1. Besides the proof using interpolation spaces, we can also prove this directly using Fourier transforms.
15 Bounded mean oscillation Let f Q := 1 f(x) dx denote the average of f over a cube Q Q Q Rn. Define f BMO := sup Q 1 Q Q f(x) f Q dx, where the supremum is taken over all cubes Q R n. Let BMO(R n ) denote the set of functions f : R n R for which f BMO <. f BMO = 0 = f is constant (almost everywhere). L BMO and f BMO 2 f ; log x BMO but is unbounded. Ḣ n/2 BMO and f BMO c f Ḣn/2 in R n, even though Ḣ n/2 L. W 1,n BMO, by Poincaré s inequality: let B be a ball of radius r, then 1 f(x) f B dx cr ( 1/n Df dx c Df dx) n c f B B r n W 1,n, B B so f BMO c f W 1,n.
16 Interpolation spaces For 0 θ 1 one can define an interpolation space X θ := [X 0, X 1 ] θ in such a way that f Xθ c f 1 θ X 0 f θ X 1. (Note that f X1 c f X 1.) Theorem (Bennett & Sharpley, 1988) L p, = [L 1, BMO] 1 (1/p) for 1 < p < ; so L 2, = [L 1, BMO] 1/2. Write B = [L 1, BMO] 1 and note that f B c f BMO. Reiteration Theorem If A 0 = [X 0, X 1 ] θ0 and A 1 = [X 0, X 1 ] θ1 then for 0 < θ < 1 [A 0, A 1 ] θ = [X 0, X 1 ] (1 θ)θ0 +θθ 1. So L 3, = [L 2,, B] 1/3 and L 6, = [L 2,, B] 2/3, and hence f L 4 c f 1/2 L 3, f 1/2 L 6, c[c f 2/3 L 2, f 1/3 B ]1/2 [c f 1/3 L 2, f 2/3 B ]1/2 = c f 1/2 f 1/2 L 2, B c f 1/2 f 1/2 L 2, BMO.
17 Generalised Ladyzhenskaya inequality: direct method There exists a Schwartz function φ such that if ˆf is supported in B(0, R), f = φ 1/R f, where φ 1/R (x) = R n φ(rx). (Take φ with ˆφ = 1 on B(0, 1); then F [φ 1/R f] = F [φ 1/R ]ˆf = ˆf.) Lemma (Weak-strong Bernstein inequality) Suppose that supp(ˆf) B(0, R). Then for 1 p < q < f L q cr n(1/p 1/q) f L p,. Using the weak form of Young s inequality, for = 1 + 1, q r p f L q, = φ 1/R f L q, c φ 1/R L r f L p,, and noting that φ 1/R L r = cr n(1 1/r), it follows that f L 1, cr n(1/p 1) f L p, and f L 2q, cr n(1/p 1/2q) f L p,. Finally interpolate L q between L 1, and L 2q,.
18 Generalised Ladyzhenskaya inequality: direct method To show write f(x) = k R f L 4 c f 1/2 L 2, f 1/2 Ḣ 1 ˆf(k)e 2πik x dk } {{ } f + k R Now using the weak-strong Bernstein inequality and using the embedding Ḣ 1/2 L 4, Thus f L 4 cr 1/2 f L 2,, f + 2 L 4 c f+ 2 Ḣ 1/2 = c c R ˆf(k)e 2πik x dk. } {{ } f + k R k R c R f 2 Ḣ 1. k ˆf(k) 2 dk k 2 ˆf(k) 2 dk f L 4 cr 1/2 f L 2, + cr 1/2 f Ḣ1, and choosing R = f Ḣ1/ f L 2, yields the inequality.
19 Generalised Gagliardo Nirenberg inequalities It is not hard to generalise this direct proof to prove the following: Theorem Take 1 q < p < and s > n if f L q, Ḣ s then f L p and ( ) 1 1. There exists a constant c 2 p p,q,s such that f L p c p,q,s f θ L q, f 1 θ H s for every f L q, Ḣ s, where 1 p = θ q + (1 θ) ( 1 2 s n ). With a little work, in the case s = n/2 we can generalise this: Theorem Take 1 q < p <. There exists a constant c p,q such that if f L q, BMO then f L p and f L p c p,q f q/p L q, f 1 q/p BMO for every f L q, BMO.
20 Global existence of weak solutions Take B m (0) = P m B(0) and consider the Galerkin approximations: u m + p = P m (B m )B m tb m ε B m + P m (u m )B m = P m (B m )u m. The B m equation is a Lipschitz ODE on a finite-dimensional space, so it has a unique solution. By repeating the a priori estimates on these smooth equations (now rigorous) we can obtain estimates uniform in n: and B m L (0, T; L 2 ) L 2 (0, T; H 1 ), tb m L 2 (0, T; H 1 ) u m L (0, T; L 2, ) L 2 (0, T; H 1 ). By Aubin Lions, a subsequence of the Galerkin approximations B m B strongly in L 2 (0, T; L 2 ). Hence, by elliptic regularity, u m u L 2, B m B m B B L 1 B m B L 2( B m L 2 + B L 2), hence u m u strongly in L 2 (0, T; L 2, ). This is enough to show the nonlinear terms converge weak- in L 2 (0, T; H 1 ), and hence that (u, B) is a weak solution (i.e. a solution with equality in L 2 (0, T; H 1 )).
21 Conclusion Similar arguments to the a priori estimates show uniqueness of weak solutions, and so: Theorem Given u 0, B 0 L 2 (Ω) with u 0 = B 0 = 0, for any T > 0 there exists a unique weak solution (u, B) with u L (0, T; L 2, ) L 2 (0, T; H 1 ) and B L (0, T; L 2 ) L 2 (0, T; H 1 ). What about ε = 0? Try looking at more regular solutions and taking the limit ε 0 to get local existence Assume regularity and show that u(t) 0 as t (Moffatt)?
22 Selected references D. S. McCormick, J. C. Robinson and J. L. Rodrigo, Generalised Gagliardo Nirenberg inequalities using weak Lebesgue spaces and BMO, submitted. D. S. McCormick, J. C. Robinson and J. L. Rodrigo, Existence and uniqueness for a coupled parabolic-elliptic model with applications to magnetic relaxation, in preparation. H. K. Moffatt (1985), Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology. I. Fundamentals, J. Fluid Mech. 159, H. K. Moffatt (2009), Relaxation routes to steady Euler flows of complex topology, slides of talk given during MIRaW Day, Warwick, June M. Núñez (2007), The limit states of magnetic relaxation, J. Fluid Mech. 580, L. Grafakos (2008), Classical Fourier Analysis and Modern Fourier Analysis. Springer, New York, GTM 249 and 250. C. Bennett and R. Sharpley (1988), Interpolation of Operators. Academic Press, New York.
Decay in Time of Incompressible Flows
J. math. fluid mech. 5 (23) 231 244 1422-6928/3/3231-14 c 23 Birkhäuser Verlag, Basel DOI 1.17/s21-3-79-1 Journal of Mathematical Fluid Mechanics Decay in Time of Incompressible Flows Heinz-Otto Kreiss,
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 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 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 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 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 informationTHE L 2 -HODGE THEORY AND REPRESENTATION ON R n
THE L 2 -HODGE THEORY AND REPRESENTATION ON R n BAISHENG YAN Abstract. We present an elementary L 2 -Hodge theory on whole R n based on the minimization principle of the calculus of variations and some
More informationOn 2 d incompressible Euler equations with partial damping.
On 2 d incompressible Euler equations with partial damping. Wenqing Hu 1. (Joint work with Tarek Elgindi 2 and Vladimir Šverák 3.) 1. Department of Mathematics and Statistics, Missouri S&T. 2. Department
More informationSobolev spaces. May 18
Sobolev spaces May 18 2015 1 Weak derivatives The purpose of these notes is to give a very basic introduction to Sobolev spaces. More extensive treatments can e.g. be found in the classical references
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 informationu( x) = g( y) ds y ( 1 ) U solves u = 0 in U; u = 0 on U. ( 3)
M ath 5 2 7 Fall 2 0 0 9 L ecture 4 ( S ep. 6, 2 0 0 9 ) Properties and Estimates of Laplace s and Poisson s Equations In our last lecture we derived the formulas for the solutions of Poisson s equation
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 informationEXISTENCE AND REGULARITY RESULTS FOR SOME NONLINEAR PARABOLIC EQUATIONS
EXISTECE AD REGULARITY RESULTS FOR SOME OLIEAR PARABOLIC EUATIOS Lucio BOCCARDO 1 Andrea DALL AGLIO 2 Thierry GALLOUËT3 Luigi ORSIA 1 Abstract We prove summability results for the solutions of nonlinear
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 informationfor all subintervals I J. If the same is true for the dyadic subintervals I D J only, we will write ϕ BMO d (J). In fact, the following is true
3 ohn Nirenberg inequality, Part I A function ϕ L () belongs to the space BMO() if sup ϕ(s) ϕ I I I < for all subintervals I If the same is true for the dyadic subintervals I D only, we will write ϕ BMO
More informationThe incompressible Navier-Stokes equations in vacuum
The incompressible, Université Paris-Est Créteil Piotr Bogus law Mucha, Warsaw University Journées Jeunes EDPistes 218, Institut Elie Cartan, Université de Lorraine March 23th, 218 Incompressible Navier-Stokes
More informationThe continuity method
The continuity method The method of continuity is used in conjunction with a priori estimates to prove the existence of suitably regular solutions to elliptic partial differential equations. One crucial
More informationThe Dirichlet problem for non-divergence parabolic equations with discontinuous in time coefficients in a wedge
The Dirichlet problem for non-divergence parabolic equations with discontinuous in time coefficients in a wedge Vladimir Kozlov (Linköping University, Sweden) 2010 joint work with A.Nazarov Lu t u a ij
More informationPartial Differential Equations
Part II Partial Differential Equations Year 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2015 Paper 4, Section II 29E Partial Differential Equations 72 (a) Show that the Cauchy problem for u(x,
More informationMath The Laplacian. 1 Green s Identities, Fundamental Solution
Math. 209 The Laplacian Green s Identities, Fundamental Solution Let be a bounded open set in R n, n 2, with smooth boundary. The fact that the boundary is smooth means that at each point x the external
More informationExistence of Weak Solutions to a Class of Non-Newtonian Flows
Existence of Weak Solutions to a Class of Non-Newtonian Flows 1. Introduction and statement of the result. Ladyzhenskaya [8]. Newtonian : Air, other gases, water, motor oil, alcohols, simple hydrocarbon
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 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 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 informationJUHA KINNUNEN. Harmonic Analysis
JUHA KINNUNEN Harmonic Analysis Department of Mathematics and Systems Analysis, Aalto University 27 Contents Calderón-Zygmund decomposition. Dyadic subcubes of a cube.........................2 Dyadic cubes
More informationUniversität des Saarlandes. Fachrichtung 6.1 Mathematik
Universität des Saarlandes U N I V E R S I T A S S A R A V I E N I S S Fachrichtung 6.1 Mathematik Preprint Nr. 155 A posteriori error estimates for stationary slow flows of power-law fluids Michael Bildhauer,
More informationON LIQUID CRYSTAL FLOWS WITH FREE-SLIP BOUNDARY CONDITIONS. Chun Liu and Jie Shen
DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS Volume 7, Number2, April2001 pp. 307 318 ON LIQUID CRYSTAL FLOWS WITH FREE-SLIP BOUNDARY CONDITIONS Chun Liu and Jie Shen Department
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 informationPartial Differential Equations, 2nd Edition, L.C.Evans Chapter 5 Sobolev Spaces
Partial Differential Equations, nd Edition, L.C.Evans Chapter 5 Sobolev Spaces Shih-Hsin Chen, Yung-Hsiang Huang 7.8.3 Abstract In these exercises always denote an open set of with smooth boundary. As
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 informationA dual form of the sharp Nash inequality and its weighted generalization
A dual form of the sharp Nash inequality and its weighted generalization Elliott Lieb Princeton University Joint work with Eric Carlen, arxiv: 1704.08720 Kato conference, University of Tokyo September
More information1. Introduction In this paper we consider the solutions to the three-dimensional steady state Navier Stokes equations in the whole space R 3,
L p -SOLUTIONS OF THE STEADY-STATE NAVIER STOKES WITH ROUGH EXTERNAL FORCES CLAYTON BJORLAND, LORENZO BRANDOLESE, DRAGOŞ IFTIMIE, AND MARIA E. SCHONBEK Abstract. In this paper we address the existence,
More informationA note on some approximation theorems in measure theory
A note on some approximation theorems in measure theory S. Kesavan and M. T. Nair Department of Mathematics, Indian Institute of Technology, Madras, Chennai - 600 06 email: kesh@iitm.ac.in and mtnair@iitm.ac.in
More informationR. M. Brown. 29 March 2008 / Regional AMS meeting in Baton Rouge. Department of Mathematics University of Kentucky. The mixed problem.
mixed R. M. Department of Mathematics University of Kentucky 29 March 2008 / Regional AMS meeting in Baton Rouge Outline mixed 1 mixed 2 3 4 mixed We consider the mixed boundary value Lu = 0 u = f D u
More informationON WEAKLY NONLINEAR BACKWARD PARABOLIC PROBLEM
ON WEAKLY NONLINEAR BACKWARD PARABOLIC PROBLEM OLEG ZUBELEVICH DEPARTMENT OF MATHEMATICS THE BUDGET AND TREASURY ACADEMY OF THE MINISTRY OF FINANCE OF THE RUSSIAN FEDERATION 7, ZLATOUSTINSKY MALIY PER.,
More informationHeat kernels of some Schrödinger operators
Heat kernels of some Schrödinger operators Alexander Grigor yan Tsinghua University 28 September 2016 Consider an elliptic Schrödinger operator H = Δ + Φ, where Δ = n 2 i=1 is the Laplace operator in R
More informationFOURIER SERIES WITH THE CONTINUOUS PRIMITIVE INTEGRAL
Real Analysis Exchange Summer Symposium XXXVI, 2012, pp. 30 35 Erik Talvila, Department of Mathematics and Statistics, University of the Fraser Valley, Chilliwack, BC V2R 0N3, Canada. email: Erik.Talvila@ufv.ca
More informationThe Dirichlet s P rinciple. In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation:
Oct. 1 The Dirichlet s P rinciple In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation: 1. Dirichlet s Principle. u = in, u = g on. ( 1 ) If we multiply
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 informationSobolev Spaces. Chapter Hölder spaces
Chapter 2 Sobolev Spaces Sobolev spaces turn out often to be the proper setting in which to apply ideas of functional analysis to get information concerning partial differential equations. Here, we collect
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 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 informationINSTITUTE OF MATHEMATICS THE CZECH ACADEMY OF SCIENCES. Note on the fast decay property of steady Navier-Stokes flows in the whole space
INSTITUTE OF MATHEMATICS THE CZECH ACADEMY OF SCIENCES Note on the fast decay property of stea Navier-Stokes flows in the whole space Tomoyuki Nakatsuka Preprint No. 15-017 PRAHA 017 Note on the fast
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 informationRegularization by noise in infinite dimensions
Regularization by noise in infinite dimensions Franco Flandoli, University of Pisa King s College 2017 Franco Flandoli, University of Pisa () Regularization by noise King s College 2017 1 / 33 Plan of
More informationExistence and Continuation for Euler Equations
Existence and Continuation for Euler Equations David Driver Zhuo Min Harold Lim 22 March, 214 Contents 1 Introduction 1 1.1 Physical Background................................... 1 1.2 Notation and Conventions................................
More informationResearch Article Remarks on the Regularity Criterion of the Navier-Stokes Equations with Nonlinear Damping
Mathematical Problems in Engineering Volume 15, Article ID 194, 5 pages http://dx.doi.org/1.1155/15/194 Research Article Remarks on the Regularity Criterion of the Navier-Stokes Equations with Nonlinear
More informationCONNECTIONS BETWEEN A CONJECTURE OF SCHIFFER S AND INCOMPRESSIBLE FLUID MECHANICS
CONNECTIONS BETWEEN A CONJECTURE OF SCHIFFER S AND INCOMPRESSIBLE FLUID MECHANICS JAMES P. KELLIHER Abstract. We demonstrate connections that exists between a conjecture of Schiffer s (which is equivalent
More informationThe Dirichlet boundary problems for second order parabolic operators satisfying a Carleson condition
The Dirichlet boundary problems for second order parabolic operators satisfying a Carleson condition Sukjung Hwang CMAC, Yonsei University Collaboration with M. Dindos and M. Mitrea The 1st Meeting of
More informationHI CAMBRIDGE n S P UNIVERSITY PRESS
Infinite-Dimensional Dynamical Systems An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors JAMES C. ROBINSON University of Warwick HI CAMBRIDGE n S P UNIVERSITY PRESS Preface
More informationbe the set of complex valued 2π-periodic functions f on R such that
. Fourier series. Definition.. Given a real number P, we say a complex valued function f on R is P -periodic if f(x + P ) f(x) for all x R. We let be the set of complex valued -periodic functions f on
More informationTakens embedding theorem for infinite-dimensional dynamical systems
Takens embedding theorem for infinite-dimensional dynamical systems James C. Robinson Mathematics Institute, University of Warwick, Coventry, CV4 7AL, U.K. E-mail: jcr@maths.warwick.ac.uk Abstract. Takens
More informationA Critical Parabolic Sobolev Embedding via Littlewood-Paley Decomposition
International Mathematical Forum, Vol. 7, 01, no. 35, 1705-174 A Critical Parabolic Sobolev Embedding via Littlewood-Paley Decomposition H. Ibrahim Lebanese University, Faculty of Sciences Mathematics
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 informationTopics in Fourier analysis - Lecture 2.
Topics in Fourier analysis - Lecture 2. Akos Magyar 1 Infinite Fourier series. In this section we develop the basic theory of Fourier series of periodic functions of one variable, but only to the extent
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 informationDETERMINATION OF THE BLOW-UP RATE FOR THE SEMILINEAR WAVE EQUATION
DETERMINATION OF THE LOW-UP RATE FOR THE SEMILINEAR WAVE EQUATION y FRANK MERLE and HATEM ZAAG Abstract. In this paper, we find the optimal blow-up rate for the semilinear wave equation with a power nonlinearity.
More informationLiquid crystal flows in two dimensions
Liquid crystal flows in two dimensions Fanghua Lin Junyu Lin Changyou Wang Abstract The paper is concerned with a simplified hydrodynamic equation, proposed by Ericksen and Leslie, modeling the flow of
More informationAn introduction to the classical theory of the Navier Stokes equations. IMECC Unicamp, January 2010
An introduction to the classical theory of the Navier Stokes equations IMECC Unicamp, January 2 James C. Robinson Mathematics Institute University of Warwick Coventry CV4 7AL. UK. Email: j.c.robinson@warwick.ac.uk
More informationTHE INVISCID LIMIT FOR TWO-DIMENSIONAL INCOMPRESSIBLE FLUIDS WITH UNBOUNDED VORTICITY. James P. Kelliher
Mathematical Research Letters 11, 519 528 (24) THE INVISCID LIMIT FOR TWO-DIMENSIONAL INCOMPRESSIBLE FLUIDS WITH UNBOUNDED VORTICITY James P. Kelliher Abstract. In [C2], Chemin shows that solutions of
More informationHARMONIC ANALYSIS. Date:
HARMONIC ANALYSIS Contents. Introduction 2. Hardy-Littlewood maximal function 3. Approximation by convolution 4. Muckenhaupt weights 4.. Calderón-Zygmund decomposition 5. Fourier transform 6. BMO (bounded
More informationIMA Preprint Series # 2260
LONG-TIME BEHAVIOR OF HYDRODYNAMIC SYSTEMS MODELING THE NEMATIC LIUID CRYSTALS By Hao Wu Xiang Xu and Chun Liu IMA Preprint Series # 2260 ( June 2009 ) INSTITUTE FOR MATHEMATICS AND ITS APPLICATIONS UNIVERSITY
More informationHarmonic Analysis Homework 5
Harmonic Analysis Homework 5 Bruno Poggi Department of Mathematics, University of Minnesota November 4, 6 Notation Throughout, B, r is the ball of radius r with center in the understood metric space usually
More informationEuler Equations: local existence
Euler Equations: local existence Mat 529, Lesson 2. 1 Active scalars formulation We start with a lemma. Lemma 1. Assume that w is a magnetization variable, i.e. t w + u w + ( u) w = 0. If u = Pw then u
More informationEXISTENCE OF WEAK SOLUTIONS FOR A NONUNIFORMLY ELLIPTIC NONLINEAR SYSTEM IN R N. 1. Introduction We study the nonuniformly elliptic, nonlinear system
Electronic Journal of Differential Equations, Vol. 20082008), No. 119, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu login: ftp) EXISTENCE
More informationSolution Sheet 3. Solution Consider. with the metric. We also define a subset. and thus for any x, y X 0
Solution Sheet Throughout this sheet denotes a domain of R n with sufficiently smooth boundary. 1. Let 1 p
More informationSELF-ADJOINTNESS OF SCHRÖDINGER-TYPE OPERATORS WITH SINGULAR POTENTIALS ON MANIFOLDS OF BOUNDED GEOMETRY
Electronic Journal of Differential Equations, Vol. 2003(2003), No.??, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) SELF-ADJOINTNESS
More informationZdzislaw Brzeźniak. Department of Mathematics University of York. joint works with Mauro Mariani (Roma 1) and Gaurav Dhariwal (York)
Navier-Stokes equations with constrained L 2 energy of the solution Zdzislaw Brzeźniak Department of Mathematics University of York joint works with Mauro Mariani (Roma 1) and Gaurav Dhariwal (York) Stochastic
More informationSHARP BOUNDARY TRACE INEQUALITIES. 1. Introduction
SHARP BOUNDARY TRACE INEQUALITIES GILES AUCHMUTY Abstract. This paper describes sharp inequalities for the trace of Sobolev functions on the boundary of a bounded region R N. The inequalities bound (semi-)norms
More informationHigher derivatives estimate for the 3D Navier-Stokes equation
Higher derivatives estimate for the 3D Navier-Stokes equation Alexis Vasseur Abstract: In this article, a non linear family of spaces, based on the energy dissipation, is introduced. This family bridges
More informationVANISHING VISCOSITY IN THE PLANE FOR VORTICITY IN BORDERLINE SPACES OF BESOV TYPE
VANISHING VISCOSITY IN THE PLANE FOR VORTICITY IN BORDERLINE SPACES OF BESOV TYPE ELAINE COZZI AND JAMES P. KELLIHER Abstract. The existence and uniqueness of solutions to the Euler equations for initial
More informationSYMMETRY OF POSITIVE SOLUTIONS OF SOME NONLINEAR EQUATIONS. M. Grossi S. Kesavan F. Pacella M. Ramaswamy. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 12, 1998, 47 59 SYMMETRY OF POSITIVE SOLUTIONS OF SOME NONLINEAR EQUATIONS M. Grossi S. Kesavan F. Pacella M. Ramaswamy
More informationOn Friedrichs inequality, Helmholtz decomposition, vector potentials, and the div-curl lemma. Ben Schweizer 1
On Friedrichs inequality, Helmholtz decomposition, vector potentials, and the div-curl lemma Ben Schweizer 1 January 16, 2017 Abstract: We study connections between four different types of results that
More informationON THE STRONG SOLUTIONS OF THE INHOMOGENEOUS INCOMPRESSIBLE NAVIER-STOKES EQUATIONS IN A THIN DOMAIN
ON THE STRONG SOLUTIONS OF THE INHOMOGENEOUS INCOMPRESSIBLE NAVIER-STOKES EQUATIONS IN A THIN DOMAIN XIAN LIAO Abstract. In this work we will show the global existence of the strong solutions of the inhomogeneous
More informationEXISTENCE OF SOLUTIONS TO THE CAHN-HILLIARD/ALLEN-CAHN EQUATION WITH DEGENERATE MOBILITY
Electronic Journal of Differential Equations, Vol. 216 216), No. 329, pp. 1 22. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS TO THE CAHN-HILLIARD/ALLEN-CAHN
More informationhal , version 6-26 Dec 2012
ON THE UNIQUENESS IN THE 3D NAVIER-STOKES EQUATIONS ABDEHAFID YOUNSI Abstract. In this paper, we give a new regularity criterion on the uniqueness results of weak solutions for the 3D Navier-Stokes equations
More informationON THE UNIQUENESS IN THE 3D NAVIER-STOKES EQUATIONS
ON THE UNIQUENESS IN THE 3D NAVIER-STOKES EQUATIONS Abdelhafid Younsi To cite this version: Abdelhafid Younsi. ON THE UNIQUENESS IN THE 3D NAVIER-STOKES EQUATIONS. 4 pages. 212. HAL Id:
More informationarxiv: v2 [math.ap] 14 May 2016
THE MINKOWSKI DIMENSION OF INTERIOR SINGULAR POINTS IN THE INCOMPRESSIBLE NAVIER STOKES EQUATIONS YOUNGWOO KOH & MINSUK YANG arxiv:16.17v2 [math.ap] 14 May 216 ABSTRACT. We study the possible interior
More informationZdzislaw Brzeźniak. Department of Mathematics University of York. joint works with Mauro Mariani (Roma 1) and Gaurav Dhariwal (York)
Navier-Stokes equations with constrained L 2 energy of the solution Zdzislaw Brzeźniak Department of Mathematics University of York joint works with Mauro Mariani (Roma 1) and Gaurav Dhariwal (York) LMS
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 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 information1 Math 241A-B Homework Problem List for F2015 and W2016
1 Math 241A-B Homework Problem List for F2015 W2016 1.1 Homework 1. Due Wednesday, October 7, 2015 Notation 1.1 Let U be any set, g be a positive function on U, Y be a normed space. For any f : U Y let
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 informationA Priori Bounds, Nodal Equilibria and Connecting Orbits in Indefinite Superlinear Parabolic Problems
A Priori Bounds, Nodal Equilibria and Connecting Orbits in Indefinite Superlinear Parabolic Problems Nils Ackermann Thomas Bartsch Petr Kaplický Pavol Quittner Abstract We consider the dynamics of the
More informationPOINCARÉ S INEQUALITY AND DIFFUSIVE EVOLUTION EQUATIONS
POINCARÉ S INEQUALITY AND DIFFUSIVE EVOLUTION EQUATIONS CLAYTON BJORLAND AND MARIA E. SCHONBEK Abstract. This paper addresses the question of change of decay rate from exponential to algebraic for diffusive
More informationMathematical analysis of the stationary Navier-Stokes equations
Mathematical analysis of the Department of Mathematics, Sogang University, Republic of Korea The 3rd GCOE International Symposium Weaving Science Web beyond Particle Matter Hierarchy February 17-19, 2011,
More informationA REMARK ON MINIMAL NODAL SOLUTIONS OF AN ELLIPTIC PROBLEM IN A BALL. Olaf Torné. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 24, 2004, 199 207 A REMARK ON MINIMAL NODAL SOLUTIONS OF AN ELLIPTIC PROBLEM IN A BALL Olaf Torné (Submitted by Michel
More informationFree energy estimates for the two-dimensional Keller-Segel model
Free energy estimates for the two-dimensional Keller-Segel model dolbeaul@ceremade.dauphine.fr CEREMADE CNRS & Université Paris-Dauphine in collaboration with A. Blanchet (CERMICS, ENPC & Ceremade) & B.
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 informationHilbert Spaces. Contents
Hilbert Spaces Contents 1 Introducing Hilbert Spaces 1 1.1 Basic definitions........................... 1 1.2 Results about norms and inner products.............. 3 1.3 Banach and Hilbert spaces......................
More informationLocal Well-Posedness for the Hall-MHD Equations with Fractional Magnetic Diffusion
J. Math. Fluid Mech. 17 (15), 67 638 c 15 Springer Basel 14-698/15/467-1 DOI 1.17/s1-15--9 Journal of Mathematical Fluid Mechanics Local Well-Posedness for the Hall-MHD Equations with Fractional Magnetic
More informationP(E t, Ω)dt, (2) 4t has an advantage with respect. to the compactly supported mollifiers, i.e., the function W (t)f satisfies a semigroup law:
Introduction Functions of bounded variation, usually denoted by BV, have had and have an important role in several problems of calculus of variations. The main features that make BV functions suitable
More informationThe Navier-Stokes Equations with Time Delay. Werner Varnhorn. Faculty of Mathematics University of Kassel, Germany
The Navier-Stokes Equations with Time Delay Werner Varnhorn Faculty of Mathematics University of Kassel, Germany AMS: 35 (A 35, D 5, K 55, Q 1), 65 M 1, 76 D 5 Abstract In the present paper we use a time
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 informationREMARKS ON THE VANISHING OBSTACLE LIMIT FOR A 3D VISCOUS INCOMPRESSIBLE FLUID
REMARKS ON THE VANISHING OBSTACLE LIMIT FOR A 3D VISCOUS INCOMPRESSIBLE FLUID DRAGOŞ IFTIMIE AND JAMES P. KELLIHER Abstract. In [Math. Ann. 336 (2006), 449-489] the authors consider the two dimensional
More informationOn Pressure Stabilization Method and Projection Method for Unsteady Navier-Stokes Equations 1
On Pressure Stabilization Method and Projection Method for Unsteady Navier-Stokes Equations 1 Jie Shen Department of Mathematics, Penn State University University Park, PA 1682 Abstract. We present some
More informationA RECURRENCE THEOREM ON THE SOLUTIONS TO THE 2D EULER EQUATION
ASIAN J. MATH. c 2009 International Press Vol. 13, No. 1, pp. 001 006, March 2009 001 A RECURRENCE THEOREM ON THE SOLUTIONS TO THE 2D EULER EQUATION Y. CHARLES LI Abstract. In this article, I will prove
More informationJ.-L. Guermond 1 FOR TURBULENT FLOWS LARGE EDDY SIMULATION MODEL A HYPERVISCOSITY SPECTRAL
A HYPERVISCOSITY SPECTRAL LARGE EDDY SIMULATION MODEL FOR TURBULENT FLOWS J.-L. Guermond 1 Collaborator: S. Prudhomme 2 Mathematical Aspects of Computational Fluid Dynamics Oberwolfach November 9th to
More informationSpace-time Finite Element Methods for Parabolic Evolution Problems
Space-time Finite Element Methods for Parabolic Evolution Problems with Variable Coefficients Ulrich Langer, Martin Neumüller, Andreas Schafelner Johannes Kepler University, Linz Doctoral Program Computational
More informationRANDOM PROPERTIES BENOIT PAUSADER
RANDOM PROPERTIES BENOIT PAUSADER. Quasilinear problems In general, one consider the following trichotomy for nonlinear PDEs: A semilinear problem is a problem where the highest-order terms appears linearly
More information