ANDREJ ZLATOŠ. 2π (x 2, x 1 ) x 2 on R 2 and extending ω
|
|
- Virginia Hardy
- 5 years ago
- Views:
Transcription
1 EXPONENTIAL GROWTH OF THE VORTIITY GRADIENT FOR THE EULER EQUATION ON THE TORUS ANDREJ ZLATOŠ Abstract. We prove that there are solutions to the Euler equation on the torus with 1,α vorticity and smooth except at one point such that the vorticity gradient grows in L at least exponentially as t. The same result is shown to hold for the vorticity Hessian and smooth solutions. Our proofs use a version of a recent result by Kiselev and Šverák [5]. 1. Introduction Let (T = [ 1, 1 be the two-dimensional torus (i.e., we identify opposite sides of the square and consider the Euler equation on (T, in vorticity formulation: ω t + u ω =, ω(, = ω. (1.1 The velocity u is found from the vorticity ω via the Biot-Savart law u(t, x = 1 (x y n, + y 1 + n 1 ω(t, ydy, [ 1,1] x y n n Z obtained by taking the Biot-Savart kernel K(x = 1 (x, x on R and extending ω periodically. Initial data ω will here be 1 and odd in both and x, hence the latter property will hold for all t, as well as ω(t, L = ω L. Global regularity of bounded solutions to (1.1 was first proved by Wolibner [8] and Hölder [4]. We consider here the question of how fast the gradient of ω can grow in L as t. The well known upper bound is double-exponential ω(t, L e et but it has been a long standing open question whether this is attainable. The best infinite time result in the plane or on the torus (i.e., domains without a boundary so far has been the proof of the possibility of super-linear growth for smooth solutions by Denisov [1]. He also proved that doubleexponential growth is possible on arbitrarily long finite time intervals [], and constructed patch solutions to the D Euler equation with a regular prescribed stirring for which the distance between two approaching patches decreases double-exponentially in time [3]. For domains with boundaries (and no flow boundary condition, Yudovich [9, 1] and Nadirashvili [7] provided examples with unbounded growth and at least linear growth, respectively. These results have been dramatically improved in a striking recent work by Kiselev and Šverák [5], who proved the possibility of infinite time double-exponential growth of the vorticity gradient in a disc, thereby answering in the affirmative the above open question in 1
2 ANDREJ ZLATOŠ this setting. The boundary is crucial in [5] and the double-exponential growth is proved to occur on it as well. Related to this, we note that the double-exponential upper bound is only known for D domains with regular boundaries. In fact, Kiselev and the author [6] proved that there are domains whose boundary is smooth except at two points, and on which some solutions to the Euler equation with smooth initial data blow up in finite time. We refer the reader to [5] for more history and further references related to (1.1. In the present paper we prove that on the torus, at least exponential growth of the vorticity gradient happens for some 1,α initial data, as well as that such growth is possible for the vorticity Hessian for smooth initial data. Our proof uses a sharper version of a key result from [5] (see Lemma.1 below, applied on the torus instead of the disc. Theorem 1.1. (i For any α (, 1 and A <, there is ω 1,α ((T with ω L 1 and there is T such that the solution of (1.1 satisfies for all T T, sup ω(t, L ([, exp( AT ] e AT. t T (ii For any A <, there is ω ((T with ω L 1 and there is T such that the solution of (1.1 satisfies for all T T, sup D ω(t, L ([, exp( AT ] e AT. t T Our ω will be very simple. For instance, in (i it can be smooth except at the origin, with ω (r, φ = r 1+α sin(φ (in polar coordinates near the origin. Then ω remains in 1,α, and since u(t, = for all t > by symmetry (oddness of ω in, x, it follows that both u, ω are smooth except at the origin at all times. We also note that we will have ω L = 1, but as in [5], this can be arbitrarily small. Acknowledgements. The author thanks Sergey Denisov and Alexander Kiselev for useful discussions and comments. He also acknowledges partial support by NSF grants DMS and DMS Proof of Theorem 1.1 For x [, 1] we let x := (, x, x := (, x, and Q(x := [, 1] [x, 1]. In what follows, will always be some universal constant which may change between inequalities. Lemma.1. Let ω(t, L ((T be odd in both and x. If, x [, 1 ], then ( 4 u j (t, x = ( 1 j y 1 y y ω(t, ydy + B j(t, x x 4 j (j = 1, (.1 Q(x where, with some universal, ( { ( B 1 (t, x ω(t, L 1 + min log 1 + x ( { ( B (t, x ω(t, L 1 + min log 1 + x } ω(t, L, x ([,x ] ω(t, L, ω(t, L ([, ] ω(t, L, (. }. (.3
3 EXPONENTIAL GROWTH FOR EULER EQUATIONS 3 Remark. If c x (resp. cx for some c <, then the min can obviously be dropped in (. (resp. (.3 as long as = (c. This version of these formulas was proved in [5] on the disc. In that case Q(x can be replaced by Q(x as well, which is done in [5]. Proof. Let us only consider j = 1 because j = follows by symmetry: K shows that if ω(t, x := ω(t, x,, then ũ(t, x = (u (t, x,, u 1 (t, x,. The Biot-Savart law gives u 1 (t, x = [ y1 ( n 1 (x y n x (y + n x (ỹ + n y ] 1( n 1 (x + y n ω(t, ydy, x (ȳ + n x ( y + n n Z [,1] (.4 by using the symmetries ω(t, ỹ = ω(t, y and then ω(t, ȳ = ω(t, y to express the integral over [ 1, 1] via that over [, 1]. Let us first consider the right-hand side of (.4 with the term n = (, removed. The first term in the integral equals (recall that, x [, 1 ] y 1 n 1 (x y n x (y + n x (ỹ + n + O( n 3. We combine it with the same term for ñ = ( n 1, n to obtain 3y 1 n 1 (x y n [n 1 (x y n + x 1 y 1 + 4n 1] x (y + n x (ỹ + n x (y + ñ x (ỹ + ñ + O( n 3 = O( n 3. This means that n (, y 1 ( n 1 (x y n ω(t, ydy [,1] x (y + n x (ỹ + n ω(t, L. An identical argument proves this also for the second term in (.4. We therefore only need to consider the term with n = (,, which is times [ y1 (x y x y x ỹ y ] 1(x + y ω(t, ydy. (.5 x ȳ x + y Q(x [,1] We will show that this equals to 4 times the integral in (.1, plus an error controlled by the right-hand side of (., thus proving (.1. We will again only consider the first term in the integral since the second will be handled in the same way. We separate the integral into either 3 or 4 regions. If x, then these regions will be Q(x, [, ] [, 1], and [, 1] [, x ]. If < x, we also split the last region into [, x ] [, x ] and [x, 1] [, x ]. The 3 region case is parallel to the treatment in [5] (where the domain is a disc. In the 4 region case we need to obtain an extra estimate for the integral over [, x ] [, x ] (this is not necessary for the second term in (.5. We start with Q(x, where we have y 1 x x y x ỹ dy x 1 Q(x y dy x dr. 3 x r
4 4 ANDREJ ZLATOŠ In Q(x also y 1 y x y x ỹ y 1y y = O( x y 5 = O( x y 3, 4 y 8 so the integral of the absolute value of this difference over Q(x is also bounded by 1 x x dr. r Hence integration of the first term in (.5 over Q(x gives ( times the integral in (.1, plus an error bounded by ω(t, L. (Integration of the second term in (.5 gives the same, whence the factor of 4 in (.1. Next integrate over [, ] [, 1]. After substituting z j := y j x j, the absolute value of the integral of the first term in (.5 can be bounded by ω(t, L times x1 1 z (z 1 + z (x 1 + z dz dz 1 = 1 x 1 + z x 1 + z arctan z dz arctan 1. Finally, the corresponding integral over [, 1] [, x ] is bounded by ω(t, L 1 x z 1 z x ( (z1 + z dz z dz 1 log 1 + x dz log x 1 + x x 1 times This gives the first term in the min in (.. If x, then we are done because the min is 1 and can be absorbed in the 1 in (.. So let us assume < x, and perform the above integration over [x, 1] [, x ] instead of [, 1] [, x ]. We see that the integral is bounded by ω(t, L log(1 + x x = ω(t, L log, and therefore it only remains to consider the integral over the remaining set [, x ] [, x ]. Let us denote M := ω(t, L ([,x ] and write ω(t, y = v(t, y 1 + w(t, y, where v(t, y 1 := ω(t, y 1, x and so w(t, y = ω(t, y 1, y ω(t, y 1, x M x y. For y 1 [, x ] we have with z := y x, x y 1 (x y x y x ỹ v(t, y 1dy = x x and we also have x x y 1 (x y w(t, y x y x ỹ dy x dy 1 M. y 1 z [( y 1 + z ][( + y 1 + z ] v(t, y 1dz = x y 1 z x (y1 + z dzdy 1 M This gives the second term in the min in (.. We note that for the second term in (.5, one can always integrate over [, 1] [, x ] because the substitution z 1 := y 1 and z := y + x yields 1 3x z 1 z x (z1 + z dz dz 1 3x z x x 1 + z dz log x 1 + 3x x 1 + x. z x 1 + z dz Mx.
5 EXPONENTIAL GROWTH FOR EULER EQUATIONS 5 Proof of Theorem 1.1. (i Given α and A, pick a function ω : (T [ 1, 1] which is odd in both and x, non-negative on [, 1] and equal to 1 on a subset of [, 1] of measure 1 δ (for some δ (, 1 to be chosen later, with ω 1 1,α ((T loc ((T \ {} and ω (s, s = s 1+α for s [, δ]. For instance, on B δ ( we could have in polar coordinates ω (r, φ = (r/ 1+α sin(φ. Take any T T := 1 log δ so that e AT δ, let X(t solve X (t = u(t, X(t with A X( = (e aat, e aat for some a > 1 to be chosen later, and let T := min{t, T }, with T the exit time of X from the square [, e AT ]. Obviously, for all t. Let us also assume that ω(t, X(t = ω (X( = e a(1+αat (.6 sup ω(t, L ([, exp( AT ] e AT (.7 t T because otherwise we are done. Since X(t [, e AT ] for t T, x ω(t, L ([,x ] 1 (.8 in (. when t T and x = X(t (and the same estimate applies to (.3. We then have (.1 with B j (t, X(t for t T (recall that ω(t, L = ω L = 1. A crucial observation of [5] is that ω on [, 1] and ω = 1 on a subset of [, 1] of measure 1 δ (along with the distribution function of ω(t, being the same for all t guarantees that the integral in (.1 is no less than 1 1 log δ when δ < and x [, 1 δ], for some universal >. (If instead we had odd ω : (T [ ε, ε] equal to ε on a subset of [, 1] of measure 1 δ, then this would be ε log δ, and our proof would be unchanged. So if we denote by k(t the value of the integral in (.1 for x = X(t, multiplied by 4, then k(t 1 log δ for t T. Hence we have for t T, ( 1 u 1 (t, X(t log δ ( 1 u (t, X(t log δ X 1 (t, (.9 X (t. (.1 If we take δ < e, it follows that X 1 (T < e AT and T (a 1AT ( 1 log δ 1. We will in fact pick δ e (a 1A so that also (a 1A( 1 log δ 1 < 1. Hence T = T < T and X (T = e AT. In addition, d dt [log X 1(t + log X (t] (.11 for t T by Lemma.1 and (.8. Therefore, log X 1 (T log X 1 ( log X (T + log X ( + T [ aa + A + ]T. (.1 But this, (.6, and ω(t,, e AT = give log ω(t, L ([,exp( AT ] log ω(t, X(T X 1 (T [a(1 αa A ]T,
6 6 ANDREJ ZLATOŠ which equals AT if we pick a := A + (1 αa and then δ as above. The proof of (i is finished because T T. (ii Given A, pick ω : (T [ 1, 1] which is odd in both and x, non-negative on [, 1] and equal to 1 on a subset of [, 1] of measure 1 δ (for some δ (, 1 to be chosen 1 later, smooth, and with ω (, x = sin 3 ( sin(x when min{, x } δ. 4 Take any T T := 1 log δ so that e AT δ, let X(t solve A 4 4 X (t = u(t, X(t with X( = (e AT, e (a 1AT for some a > 1 to be chosen later, and let T := min{t, T }, with T the exit time of X from the square [, e AT ]. Obviously, ω(t, X(t = ω (X( = sin 3 (e AT sin(e (a 1AT e (a+at (.13 for all t. Let us also assume (.7 (because otherwise we are done, using ω(t, =. As above, we obtain (.1 with B j (t, X(t for t T. We thus again have (.9 (.11 for t T, as well as X (T = e AT > X 1 (T and T = T < T, provided we pick δ < e (a A. So (.1 follows as well and then by (.13, log sup s [,exp([ aa+a+]t ] ω x1 (T, s, e AT log ω(t, X(T X 1 (T [3A + ]T. (.14 The result will follow if we can show that ω x1 (T,, e AT =, because this and (.14 imply provided a 1 + A Let v(t, x := ω x1 (t, x. Then log D ω(t, L ([,exp( AT ] [aa 4A ]T, (so that aa + A + A. Then we only need to pick a := 5A + A. v t + u 1 v x1 + u v x + (u 1 x1 v + (u x1 ω x =. We have u 1 (t,, x = = ω(t,, x by symmetry, so also ω x (t,, x =. This shows that if we denote w(t, s := v(t,, s for s T, then for (t, s R + T, w t + u (t,, sw s + (u 1 x1 (t,, sw =. Since w(, s = (ω x1 (, s = and u is smooth, it follows that w. Thus we indeed obtain ω x1 (T,, e AT =, and the proof of (ii is finished. References [1] S. Denisov, Infinite superlinear growth of the gradient for the two-dimensional Euler equation. Discrete ontin. Dyn. Syst. A 3 (9, [] S. Denisov, Double-exponential growth of the vorticity gradient for the two-dimensional Euler equation Proceedings of the AMS, to appear. [3] S. Denisov, The sharp corner formation in D Euler dynamics of patches: infinite double exponential rate of merging, Arch. Rational Mech. Anal., to appear.
7 EXPONENTIAL GROWTH FOR EULER EQUATIONS 7 [4] E. Hölder, Über die unbeschränkte Fortsetzbarkeit einer stetigen ebenen Bewegung in einer unbegrentzten inkompressiblen Flüssigkeit (German, Math. Z. 37 (1933, [5] A. Kiselev and V. Šverák, Small scale creation for solutions of the incompressible two dimensional Euler equation, preprint. [6] A. Kiselev and A. Zlatoš, Blow up for the D Euler equation on some bounded domains, preprint. [7] N. S. Nadirashvili, Wandering solutions of the two-dimensional Euler equation (Russian, Funktsional. Anal. i Prilozhen. 5 (1991, 7 71; translation in Funct. Anal. Appl. 5 (1991, 1. [8] W. Wolibner, Un theorème sur l existence du mouvement plan d un fluide parfait, homogène, incompressible, pendant un temps infiniment long (French, Mat. Z. 37 (1933, [9] V. I. Judovič, The loss of smoothness of the solutions of Euler equations with time (Russian, Dinamika Splošn. Sredy 16 (1974, [1] V. I. Yudovich, On the loss of smoothness of the solutions of the Euler equations and the inherent instability of flows of an ideal fluid, haos 1 (, Department of Mathematics, University of Wisconsin, Madison, WI 5376, USA zlatos@math.wisc.edu
Infinite-time Exponential Growth of the Euler Equation on Two-dimensional Torus
Infinite-time Exponential Growth of the Euler Equation on Two-dimensional Torus arxiv:1608.07010v1 [math.ap] 5 Aug 016 Zhen Lei Jia Shi August 6, 016 Abstract For any A >, we construct solutions to the
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 informationLACK OF HÖLDER REGULARITY OF THE FLOW FOR 2D EULER EQUATIONS WITH UNBOUNDED VORTICITY. 1. Introduction
LACK OF HÖLDER REGULARITY OF THE FLOW FOR 2D EULER EQUATIONS WITH UNBOUNDED VORTICITY JAMES P. KELLIHER Abstract. We construct a class of examples of initial vorticities for which the solution to the Euler
More informationON THE EVOLUTION OF COMPACTLY SUPPORTED PLANAR VORTICITY
ON THE EVOLUTION OF COMPACTLY SUPPORTED PLANAR VORTICITY Dragoş Iftimie Thomas C. Sideris IRMAR Department of Mathematics Université de Rennes University of California Campus de Beaulieu Santa Barbara,
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 informationFinite time singularity for the modified SQG patch equation
Finite time singularity for the modified SQG patch equation Alexander Kiselev Lenya Ryzhik Yao Yao Andrej Zlatoš September 4, 2015 Abstract It is well known that the incompressible Euler equations in two
More informationOn the Uniqueness of Weak Solutions to the 2D Euler Equations
On the Uniqueness of Weak Solutions to the 2D Euler Equations (properties of the flow) 2009 SIAM Conference on Analysis of PDEs University of California, Riverside 9 December 2009 Bounded vorticity and
More informationTHE POINCARÉ RECURRENCE PROBLEM OF INVISCID INCOMPRESSIBLE FLUIDS
ASIAN J. MATH. c 2009 International Press Vol. 13, No. 1, pp. 007 014, March 2009 002 THE POINCARÉ RECURRENCE PROBLEM OF INVISCID INCOMPRESSIBLE FLUIDS Y. CHARLES LI Abstract. Nadirashvili presented a
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 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 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 informationPeriodic orbits of the ABC flow with A = B = C = 1
Periodic orbits of the ABC flow with A = B = C = 1 Jack Xin, Yifeng Yu, Andrej Zlatoš Abstract In this paper, we prove that the celebrated Arnold-Beltrami-Childress (ABC) flow with parameters A = B = C
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 informationStriated Regularity of Velocity for the Euler Equations
Striated Regularity of Velocity for the Euler Equations JIM KELLIHER 1 with HANTAEK BAE 2 1 University of California Riverside 2 Ulsan National Institute of Science and Technology, Korea 2015 SIAM Conference
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 informationON THE BEHAVIOR OF A CAPILLARY SURFACE AT A RE-ENTRANT CORNER
PACIFIC JOURNAL OF MATHEMATICS Vol 88, No 2, 1980 ON THE BEHAVIOR OF A CAPILLARY SURFACE AT A RE-ENTRANT CORNER NICHOLAS J KOREVAAR Changes in a domain's geometry can force striking changes in the capillary
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 informationSome recent results on controllability of coupled parabolic systems: Towards a Kalman condition
Some recent results on controllability of coupled parabolic systems: Towards a Kalman condition F. Ammar Khodja Clermont-Ferrand, June 2011 GOAL: 1 Show the important differences between scalar and non
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 informationTWO EXISTENCE RESULTS FOR THE VORTEX-WAVE SYSTEM
TWO EXISTENCE RESULTS FOR THE VORTEX-WAVE SYSTEM EVELYNE MIOT Abstract. The vortex-wave system is a coupling of the two-dimensional Euler equations for the vorticity together with the point vortex system.
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 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 informationarxiv: v1 [math.ap] 5 Nov 2018
STRONG CONTINUITY FOR THE 2D EULER EQUATIONS GIANLUCA CRIPPA, ELIZAVETA SEMENOVA, AND STEFANO SPIRITO arxiv:1811.01553v1 [math.ap] 5 Nov 2018 Abstract. We prove two results of strong continuity with respect
More informationPiecewise Smooth Solutions to the Burgers-Hilbert Equation
Piecewise Smooth Solutions to the Burgers-Hilbert Equation Alberto Bressan and Tianyou Zhang Department of Mathematics, Penn State University, University Park, Pa 68, USA e-mails: bressan@mathpsuedu, zhang
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 local existence for an active scalar equation in critical regularity setting
On the local existence for an active scalar equation in critical regularity setting Walter Rusin Department of Mathematics, Oklahoma State University, Stillwater, OK 7478 Fei Wang Department of Mathematics,
More informationControl, Stabilization and Numerics for Partial Differential Equations
Paris-Sud, Orsay, December 06 Control, Stabilization and Numerics for Partial Differential Equations Enrique Zuazua Universidad Autónoma 28049 Madrid, Spain enrique.zuazua@uam.es http://www.uam.es/enrique.zuazua
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 informationREGULARITY FOR 3D NAVIER-STOKES EQUATIONS IN TERMS OF TWO COMPONENTS OF THE VORTICITY
lectronic Journal of Differential quations, Vol. 2010(2010), No. 15, pp. 1 7. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu RGULARITY FOD NAVIR-STOKS
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 informationQUENCHING AND PROPAGATION OF COMBUSTION WITHOUT IGNITION TEMPERATURE CUTOFF
QUENCHING AND PROPAGATION OF COMBUSTION WITHOUT IGNITION TEMPERATURE CUTOFF ANDREJ ZLATOŠ Abstract. We study a reaction-diffusion equation in the cylinder Ω = R T m, with combustion-type reaction term
More informationOn the local well-posedness of compressible viscous flows with bounded density
On the local well-posedness of compressible viscous flows with bounded density Marius Paicu University of Bordeaux joint work with Raphaël Danchin and Francesco Fanelli Mathflows 2018, Porquerolles September
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 informationMemoirs on Differential Equations and Mathematical Physics
Memoirs on Differential Equations and Mathematical Physics Volume 41, 27, 27 42 Robert Hakl and Sulkhan Mukhigulashvili ON A PERIODIC BOUNDARY VALUE PROBLEM FOR THIRD ORDER LINEAR FUNCTIONAL DIFFERENTIAL
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 informationRegularity of the density for the stochastic heat equation
Regularity of the density for the stochastic heat equation Carl Mueller 1 Department of Mathematics University of Rochester Rochester, NY 15627 USA email: cmlr@math.rochester.edu David Nualart 2 Department
More informationOn Liouville type theorems for the steady Navier-Stokes equations in R 3
On Liouville type theorems for the steady Navier-Stokes equations in R 3 arxiv:604.07643v [math.ap] 6 Apr 06 Dongho Chae and Jörg Wolf Department of Mathematics Chung-Ang University Seoul 56-756, Republic
More informationAn Inverse Problem for Gibbs Fields with Hard Core Potential
An Inverse Problem for Gibbs Fields with Hard Core Potential Leonid Koralov Department of Mathematics University of Maryland College Park, MD 20742-4015 koralov@math.umd.edu Abstract It is well known that
More informationh(x) lim H(x) = lim Since h is nondecreasing then h(x) 0 for all x, and if h is discontinuous at a point x then H(x) > 0. Denote
Real Variables, Fall 4 Problem set 4 Solution suggestions Exercise. Let f be of bounded variation on [a, b]. Show that for each c (a, b), lim x c f(x) and lim x c f(x) exist. Prove that a monotone function
More informationA RELATIONSHIP BETWEEN THE DIRICHLET AND REGULARITY PROBLEMS FOR ELLIPTIC EQUATIONS. Zhongwei Shen
A RELATIONSHIP BETWEEN THE DIRICHLET AND REGULARITY PROBLEMS FOR ELLIPTIC EQUATIONS Zhongwei Shen Abstract. Let L = diva be a real, symmetric second order elliptic operator with bounded measurable coefficients.
More informationNotes on uniform convergence
Notes on uniform convergence Erik Wahlén erik.wahlen@math.lu.se January 17, 2012 1 Numerical sequences We begin by recalling some properties of numerical sequences. By a numerical sequence we simply mean
More informationThe Euler Equations in Planar Domains with Corners
The Euler Equations in Planar Domains with Corners Christophe Lacave and Andrej Zlatoš March 18, 219 Abstract When the velocity field is not a priori known to be globally almost Lipschitz, global uniqueness
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 informationHOMEOMORPHISMS OF BOUNDED VARIATION
HOMEOMORPHISMS OF BOUNDED VARIATION STANISLAV HENCL, PEKKA KOSKELA AND JANI ONNINEN Abstract. We show that the inverse of a planar homeomorphism of bounded variation is also of bounded variation. In higher
More informationTRAVELING WAVES IN 2D REACTIVE BOUSSINESQ SYSTEMS WITH NO-SLIP BOUNDARY CONDITIONS
TRAVELING WAVES IN 2D REACTIVE BOUSSINESQ SYSTEMS WITH NO-SLIP BOUNDARY CONDITIONS PETER CONSTANTIN, MARTA LEWICKA AND LENYA RYZHIK Abstract. We consider systems of reactive Boussinesq equations in two
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 informationGlobal well-posedness for the critical 2D dissipative quasi-geostrophic equation
Invent. math. 167, 445 453 (7) DOI: 1.17/s-6--3 Global well-posedness for the critical D dissipative quasi-geostrophic equation A. Kiselev 1, F. Nazarov, A. Volberg 1 Department of Mathematics, University
More informationNonlinear aspects of Calderón-Zygmund theory
Ancona, June 7 2011 Overture: The standard CZ theory Consider the model case u = f in R n Overture: The standard CZ theory Consider the model case u = f in R n Then f L q implies D 2 u L q 1 < q < with
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 informationRichard F. Bass Krzysztof Burdzy University of Washington
ON DOMAIN MONOTONICITY OF THE NEUMANN HEAT KERNEL Richard F. Bass Krzysztof Burdzy University of Washington Abstract. Some examples are given of convex domains for which domain monotonicity of the Neumann
More informationarxiv: v2 [math.fa] 17 May 2016
ESTIMATES ON SINGULAR VALUES OF FUNCTIONS OF PERTURBED OPERATORS arxiv:1605.03931v2 [math.fa] 17 May 2016 QINBO LIU DEPARTMENT OF MATHEMATICS, MICHIGAN STATE UNIVERSITY EAST LANSING, MI 48824, USA Abstract.
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 informationThe first order quasi-linear PDEs
Chapter 2 The first order quasi-linear PDEs The first order quasi-linear PDEs have the following general form: F (x, u, Du) = 0, (2.1) where x = (x 1, x 2,, x 3 ) R n, u = u(x), Du is the gradient of u.
More informationESTIMATES OF LOWER ORDER DERIVATIVES OF VISCOUS FLUID FLOW PAST A ROTATING OBSTACLE
REGULARITY AND OTHER ASPECTS OF THE NAVIER STOKES EQUATIONS BANACH CENTER PUBLICATIONS, VOLUME 7 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 25 ESTIMATES OF LOWER ORDER DERIVATIVES OF
More informationIMA Preprint Series # 2143
A BASIC INEQUALITY FOR THE STOKES OPERATOR RELATED TO THE NAVIER BOUNDARY CONDITION By Luan Thach Hoang IMA Preprint Series # 2143 ( November 2006 ) INSTITUTE FOR MATHEMATICS AND ITS APPLICATIONS UNIVERSITY
More information2tdt 1 y = t2 + C y = which implies C = 1 and the solution is y = 1
Lectures - Week 11 General First Order ODEs & Numerical Methods for IVPs In general, nonlinear problems are much more difficult to solve than linear ones. Unfortunately many phenomena exhibit nonlinear
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 informationRelaxation methods and finite element schemes for the equations of visco-elastodynamics. Chiara Simeoni
Relaxation methods and finite element schemes for the equations of visco-elastodynamics Chiara Simeoni Department of Information Engineering, Computer Science and Mathematics University of L Aquila (Italy)
More informationON PARABOLIC HARNACK INEQUALITY
ON PARABOLIC HARNACK INEQUALITY JIAXIN HU Abstract. We show that the parabolic Harnack inequality is equivalent to the near-diagonal lower bound of the Dirichlet heat kernel on any ball in a metric measure-energy
More informationFractal Conservation Laws: Global Smooth Solutions and Vanishing Regularization
Progress in Nonlinear Differential Equations and Their Applications, Vol. 63, 217 224 c 2005 Birkhäuser Verlag Basel/Switzerland Fractal Conservation Laws: Global Smooth Solutions and Vanishing Regularization
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 informationLocal regularity for the modified SQG patch equation
Local regularity for the modified SQG patch equation Alexander Kiselev Yao Yao Andrej Zlatoš September 4, 2015 Abstract We study the patch dynamics for a family of active scalars called modified SQG equations,
More informationEXISTENCE OF SOLUTIONS FOR CROSS CRITICAL EXPONENTIAL N-LAPLACIAN SYSTEMS
Electronic Journal of Differential Equations, Vol. 2014 (2014), o. 28, pp. 1 10. ISS: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTECE OF SOLUTIOS
More informationOn Antichains in Product Posets
On Antichains in Product Posets Sergei L. Bezrukov Department of Math and Computer Science University of Wisconsin - Superior Superior, WI 54880, USA sb@mcs.uwsuper.edu Ian T. Roberts School of Engineering
More informationUNIFORM DECAY OF SOLUTIONS FOR COUPLED VISCOELASTIC WAVE EQUATIONS
Electronic Journal of Differential Equations, Vol. 16 16, No. 7, pp. 1 11. ISSN: 17-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu UNIFORM DECAY OF SOLUTIONS
More information1.5 Approximate Identities
38 1 The Fourier Transform on L 1 (R) which are dense subspaces of L p (R). On these domains, P : D P L p (R) and M : D M L p (R). Show, however, that P and M are unbounded even when restricted to these
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 informationWiener Measure and Brownian Motion
Chapter 16 Wiener Measure and Brownian Motion Diffusion of particles is a product of their apparently random motion. The density u(t, x) of diffusing particles satisfies the diffusion equation (16.1) u
More informationIntroduction to Bases in Banach Spaces
Introduction to Bases in Banach Spaces Matt Daws June 5, 2005 Abstract We introduce the notion of Schauder bases in Banach spaces, aiming to be able to give a statement of, and make sense of, the Gowers
More informationThe KPP minimal speed within large drift in two dimensions
The KPP minimal speed within large drift in two dimensions Mohammad El Smaily Joint work with Stéphane Kirsch University of British Columbia & Pacific Institute for the Mathematical Sciences Banff, March-2010
More informationChapter III. Stability of Linear Systems
1 Chapter III Stability of Linear Systems 1. Stability and state transition matrix 2. Time-varying (non-autonomous) systems 3. Time-invariant systems 1 STABILITY AND STATE TRANSITION MATRIX 2 In this chapter,
More informationMATH 220 solution to homework 4
MATH 22 solution to homework 4 Problem. Define v(t, x) = u(t, x + bt), then v t (t, x) a(x + u bt) 2 (t, x) =, t >, x R, x2 v(, x) = f(x). It suffices to show that v(t, x) F = max y R f(y). We consider
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 informationGaussian Measure of Sections of convex bodies
Gaussian Measure of Sections of convex bodies A. Zvavitch Department of Mathematics, University of Missouri, Columbia, MO 652, USA Abstract In this paper we study properties of sections of convex bodies
More informationSome asymptotic properties of solutions for Burgers equation in L p (R)
ARMA manuscript No. (will be inserted by the editor) Some asymptotic properties of solutions for Burgers equation in L p (R) PAULO R. ZINGANO Abstract We discuss time asymptotic properties of solutions
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 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 informationEXIT TIMES OF DIFFUSIONS WITH INCOMPRESSIBLE DRIFT
EXIT TIMES OF DIFFUSIONS WITH INCOMPRESSIBLE DRIFT GAUTAM IYER, ALEXEI NOVIKOV, LENYA RYZHIK, AND ANDREJ ZLATOŠ Abstract. Let R n be a bounded domain and for x let τ(x) be the expected exit time from of
More informationNon-radial solutions to a bi-harmonic equation with negative exponent
Non-radial solutions to a bi-harmonic equation with negative exponent Ali Hyder Department of Mathematics, University of British Columbia, Vancouver BC V6TZ2, Canada ali.hyder@math.ubc.ca Juncheng Wei
More informationEXIT TIMES OF DIFFUSIONS WITH INCOMPRESSIBLE DRIFTS
EXIT TIMES OF DIFFUSIONS WITH INCOMPRESSIBLE DRIFTS GAUTAM IYER, ALEXEI NOVIKOV, LENYA RYZHIK, AND ANDREJ ZLATOŠ Abstract. Let R n be a bounded domain and for x let τ(x) be the expected exit time from
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 informationRegularity estimates for fully non linear elliptic equations which are asymptotically convex
Regularity estimates for fully non linear elliptic equations which are asymptotically convex Luis Silvestre and Eduardo V. Teixeira Abstract In this paper we deliver improved C 1,α regularity estimates
More informationMath 207 Honors Calculus III Final Exam Solutions
Math 207 Honors Calculus III Final Exam Solutions PART I. Problem 1. A particle moves in the 3-dimensional space so that its velocity v(t) and acceleration a(t) satisfy v(0) = 3j and v(t) a(t) = t 3 for
More informationMem. Differential Equations Math. Phys. 36 (2005), T. Kiguradze
Mem. Differential Equations Math. Phys. 36 (2005), 42 46 T. Kiguradze and EXISTENCE AND UNIQUENESS THEOREMS ON PERIODIC SOLUTIONS TO MULTIDIMENSIONAL LINEAR HYPERBOLIC EQUATIONS (Reported on June 20, 2005)
More informationVANISHING-CONCENTRATION-COMPACTNESS ALTERNATIVE FOR THE TRUDINGER-MOSER INEQUALITY IN R N
VAISHIG-COCETRATIO-COMPACTESS ALTERATIVE FOR THE TRUDIGER-MOSER IEQUALITY I R Abstract. Let 2, a > 0 0 < b. Our aim is to clarify the influence of the constraint S a,b = { u W 1, (R ) u a + u b = 1 } on
More information2 (Bonus). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?
MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due 9/5). Prove that every countable set A is measurable and µ(a) = 0. 2 (Bonus). Let A consist of points (x, y) such that either x or y is
More informationTopic # /31 Feedback Control Systems. Analysis of Nonlinear Systems Lyapunov Stability Analysis
Topic # 16.30/31 Feedback Control Systems Analysis of Nonlinear Systems Lyapunov Stability Analysis Fall 010 16.30/31 Lyapunov Stability Analysis Very general method to prove (or disprove) stability of
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 informationThe Lusin Theorem and Horizontal Graphs in the Heisenberg Group
Analysis and Geometry in Metric Spaces Research Article DOI: 10.2478/agms-2013-0008 AGMS 2013 295-301 The Lusin Theorem and Horizontal Graphs in the Heisenberg Group Abstract In this paper we prove that
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 informationSolution of Additional Exercises for Chapter 4
1 1. (1) Try V (x) = 1 (x 1 + x ). Solution of Additional Exercises for Chapter 4 V (x) = x 1 ( x 1 + x ) x = x 1 x + x 1 x In the neighborhood of the origin, the term (x 1 + x ) dominates. Hence, the
More informationKPP Pulsating Traveling Fronts within Large Drift
KPP Pulsating Traveling Fronts within Large Drift Mohammad El Smaily Joint work with Stéphane Kirsch University of British olumbia & Pacific Institute for the Mathematical Sciences September 17, 2009 PIMS
More informationExistence of Secondary Bifurcations or Isolas for PDEs
Existence of Secondary Bifurcations or Isolas for PDEs Marcio Gameiro Jean-Philippe Lessard Abstract In this paper, we introduce a method to conclude about the existence of secondary bifurcations or isolas
More informationFundamentals of Atmospheric Modelling
M.Sc. in Computational Science Fundamentals of Atmospheric Modelling Peter Lynch, Met Éireann Mathematical Computation Laboratory (Opp. Room 30) Dept. of Maths. Physics, UCD, Belfield. January April, 2004.
More informationRATIONAL LINEAR SPACES ON HYPERSURFACES OVER QUASI-ALGEBRAICALLY CLOSED FIELDS
RATIONAL LINEAR SPACES ON HYPERSURFACES OVER QUASI-ALGEBRAICALLY CLOSED FIELDS TODD COCHRANE, CRAIG V. SPENCER, AND HEE-SUNG YANG Abstract. Let k = F q (t) be the rational function field over F q and f(x)
More informationSubelliptic mollifiers and a basic pointwise estimate of Poincaré type
Math. Z. 226, 147 154 (1997) c Springer-Verlag 1997 Subelliptic mollifiers and a basic pointwise estimate of Poincaré type Luca Capogna, Donatella Danielli, Nicola Garofalo Department of Mathematics, Purdue
More informationIntegro-differential equations: Regularity theory and Pohozaev identities
Integro-differential equations: Regularity theory and Pohozaev identities Xavier Ros Oton Departament Matemàtica Aplicada I, Universitat Politècnica de Catalunya PhD Thesis Advisor: Xavier Cabré Xavier
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 informationODE Final exam - Solutions
ODE Final exam - Solutions May 3, 018 1 Computational questions (30 For all the following ODE s with given initial condition, find the expression of the solution as a function of the time variable t You
More informationExercises for algebraic curves
Exercises for algebraic curves Christophe Ritzenthaler February 18, 2019 1 Exercise Lecture 1 1.1 Exercise Show that V = {(x, y) C 2 s.t. y = sin x} is not an algebraic set. Solutions. Let us assume that
More information