Lenya Ryzhik
How to reach me
Summer School at Stanford, August 518, 2013
Teaching
 Winter 2023:
Stanford Winter 2023
Topics in PDE lecture notes These notes will be continuously updatd back and forth.
 Fall 2021:
Stanford Fall 2021 BBM and
PDE minicourse
lecture notes These notes will be continuously updatd back and forth.
 Winter 2021:
Lecture notes for the topics in PDE class
 Math 63 CM, Winter 2020
 Stanford Spring 2020 PDE minicourse
lecture notes
 Technion summer school lecture notes
 Wien lecture notes on waves in random media
 Toulousain KPP lecture notes
 Santa Barbara lecture notes
 Lecture Notes on Complex Analysis
 translation of several chapters
from "Introduction to Complex Analysis" by B. Shabat
 Winter 2019:
Lecture notes for the fluids PDE class
 Winter 2018:
Lecture notes for a reading course
 Spring 2017:
Lecture Notes for Topics in PDE
 Spring 2016:
Lecture notes for the Introduction to SPDEs
 Fall 2015:
Lecture Notes for Topics in PDES, Math 272 in Fall 2015
 Winter 2014:
Lecture Notes for Fluid PDEs Math 256B
Lecture Notes for Topics in PDEs
 Fall 2011:
Technion Lecture Notes.
 Winter 2003:
(Applied Analysis Lecture Notes).
Curriculum Vitae
Click here to see a more or less up to date vita.
(pdf)
Preprints and publications (preliminary versions before refereeing,
with misprints and possibly mistakes):
 The speed of a random front for stochastic reactiondiffusion
equations with strong noise (with C. Mueller and L. Mytnik),
(pdf), Preprint, 2019.

Global wellposedness for the Euler alignment system with mildly singular interactions (with J. An), (arxiv link), Preprint, 2019.
 Fluctuations of the KPZ equation in dimensions three and higher
(with A. Dunlap, Y. Gu and O. Zeitouni),
(arxiv link),
Preprint, 2018.

The random heat equation in dimensions three and higher: the homogenization
viewpoint (with A. Dunlap, Y. Gu and O. Zeitouni),
(arxiv link),
Preprint, 2018.
 High frequency limit for a chain of harmonic oscillators with
a point Langevin thermostat (with T. Komorowski, S. Olla,
and H. Spohn), (arxiv link), Preprint, 2018.
 Fluctuations of random semilinear advection equations
(with Y. Gu and T. Komorowski),
(pdf),
Preprint, 2018.
 The Bramson delay in the nonlocal
FisherKPP equation (with E. Bouin and C. Henderson),
(pdf),
Preprint, 2017.
 The EdwardsWilkinson limit
of the random heat equation in dimensions three and higher
(with Y. Gu and O. Zeitouni), (pdf),
Preprint, 2017.
 The Schroedinger equation with
spatial white noise: the average wave function (with Y. Gu and
T. Komorowski), (pdf),
Preprint, 2017.
 Global regularity for the fractional Euler alignment system
(with T. Do, A. Kiselev and C. Tan),
(pdf), Preprint, 2017.
 The weak coupling limit for the random SchrÃ¶dinger equation: The average wave function (with T. Chen and T. Komorowski),
(pdf),
Preprint, 2017.
 The Bramson logarithmic delay in the cane toads equations
(with E. Bouin and C. Henderson),(pdf),
Preprint, 2016.
 Refined long time asymptotics for the FisherKPP fronts
(with J. Nolen and J.M. Roquejoffre),
(pdf), Preprint, 2016.
 Convergence to a single wave in the FisherKPP equation
(with J. Nolen and J.M. Roquejoffre),
(pdf), Preprint, 2016.

Superlinear spreading in local and nonlocal cane toads equations
(with E. Bouin and C. Henderson),(pdf),
Preprint, 2015.

Radiative transfer with longrange interactions: regularity and asymptotics
(with C. Gomez and O. Pinaud),
(pdf), Preprint, 2015.
 Finite time singularity for the modified SQG patch equation
(with A. Kiselev, Y. Yao and A. Zlatos),
(pdf), Preprint, 2015.
 The random Schroedinger equation: slowly decorrelating
timedependent potentials
(with Y. Gu),
(pdf), Preprint, 2015.
 The random Schroedinger equation: homogenization in
timedependent potentials (with Y. Gu),
(pdf), Preprint, 2015.
 Hypoelliptic estimates in radiative transfer (with C. Gomez and O. Pinaud),
(pdf), Preprint, 2015.
 Anisotropic interactions in an aggregation model
(with J.H.M. Evers and R. Fetecau),
(pdf), Preprint, 2014.
 Exploring the effects of social preference, economic disparity, and
heterogeneous environments on segregation (with N. Rodriguez),
(pdf), Preprint, 2014.
 Precursors for Waves in Random Media (with G. Bal O. Pinaud and
K. Solna), (pdf), Preprint, 2013.
 On the nonlocal FisherKPP equation: steady states,
spreading speed and
global bounds (with Francois Hamel), (pdf),
Preprint, 2013.
 Powerlike delay in time inhomogeneous FisherKPP equations
(with J. Nolen and J.M. Roquejoffre),
(pdf), Preprint, 2013.

Homogenization driven by
a fractional Brownian motion: the shear layer case
(with T. Komorowski and A. Novikov), (pdf),
Preprint, 2013.
 The Harnack inequality for secondorder parabolic equations
with divergencefree drifts of low regularity (with I. Kukavica and Mi. Ignatova), (pdf), Preprint, 2013.
 Long time energy transfer in the random Schroedinger equation
(with T. Komorowski),
(pdf), Preprint, 2013.
 Traveling wave solutions in a reactiondiffusion model for criminal activity,
(with H. Berestycki and N. Rodriguez),
(pdf), Preprint, 2012.

Wavelike solutions for nonlocal reactiondiffusion equations:
a toy model (with G. Nadin, B. Perthame and L. Rossi)
(pdf), to appear in Math. Mod. Nat. Phenom., 2013.

The logarithmic delay of KPP fronts in a periodic medium
(with F. Hamel, J. Nolen and J.M. Roquejoffre),
(pdf),
Preprint, 2012.
 The Harnack inequality for secondorder elliptic equations with divergencefree drifts (with I. Kukavica and M. Ignatova),
(pdf), to appear in Comm. Math. Sci., 2013.

A short proof of the logarithmic Bramson correction in FisherKPP equations
(with F. Hamel, J. Nolen and J.M. Roquejoffre),
(pdf),
to appear in Net. Heter. Med., 2013.
 Biomixing by chemotaxis and efficiency of
biological reactions: the critical reaction case (with A. Kiselev),
(pdf),
to appear in Jour. Math. Phys., 2013.

Asymptotics of the solutions of the stochastic lattice wave equation
(with T. Komorowski and S. Olla), (pdf),
to appear in Arch. Rat. Mech. Anal., 2013.

A differential equations approach to
l_1minimization with applications to array imaging (with M. Moscoso,
A. Novikov and G. Papanicolaou), (pdf),
Inverse Problems, 28,
2012, 1050012012.

From homogenization to averaging in cellular
flows (with G. Iyer, T. Komorowski and A. Novikov),
(pdf), Preprint, 2011.

Evolution of particle separation in slowly decorrelating velocity fields
(with T. Komorowski and A. Novikov), (pdf),
Comm. Math. Sci., 10, 2012, 767786.

Biomixing by chemotaxis and enhancement of biological reactions
(with A. Kiselev), (pdf),
Comm. PDEs, 37, 2012, 298318.

Existence and nonexistence of FisherKPP transition fronts
(with J. Nolen, J.M. Roquejoffre and A. Zlatos), (pdf),
Arch. Rat. Mech. Anal., 203, 2012, 217246.

Kinetic limits for waves in a random medium (with G. Bal and T. Komorowski),
(pdf), Kinet.
Relat. Models 3, 2010, 529644.

A sharp bound on the L^2 norm of the solution of a random
elliptic dierence equation (with T. Komorowski),
(pdf,
to appear in Comm. Math. Sci., 2011.

Fluctuations of solutions to Wigner equation
with an OrnsteinUhlenbeck potential (with T. Komorowski),
(pdf),
Discr. Cont. Dyn. Sys. B, 17, 2012, 871914

A sharp bound on the $L^2$ norm of the solution of a random elliptic difference equation
(with T. Komorowski), (pdf),
Comm. Math. Sci., 9, 2011, 607622.

A simple model for asset price bubble formation and collapse
(with A. Kiselev), (pdf),
Preprint, 2010.

Exit times for diffusions with incompressible drifts
(with G. Iyer, A. Novikov and A. Zlatos),
(pdf),
SIAM J. Math. Anal. {42}, 2010, 24842498.

Asymptotics of the phase of the solutions of the random Schr\"odinger equation
(with G. Bal and T. Komorowski),
(pdf),
Arch. Ration. Mech. Anal. 200, 2011, 613664.

The nonlocal FisherKPP equation: traveling waves and steady states
(with H. Berestycki, G. Nadin and B. Perthame),
(pdf),
Nonlinearity, 22, 2009,
28132844.

Limit of fluctuations of solutions of Wigner Equation
(with T. Komorowski and Sz. Peszat),
(pdf),
Communications in Mathematical Physics, 292, 2009, 479510

Travelling fronts for the thermodiffusive system with arbitrary Lewis numbers
(with F. Hamel),
(pdf),
Archive for Rational Mechanics
and Analysis, 195, 2010, 923952.

Stability of generalized transitions fronts
(with A. Mellet, J, Nolen and J.M. Roquejoffre),
(pdf),
Communications in PDE, {\bf 34}, 2009, 521552.

The explosion problem in a flow
(with H. Berestycki, A. Kiselev and A. Novikov),
(pdf),
Journal d'Analyse Math\'ematique, 110, 2010, 3165.

Traveling waves in a onedimensional random medium
(with J. Nolen),
(pdf),
to appear in Ann. de l'Inst. Henri Poincare, C. Analyse non lineaire, 2009.

Propagation and quenching in a reactive BurgersBoussinesq system
(with P. Constantin, J.M. Roquejoffre and N. Vladimirova),
(pdf),
Nonlinearity, 21, 2008, 221271.

Traveling waves for the KellerSegel system with Fisher birth terms
(with G. Nadin and B. Perthame),
(pdf),
Interfaces and Free Boundaries, 10, 2008, 517538.

On
asymptotics
of a tracer advected in a locally selfsimilar, correlated flow
(with T. Komorowski),
(pdf), Asymptotic Analysis, 53, 2007,
159187.

Relaxation in reactive flows,
(with P. Constantin and A. Novikov),
(pdf),
Geometric and Functional Analysis, 18, 2008, 11451167.

KPP pulsating front speedup by flows,
(with A. Zlatos), (pdf),
Communications in Mathematical Sciences, 5, 2007, 575593.

Moderate dispersion in scalar conservation laws
(with B. Perthame), (pdf),
Communications in Mathematical Sciences, 5, 2007, 473484.

Selfaveraging from lateral diversity in the ItoSchroedinger equation,
(with G. Papanicolaou and K. Solna), (pdf),
Multiscale Modeling and Simulation, 6, 2007, 468492.

Passive tracer in a slowly decorrelating random flow with a large
mean, (with T. Komorowski),
(pdf), Nonlinearity, 20, 2007, 12151239.

Traveling waves in 2D reactive Boussinesq
systems with noslip boundary conditions, (with P. Constantin and M. Lewicka),
(pdf), Nonlinearity, 19, 2006, 26052615.

Diffusion and Mixing in a Fluid Flow, (with P. Constantin,
A. Kiselev and A. Zlatos),(pdf),
Annals of Mathematics, 68, 2008, 643674..

Wave field correlations in weakly mismatched random media
(with G. Bal), (pdf),
Stochastics and Dynamics, 2006, 301328.

Existence of pulsating waves in a model of flames in sprays
(with P. Constantin, K. Domelevo and J.M. Roquejoffre),
(pdf),
Journal of the European Mathematical Society, 8, 2006, 555584.

The stochastic acceleration problem in two dimensions (with
T. Komorowski), (pdf), Israel Journal
of Mathematics, 155, 2006, 157204.

Nonadiabatic KPP fronts with an arbitrary Lewis number (with
F. Hamel), (pdf), Nonlinearity, 18, 2005,
28812902.

Bounds on the speed of propagation of the KPP fronts in a cellular
flow (with A. Novikov), (pdf),
Archive for Rational Mechanics and Analysis, 184, 2007, 2348.

Flame capturing with an advectionreactiondiffusion model
(with N. Vladimirova and G. Weirs),
(ps), Combustion Theory and Modeling, 10, 2006,
727747.

Traveling fronts in porous media: existence and a singular limit
(with P. Gordon),
(pdf), Proceedings of the
Royal Society London A, 462, 2006, 19651985.

Diffusion in a weakly random Hamiltonian flow
(with T. Komorowski),
(pdf), Communications
in Mathematical Physics, 263, 2006, 277323.

Quenching of reaction by cellular flows
(with A. Fannjiang and A. Kiselev),
(pdf), Geometric
and Functional Analysis, 16, 2006, 4069.

The quantum scattering limit for a regularized Wigner equation
(with B. Perthame),
(pdf),
Methods and Applications of Analysis, 11, 2004, 447464.

NonPlanar Fronts in Boussinesq
Reactive Flows
(with H. Berestycki and P. Constantin),
(pdf), Ann. de l'Inst.
Henri Poincar\'e, C. Analyse non lin\'eaire,
23, 2006, 407437.

The KPP system in a periodic flow with a heat loss
(with P. Gordon and N. Vladimirova),
(ps,pdf ), Nonlinearity,
18, 2005, 571589.

Quenching and propagation in KPP reactiondiffusion
equations with a heat loss
(with H. Berestycki, F. Hamel and A. Kiselev),
(pdf), Archive for Rational
Mechanics and Analysis, 178, 2005, 5780.

Stability of time reversed waves in changing media
(with G. Bal),
(pdf),
Discrete and Continuous Dynamical Systems A, 12, 2005, 793  815.

Time splitting for the Liouville equation in a random medium
(with G. Bal),
(pdf),
Communications in Mathematical Sciences, 3, 2004, 515  534.

Boundary layers for cellular flows at high Peclet numbers
(with A. Novikov and G. Papanicolaou),
(ps,pdf)),
Communications in Pure and Applied Mathematics, 58, 2005,
867922.

Exponential decay for the fragmentation or celldivision equation
(with B. Perthame),
(pdf), Journal
of the Differential Equations, 210, 2005, 155177.

Time splitting for wave equations in random media
(with G. Bal),
(pdf), Model. Math. Anal. Num. (M2AN), 38,
2004, 961987.

Flame Enhancement and Quenching in Fluid Flows
(with N. Vladimirova, P. Constantin, A. Kiselev and O. Ruchaiskiy),
(ps), Combustion
Theory and Modeling, 7, 2003, 487  508.

Fronts in reactive convection: bounds, stability and instability
(with P. Constantin and A. Kiselev),
(ps), Communications
in Pure and Applied Mathematics, 56, 2003, 17811803.

Selfaveraging of Wigner transforms in random media
(with G. Bal and T. Komorowski),
(ps), Communications
in Mathematical Physics, 242, 2003, 81135.

Statistical stability in time reversal
(with G. Papanicolaou and K. Solna),
(ps), SIAM Journal of
Applied Mathematics, 64, 2004, 11331155.

Stability in a nonlinear population maturation model (with
S. Mischler and B. Perthame) (ps),
M3AS (Mathematical Models & Methods in Applied Science), 12, 2002,
17511772.

Selfaveraging in time reversal for the parabolic wave
equation (with G. Bal and G. Papanicolaou),
(ps),
Stochastics and Dynamics, 2, 2002, 507531.

The parabolic wave approximation and time reversal
(with G. Papanicolaou and K. Solna),
(ps), Matematica Contemporanea, 23, 2002, 139159.

Time reversal and refocusing in random media (with G. Bal),
(ps), SIAM Journal of
Applied Mathematics, 63, 2003, 14751498.

Radiative transport limit for the random Schroedinger equation
(with G. Bal and G. Papanicolaou),
(ps), Nonlinearity, 15, 2002, 513529.

Time Reversal for Classical Waves in Random Media (with G. Bal),
(ps), Comptes rendus
de l'Academies Sciences  Serie I/Mathematique, 333, 2001, 10411046.

Wave transport for a scalar model of the Love waves (with G. Bal),
(ps), Wave Motion, 36, 2002, 4966.

An upper bound for the bulk burning rate for systems (with
A. Kiselev), (ps),
Nonlinearity, 14, 2001, 12971310.

Quenching of flames by fluid advection (with P. Constantin and
A. Kiselev), (ps),
Communications in Pure and Applied Mathematics, 54, 2001, 13201342.

Enhancement of the traveling front speeds in reactiondiffusion
equations with advection (with A. Kiselev), (ps),
Ann. de l'Inst.
Henri Poincar\'e, C. Analyse non lin\'eaire, 18, 2001, 309358.

Wave transport along surfaces with random impedance (with G. Bal,
V. Freilikher and G. Papanicolaou),
(ps) Physical Review B, 62, 2000, 62286240.

Bulk burning rate in passive  reactive diffusion (with P. Constantin,
A. Kiselev and A. Oberman), (ps),
Archive for Rational Mechanics, 154, 2000, 5391.

Probabilistic theory of transport processes with polarization
(with G.Bal and G.Papanicolaou),
(ps)
SIAM Journal of Applied Mathematics, 60, 2000, 1639  1666.

Radiative Transfer of Sound Waves in a Random Flow: Turbulent
Scattering and ModeCoupling (with A.Fannjiang), (ps), SIAM Journal of Applied
Mathematics, 61, 2001, 15451577.

Diffusive energy scattering from weakly
random surfaces (with G.Bal and G.Papanicolaou),
(ps),
Journal of Mathematical Physics, 40, 1999, 48134827.

Diffusion approximation of radiative transfer problems with
interfaces (with G.Bal),
(ps)SIAM Journal of Applied
Mathematics, 60, 2000, 18871912.

Transport theory for acoustic waves with reflection and transmission
at interfaces (with G.Bal, J.B.Keller and G.Papanicoalou),
(ps), Wave Motion, 30, 1999, 303327.

Level sets in turbulent front propagation (with P.Constantin and
A.Kiselev), (ps), Preprint, 1998.

Radiative transport in a periodic structure (with G.Bal, A.Fannjiang
and G.Papanicolaou), (ps),
Journal of Statistical Physics 95 (1/2):479494, 1999.

Waves and transport (with G.Papanicolaou), IAS/Park City
Mathematics Series, Vol. 5, L. Caffarelli and Weinan E, eds., AMS,
1998, pp. 305382.

Evolution of trajectories correlations in steady random flows
(with A.Fannjiang and G.Papanicolaou)
(ps).
In `Proceedings of Symposia in Applied Mathematics', Volume
54, R. Spigler and S. Venakides editors, American Mathematical Society, (1998), 105130.

Transport equations for waves in a half space (with J.Keller and
G.Papanicolaou)
(ps).
Comm. PDE's, 22, (1997), 18691910.

Transport equations for elastic and other waves in random media
(with J.Keller and G.Papanicolaou)
(ps).
Wave Motion 24, (1996), 327370

Stability of the P to S energy ratio in the diffusive regime
(with George
Papanicolaou and Joseph B. Keller)
(ps).
Bulletin of the Seismological Society of America 86, (1996),
11071115. Erratum. 86, (1996), p. 1997.

On the complete algebra of symmetries of integrable systems
(with E.Schulman),
Theor. Math. Phys. 95, 1993, 387392.
Grants
 NSF grant DMS9971742 "Radiative transport theory for waves",
August 1999August 2002.
 ONR grant N000140210089 "TimeReversal for Waves in
Random Media", November 20012004.
 Alfred P. Sloan Research Fellowship, 20022004.
 NSF grant DMS0203537 ``Propagation of fronts and waves in complex
media'', 20022005.
 DARPAONR grant N000140410224
"Time Reversal for Electromagnetic Waves in Random Media", 20042009.
 NSF grant DMS0604687 ``The Kinetic Theory of Waves and ReactiveDiffusive Fronts'',
20062009.
 NSF FRG grant DMS0854952 "Stochastics and Dynamics: Asymptotic problems", 20092012.
 NSF grant DMS0908507 "Waves and Fronts in Heterogeneous Media", 20092012.
 AFOSR NSSEFF Fellowship, 20102015.
 NSF FRG grant DMS115893,
"Singularities, mixing and long time behavior in nonlinear evolution",
20122015.
 NSF grant DMS1311903, "Waves, Particle Transport and Fronts in Heterogeneous Media".
Links
Here are some links related to
Israel.
Here are some links related to
Classical Music.
Lenya Ryzhik (
ryzhik@math.stanford.edu)