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in the understanding of cancer development , mathematical modeling and numerical simulations have nowadays complemented experimental and clinical observations . the field is now mature ; books and surveys are available , as for example @xcite . a first class of models , initiated in the 70 s by greenspan @xcite , has considered that cancerous cells multiplication is limited by nutrients ( glucosis , oxygen ) brought by blood vessels . models of this class rely on two kinds of descriptions ; either they describe the dynamics of cell population density @xcite or they consider the geometric motion of the tumor through a free boundary problem ; see and the references therein . this stage lasts until the tumor reaches the size of @xmath0 mm ; then lack of nutrients leads to cell necrosis which triggers neovasculatures development @xcite that supply the tumor with enough nourishment . this has motivated a new generation of models where growth is limited by the competition for space @xcite , turning the modeling effort towards mechanical concepts , considering tissues as multiphasic fluids ( the phases could be intersticial water , healthy and tumor cells , extra - cellular matrix  ) @xcite . this point of view is now sustained by experimental evidence @xcite . the term homeostatic pressure , coined recently , denotes the lower pressure that prevents cell multiplication by contact inhibition . the aim of this paper is to explain how asymptotic analysis can link the two main approaches , cell density models and free boundary models , in the context of fluid mechanics . we depart from the simplest cell population density model , proposed in  @xcite , in which the cell population density @xmath1 evolves under pressure forces and cell multiplication according to the equation @xmath2 where @xmath3 is the pressure field . pressure - limited growth is described by the term @xmath4 , which typically satisfies @xmath5 for some @xmath6 ( the homeostatic pressure ) . the pressure is assumed to be a given increasing function of the density . a representative example is @xmath7 with parameter @xmath8 . in the free boundary problem to be discussed later , the value @xmath9 represents the maximum packing density of cells , as discussed in @xcite .    nutrients consumed by the tumor cells and brought by the capillary blood network are a usual additional ingredient to the modeling . the situation is described in that case by the system @xmath10 \displaystyle \partial_t c - \delta c   = - \varrho \ ; \psi(p , c ) , \\[2 mm ] c(x , t ) = c_b > 0 \qquad \text{as } \ ; |x| \to \infty , \end{cases}\ ] ] where @xmath11 denotes the density of nutrients , and @xmath12 the far field supply of nutrients ( from blood vessels ) . the coupling functions @xmath13 , @xmath14 are assumed to be smooth and to satisfy the natural hypotheses @xmath15 \displaystyle\partial_p\psi\leq 0,\qquad & \displaystyle\partial_c\psi \geq 0 , \qquad&\displaystyle \psi(p,0 ) = 0 . \end{array}\ ] ]    variants are possible ; for instance , we could assume that nutrients are released continuously from a vasculature , then leading to an equation as @xmath16    we will consider below the purely mechanical model under assumptions  , and also the system , where nutrients are taken into account , under assumptions and . in both cases we will show that the asymptotic limit @xmath17 yields a free boundary model of hele - shaw type , as we were looking for . let us recall that the mathematical theory of the limit for an equation like in the absence of a growth term is well developed , the asymptotic limit being in this case the standard hele - shaw model for incompressible fluids with free boundaries . early papers on the subject @xcite consider situations in which mass is conserved and the limit is stationary . in order to have a non - trivial limit evolution one needs some source , either in the equation or at the boundary of the domain . a first example , in which there is an inwards flux at infinity , is given in  @xcite , where the authors study the limit @xmath17 of self - similar focusing ( hole - filling ) solutions to the porous medium equation @xmath18 . a second example , in which there is a bounded boundary with a nontrivial boundary data , was first considered in  @xcite , and later on in  @xcite . to our knowledge , the present paper is the first one in which the evolution in the limit is produced by a nonlinear source term in the equation . in order to pass to the limit , three approaches have been used : weak solutions , variational formulations ( using the so - called baiocchi variable ) , and viscosity solutions ; see @xcite for this last case . the weak formulation of hele - shaw was first introduced in @xcite , and the variational formulation in  @xcite . let us also mention that the hele - shaw graph , and hence the hele - shaw equation , can be approximated in other ways , for example by the stefan problem . this situation has also been considered in the literature , even in situations where there is an evolution in the limit ; see for instance  @xcite . we would like to stress that including the growth term is not a simple change : several powerful but specific tools do not apply any longer . more deeply , as we explain later , several approaches have no chance to work as they are . this is the reason why we will work on an equation for the cell density itself rather than the pressure . we also would like to point out that models of hele - shaw type are still an active field arising in several unexpected applications , see for instance  @xcite , and that surface tension is not covered by the present work ; see @xcite . _ organization of the paper . _ we first study the simplest mechanical model in section  [ sec : mechanical ] . the more complex model with nutrients is studied in section  [ sec : nutrients ] . the uniqueness proofs for both cases are highly technical , and are performed separately , in sections  [ sec : uniqueness ] and  [ sec : nutrient_unique ] . these two sections can safely be skipped by the readers who are mainly interested in the applications . we finally include an appendix devoted to some interesting examples for the purely mechanical model , that illustrate several phenomena .    _ some notations . _ we will use several times the abridged notations @xmath19 , @xmath20 , meaning @xmath21 , @xmath22 . given any @xmath23 , we denote @xmath24 , while @xmath25 . we start our analysis by a detailed investigation of the purely mechanical model , which does not take into account the consumption of nutrients , with a pressure field given by . a simple change of scale allows us to assume without loss of generality that @xmath26 . we arrive to the porous medium type equation , set on @xmath27 , @xmath28 we assume that the initial data @xmath29 are such that , for some @xmath30 , @xmath31 \|\varrho_m^0- \varrho^0 \|_{l^1(\mathbb{r}^n)}\underset{m \to \infty } { \longrightarrow }   0,\quad & \|\partial_{x_i } \varrho^0_m\|_{l^1(\mathbb{r}^n)}\le c , \quad i=1,\dots , n . \end{array}\right.\ ] ]      free boundary limit . let @xmath32 be the unique bounded weak solution to . we will prove that , along some subsequence , there is a limit as @xmath17 which turns out to be a solution to a free boundary problem of hele - shaw type . [ th : fb ] let @xmath13 and @xmath33 satisfy   and   respectively . then , after extraction of subsequences , both the density @xmath34 and the pressure @xmath35 converge for all @xmath23 strongly in @xmath36 as @xmath17 to limits @xmath37 , @xmath38 , that satisfy @xmath39 , @xmath40 , and @xmath41 plus the relation @xmath42 , where @xmath43 is the hele - shaw monotone graph @xmath44 [ 0,\infty),\qquad & \varrho=1 . \end{array } \right.\ ] ]    note that means that a.e . @xmath45 and @xmath46 a.e .  in @xmath47 . we will obtain also several important qualitative properties for the limit . on the one hand , the tumor is growing and the pressure increases , that is , @xmath48 on the other hand , if the initial data @xmath33 have a common compact support , then the limit solution propagates with a finite speed : @xmath49 and @xmath50 are compactly supported for all @xmath51 . finally , the constructed limit solution enjoys the monotonicity property , inherited from the case where @xmath52 is finite , @xmath53 p_\infty(t ) \geq \bar p_\infty(t ) . \end{array } \right.\ ] ]    transport equation and complementarity formula . we will also obtain @xmath54  estimates on the gradients @xmath55 that will show on the one hand that @xmath56 and on the other hand that @xmath57 solves a transport equation . [ th : transport ] the limit solution @xmath57 obtained in theorem  [ th : fb ] satisfies @xmath58 in a weak sense . a direct calculation shows that the pressure @xmath35 satisfies @xmath59 hence , if we let @xmath17 , we formally obtain the _ complementarity formula _ @xmath60 this formal computation can be made rigorous . [ thm : complementarity ] under the assumptions of theorem [ th : fb ] , the limit pressure @xmath61 satisfies the complementarity formula @xmath62    it is worth noticing that is equivalent to the strong convergence of @xmath55 in @xmath63 for all @xmath23 ; see lemma  [ lemma : equivalence.complementarity.strong.convergence ] . since the limit pressure @xmath61 is expected to be continuous in space for all positive times , the positivity set @xmath64 should be well defined for all @xmath51 . notice that it coincides almost everywhere with the set where @xmath65 ; see figure [ fig : strict ] . indeed , on the one hand , by the definition of the graph @xmath43 we have @xmath66 ; on the other hand , if we had @xmath67 and @xmath65 in some set with positive measure , then @xmath68 would continue to grow ( exponentially ) there , which is a contradiction . therefore , @xmath69 may be regarded as the _ tumor _ , while the regions where @xmath70 ( mushy regions , in the literature of phase - changes ) correspond to _ precancer _ cells . _ a solution to the mechanical model , in one dimension with @xmath71 . left : @xmath72 . right : @xmath73 . the upper line is @xmath74 ; the bottom line is @xmath3 ( scale enlarged for visibility ) . notice that the density scales are not the same in the two figures . the initial data is taken with compact support and the solution is displayed for a time large enough ( see figure  [ fig : strict+ ] below for and intermediate regime ) . , title="fig:",width=264 ]   large . _ a solution to the mechanical model , in one dimension with @xmath71 . left : @xmath72 . right : @xmath73 . the upper line is @xmath74 ; the bottom line is @xmath3 ( scale enlarged for visibility ) . notice that the density scales are not the same in the two figures . the initial data is taken with compact support and the solution is displayed for a time large enough ( see figure  [ fig : strict+ ] below for and intermediate regime ) . , title="fig:",width=264 ] +    the complementarity formula   indicates that the limit pressure at time @xmath75 , @xmath50 , should solve the elliptic equation @xmath76 a problem which is wellposed if @xmath69 is smooth enough , since @xmath77 is decreasing . this implies in particular that in general regularity in time for the pressure is missing , since time discontinuities may show up when two tumors meet ; see subsection  [ sec : timedisc ] in the appendix .    geometric motion vs.  equation on the cell number density . to complete the description of the limit problem we should be able to trace @xmath69 starting from its initial position . the pressure equation   suggests that we should have @xmath78 which leads to a geometric motion with normal velocity @xmath79 at the boundary of @xmath69 given by @xmath80 thus we have arrived to a geometric hele - shaw type problem , which is the classical one when @xmath81 . the above formal computation is expected to be true if we prescribe a fixed initial pressure @xmath82 ( which implies that the initial densities converge to the indicator function of the positivity set of @xmath83 ) . when @xmath81 , it was proved to be true in a viscosity sense in @xcite ; see also  @xcite for an earlier result in this direction using a variational formulation of baiocchi type .    however , if the initial densities @xmath29 are such that @xmath84 is below 1 in a set with positive measure , the result is no longer true . the main point is that the tumor may meet precancer zones . at a meeting point @xmath85 , the example in subsection  [ subsect : effect.of.eq.density ] in the appendix suggests that the tumor grows faster ( also for the case @xmath81 ) , with a normal velocity given by a rule of the form @xmath86 where @xmath87 is some limit of @xmath74 as @xmath88 and @xmath89 from the outside of @xmath69 . we leave open this problem , which seems to be a challenging extension of the viscosity method in @xcite . even if we consider the modified geometric motion law instead of , the geometric formulation does not carry all the information of the limit solution . indeed , the density in precancer zones evolves , with an exponential growth , until it reaches the level @xmath90 , a fact that is not captured neither by  , nor by  ; see figure  [ fig : strict+ ] and the examples in the appendix . when @xmath81 we also need   to give a full description of the limit . however , in this latter case the evolution in the mushy regions is simpler , since the density does not change until it interacts with the positivity set for the pressure .     and the initial data @xmath74 is less than @xmath91 . the solution is displayed at four different times . it shows how the smooth part of @xmath74 strictly less than @xmath91 is growing with @xmath92 ( figure on the left ) . when @xmath74 reaches the value @xmath91 , the pressure becomes positive , increases and creates a moving front that delimitates the growing domain where @xmath93 . thin line is @xmath74 and thick line is @xmath3 as functions of @xmath94 . see also figure  [ fig : tw ] in the appendix.,title="fig:",width=136 ]   and the initial data @xmath74 is less than @xmath91 . the solution is displayed at four different times . it shows how the smooth part of @xmath74 strictly less than @xmath91 is growing with @xmath92 ( figure on the left ) . when @xmath74 reaches the value @xmath91 , the pressure becomes positive , increases and creates a moving front that delimitates the growing domain where @xmath93 . thin line is @xmath74 and thick line is @xmath3 as functions of @xmath94 . see also figure  [ fig : tw ] in the appendix.,title="fig:",width=136 ]   and the initial data @xmath74 is less than @xmath91 . the solution is displayed at four different times . it shows how the smooth part of @xmath74 strictly less than @xmath91 is growing with @xmath92 ( figure on the left ) . when @xmath74 reaches the value @xmath91 , the pressure becomes positive , increases and creates a moving front that delimitates the growing domain where @xmath93 . thin line is @xmath74 and thick line is @xmath3 as functions of @xmath94 . see also figure  [ fig : tw ] in the appendix.,title="fig:",width=136 ]   and the initial data @xmath74 is less than @xmath91 . the solution is displayed at four different times . it shows how the smooth part of @xmath74 strictly less than @xmath91 is growing with @xmath92 ( figure on the left ) . when @xmath74 reaches the value @xmath91 , the pressure becomes positive , increases and creates a moving front that delimitates the growing domain where @xmath93 . thin line is @xmath74 and thick line is @xmath3 as functions of @xmath94 . see also figure  [ fig : tw ] in the appendix.,title="fig:",width=136 ] +    let us point out that , in the case of a compactly supported initial data , precancer zones disappear in finite time .    uniqueness . to complete the description of the asymptotic limit , we conclude with a uniqueness result , relying on a hilbert duality method , for the free boundary problem  . [ th : unique ] there is a unique pair @xmath95 of functions in @xmath96 , @xmath97 , @xmath98 , satisfying   and such that for all @xmath23 :    * @xmath99 is uniformly compactly supported for @xmath100 $ ] ; * @xmath101 ; * @xmath102 , @xmath103 . it is important to notice that if , in addition to  , we assume that the initial data @xmath104 are uniformly compactly supported , then the limit solution @xmath57 to   given by theorem  [ th : fb ] falls within the uniqueness class . as a consequence , convergence is not restricted to a subsequence . _ the compact support assumption in theorem  [ th : unique ] can be removed , to the cost of using rather technical arguments , in the style of the ones employed in the uniqueness proof for the case with nutrients , theorem  [ thm : uniqueness.system ] below . we keep it for the sake of clarity . the rest of this section is devoted to the proofs of the above mentioned results , except uniqueness , which is postponed to section  [ sec : uniqueness ] . we consider the solutions to and proceed to obtain the estimates that allow us to pass to the limit and prove theorem [ th : fb ] . this is a standard porous media type equation and all the manipulations below can be justified , see the monograph @xcite . the main observation here is that the lower estimate on @xmath105 that does the job when passing to the limit in porous media equations without a source , does not work in the present case . however , a control can be obtained on the quantities @xmath106 , which will be enough for our purposes . @xmath107 bounds for @xmath34 , @xmath35 . standard comparison arguments yield @xmath108 this allows in particular to avoid initial layers ( which are present when @xmath109 ; see  @xcite ) . @xmath110 bounds for @xmath34 , @xmath35 . let @xmath34 , @xmath111 be two ( non - negative ) solutions to , and @xmath112 , @xmath113 the corresponding pressures . we have @xmath114 from where we get @xmath115 this almost contraction  property yields the uniform ( in @xmath52 ) bound @xmath116 on the other hand , @xmath117 . thus , using we conclude @xmath118    a semiconvexity estimate for @xmath35 . according to the hypotheses   on the growth function , @xmath119}\left(\phi ( p)- p\phi'(p)\right ) > 0 we will prove that @xmath120 as a consequence , in the limit we will have @xmath121 . we borrow the idea to prove inequality   from  @xcite , where the case @xmath81 was considered . we write the equation as @xmath122 let us denote @xmath123 . since @xmath124 , we have @xmath125 \partial_t ( \phi(p_m))=\phi'(p_m ) \partial_t p_m=(m-1)p_m\phi'(p_m)w+ \phi'(p_m)|\nabla p_m|^2\\[10pt ] \qquad\qquad\ \ , \geq ( m-1)p_m\phi'(p_m)w+ 2 \nabla ( \phi(p_m ) ) \cdot \nabla p_m , \end{array}\ ] ] which gives @xmath126 the function @xmath127 is a subsolution to this equation , and   follows .    _ remarks . _ ( i ) the right hand side of behaves as @xmath128 for @xmath129 . \(ii ) for @xmath130 we have @xmath131 , and we recover the well - known result for the case @xmath81 , namely @xmath132 .    bounds for @xmath133 , @xmath134 . we combine   with to obtain an estimate from below for the time derivative of the pressure , @xmath135 this in turn gives an estimate from below for the time derivative of the density , @xmath136 the monotonicity inequalities   for the limit problem are then obtained just by letting @xmath17 in and . we now use @xmath137 to obtain @xmath138 this , together with  , leads to a uniform bound in space and time for time intervals of the form @xmath139 $ ] .    an analogous computation shows that @xmath140 & & \displaystyle+(m-1)r_\phi \int_{\frac1{m-1}}^t \left(\frac{\textrm{e}^{-(m-1)r_\phi t } } { 1- \textrm{e}^{-(m-1)r_\phi t}}\int_{\mathbb{r}^n } p_m(t)\right)\,dt \le c(t ) . \end{array}\ ] ]    @xmath110 bounds for @xmath141 , @xmath55 . let @xmath142}|\phi'(p)|>0 $ ] . we consider the equation for @xmath143 , multiply it by @xmath144 , and use kato s inequality ; thanks to the monotonicity of @xmath13 we obtain @xmath145 & \le & \phi(0)|\partial_{x_i}\varrho_m|-\alpha \varrho_m |\partial_{x_i } p_m| . \end{array}\ ] ] integrating in @xmath146 , we get @xmath147 which yields , on the one hand , that @xmath148 and on the other hand that @xmath149    convergence and identification of the limit . since the families @xmath34 and @xmath35 are bounded in @xmath150 , we have strong convergence in @xmath151 both for @xmath34 and @xmath35 .    to pass from local convergence to global convergence in @xmath36 , we need to prove that the mass in an initial strip @xmath152 $ ] and in the tails @xmath153 are uniformly ( in @xmath52 ) small if @xmath154 is large enough . the control on the initial strip is immediate using our uniform , in @xmath52 and @xmath75 , bounds for @xmath155 and @xmath156 . in order to control the tails , we consider @xmath157 such that @xmath158 , @xmath159 for @xmath160 and @xmath161 for @xmath153 , and define @xmath162 . then , for any @xmath163 , @xmath164 & \le & \displaystyle cr^{-2}+\phi(0)\int_{\mathbb{r}^n}\varrho_m(t)\varphi_r . \end{array}\ ] ] therefore , for any @xmath100 $ ] we have @xmath165 \displaystyle & \le&\displaystyle\textrm{e}^{\phi(0)t } \left(\|\varrho^0_m-\varrho^0\|_{l^1(\mathbb{r}^n)}+\int_{\mathbb{r}^n}\varrho^0\varphi_r+cr^{-2}t \right ) \le \displaystyle{\varepsilon}\end{array}\ ] ] for @xmath154 and @xmath52 are large enough . since @xmath166 for any @xmath167 , the tail control for the pressures follows easily . after extraction of subsequences , we can pass to the a.e .- limit in the equation @xmath168 to obtain that @xmath169 . passing to the limit in the estimates , we have also @xmath170 , @xmath171 and @xmath172 for all @xmath23 . all the above is enough to prove that the pair @xmath57 solves the hele - shaw equation but for the question of the initial data . time continuity and initial trace . we need delicate arguments that are fortunately in the folklore of the topic of nonlinear diffusion equations , and borrow also the conclusion from @xcite . because @xmath68 is non - decreasing in time , we can write for a test function @xmath173 and @xmath174 , @xmath175 \zeta = \displaystyle\int_{t_1}^{t_2 } \int_{\mathbb{r}^n } \left [ p_\infty \delta \zeta + \varrho_\infty \phi(p_\infty ) \zeta\right ] \\[20pt ] & \leq c(t ) ( t_2-t_1)\left ( \| \delta \zeta \|_\infty + 1\right ) . \end{array}\ ] ] taking a sequence of such uniformly smooth functions that converge to @xmath91 , we find that @xmath176 ( in fact with a locally uniform lipschitz constant ) .    in order to identify the initial trace , we observe that for any test function @xmath177 as above , @xmath178 letting @xmath17 , we have @xmath179 letting first @xmath180 and then @xmath181 , we conclude that @xmath182 in @xmath183 . we now prove theorems  [ th : transport ] and  [ thm : complementarity ] , both of them connected with the fact that @xmath184 for all @xmath23 .    since we have already proved that both @xmath34 and @xmath35 converge strongly in @xmath36 for all  @xmath23 , the first of these theorems just depends on obtaining a uniform bound in @xmath63 for  @xmath55 . _ proof of theorem  [ th : transport ] . _ we rewrite the pressure equation   as @xmath185 integrating in @xmath186 we obtain the required estimate , @xmath187 @xmath188    _ remark . _ combining   with  , we obtain @xmath189 for all @xmath163 . this gives a uniform estimate for @xmath190 , @xmath191 $ ] , for any fixed values @xmath192 . however , it does not allow to go down to @xmath193 .    from the equation in   and the @xmath110 bound on @xmath134 we conclude that @xmath194 this is an optimal bound since @xmath61 has corners on @xmath195 . because this only gives space compactness , it is not enough to establish the complementarity formula on the pressure when passing to the limit in . we will need to perform a time regularization argument _  la steklov_.    _ proof of theorem  [ thm : complementarity ] . _ let @xmath196 be a regularizing kernel in time with support @xmath197 . we use the notation @xmath198 . the equation gives @xmath199 from here it follows that @xmath200 is uniformly ( in @xmath52 ) smooth in time and @xmath201 in space for @xmath202 fixed . therefore @xmath203 converges strongly in @xmath151 . hence , after multiplying by @xmath35 , we may pass to the limit to obtain @xmath204 in order to estimate the right hand side we make the following decomposition , @xmath205 \qquad\qquad\qquad   + \underbrace{\frac{m}{m-1}\int_{\mathbb{r}}\left(\varrho_m^{m-1}(t ) -\varrho_m^{m-1}(s)\right )   \left(\partial_t \varrho_{m}(s ) + \frac cs\right ) \omega_{\varepsilon}(t - s)ds}_{{\text{ii } } _ m(t)}\\[5 mm ] \qquad\qquad\qquad\underbrace{-\frac{cm}{m-1}\int_{\mathbb{r } } \left(\varrho_m^{m-1}(t ) -\varrho_m^{m-1}(s)\right ) \frac{\omega_{\varepsilon}(t - s)}s\,ds}_{{\text{iii } } _ m(t ) } , \end{array}\ ] ] where @xmath206 is a constant such that @xmath207 ; see estimate  .    for the first term we have @xmath208 regarding the second term , the estimate   implies that @xmath209 . therefore , @xmath210 . let @xmath211 , and @xmath212 the smallest time in its support . then , @xmath213    the third term is easy to treat . since @xmath214 , for any test function @xmath177 as above , @xmath215 as @xmath216 . these three calculations give , in the limit @xmath217 , @xmath218 to recover the desired information , @xmath219 it remains to pass to the limit as @xmath216 after noticing that @xmath220 and that the differential term can be treated through its weak formulation ; indeed , after testing it against a test function as before , it is written as @xmath221\ ] ] which passes to the limit @xmath222 because we already know that @xmath223 . therefore , @xmath224=0.\ ] ] @xmath188    as we have already mentioned , the complementarity formula is equivalent to the strong convergence of the gradients . [ lemma : equivalence.complementarity.strong.convergence ] the complementarity formula holds if and only if @xmath225 strongly in @xmath63 for all @xmath23 ,    _ proof . _ we consider a test function @xmath226 and use and   to obtain @xmath227 this implies , after an integration by parts , that @xmath228 \underset{m \to \infty}{\longrightarrow } 0.\ ] ] let @xmath229 . thanks to fatou s lemma , @xmath230 . on the other hand , since @xmath35 converges strongly , after extraction of subsequences we have @xmath231 = r,\ ] ] and @xmath232 is equivalent to the strong convergence . @xmath188      if the initial data are compactly supported uniformly in @xmath52 , we can control the supports of @xmath233 and @xmath234 uniformly in @xmath52 and @xmath235 $ ] for all @xmath23 . this implies that the speed of propagation is finite for the limit problem . the control is performed comparing the pressures with functions of the form @xmath236 the key point is that these functions , which are viscosity solutions of the hamilton - jacobi equation @xmath237 , are supersolutions to the equation @xmath238 for some time interval which does not depend on @xmath52 . this idea was already used for the case @xmath81 in @xcite . however , in the absence of reaction , @xmath239 is a supersolution for all times , and the proof is a bit easier .    let @xmath33 satisfying   and such that all their supports are contained in a common ball @xmath240 . then , for every @xmath23 , there is a radius @xmath241 depending only on @xmath242 , @xmath154 and @xmath243 , such that the supports of @xmath244 are contained in the ball @xmath245 for all @xmath246 and @xmath100 $ ] . _ we consider @xmath239 as in   with @xmath247 and @xmath206 large enough so that @xmath248 for all @xmath249 and @xmath246 . an easy computation shows that @xmath250 for all @xmath251 $ ] . since the functions @xmath35 are subsolutions to  , we conclude that the supports of the functions @xmath34 are contained in the ball of radius @xmath252 for all @xmath251 $ ] . moreover , @xmath253 , and the argument can be iterated to reach the time @xmath243 in a finite number of steps in this section we prove , under suitable assumptions , that the limit problem   has a unique solution , theorem [ th : unique ] . the main difficulty comes from the fact that @xmath3 is not a lipschitz , single - valued function of @xmath74 . hence , we can not apply directly the ideas developed in @xcite to adapt hilbert s duality method to the porous medium equation . the technique has to be modified , in the spirit of @xcite . dual problem . consider two solutions , @xmath254 , @xmath255 . let @xmath256 be a bounded domain containing the supports of both solutions for all @xmath100 $ ] , and @xmath257 . then we have @xmath258=0\ ] ] for all suitable test functions @xmath259 . this can be rewritten as @xmath260=0,\ ] ] where , for some fixed @xmath261 , @xmath262 0 \leq \displaystyle b=\frac{p_1-p_2}{(\varrho_1-\varrho_2)+(p_1-p_2 ) } \leq   1,\\[10pt ] 0 \leq \displaystyle c= \ ; - \varrho_2\,\frac{\phi(p_1)-\phi(p_2)}{p_1-p_2}\ ;   \leq   \nu . \end{array}\ ] ] to arrive to these bounds on @xmath263 and @xmath264 , we define @xmath265 when @xmath266 , even when @xmath267 , and @xmath268 when @xmath267 , even when @xmath269 . the idea of hilbert s duality method is to solve the _ dual problem _ @xmath270 \psi=0\quad\text{in } \partial\omega\times(0,t),\quad \psi(\cdot , t)=0\quad\text{in } \omega , \end{array } \right.\ ] ] for any smooth function @xmath271 , and use @xmath259 as test function . this would yield @xmath272 from where uniqueness for the density is immediate . uniqueness for the pressure would then follow from  . however , on the one hand the coefficients of the dual problem are not smooth , and , on the other , since @xmath263 and @xmath264 are not strictly positive , the dual equation is not uniformly parabolic . a smoothing argument is required .    regularized dual problem let @xmath273 , @xmath274 , @xmath275 , @xmath276 be sequences of smooth bounded functions such that @xmath277 \|b - b_n\|_{l^2(\omega_t)}<\beta / n,\qquad&1/n < b_n\le 1,\\[3 mm ] \|c - c_n\|_{l^2(\omega_t)}<\delta_1/n,\qquad&0 \leq c_n < m_1,\qquad&\| \partial_t c_n \|_{l^1(\omega_t ) } \leq k_1 \\[3 mm ] \|\phi(p_1)-\phi_{1,n}\|_{l^2(\omega_t)}<\delta_2/n,\qquad&|\phi_{1,n}|<m_2,\qquad & \| \nabla \phi_{1,n }   \|_{l^2(\omega_t ) } \leq k_2 , \end{array } \right.\ ] ] for some constants @xmath278 .    for any smooth function @xmath271 , we consider the solution @xmath279 of the backward heat equation @xmath280 \psi_n=0\quad\text{in } \partial\omega\times(0,t),\quad \psi_n(\cdot , t)=0\quad\text{in } \omega . \end{array } \right.\ ] ] the coefficient @xmath281 is continuous , positive and bounded below away from zero . thus , the equation satisfied by @xmath279 is uniformly parabolic in @xmath282 . hence @xmath279 is smooth and can be used as a test function in .    combining and , we have @xmath283 where @xmath284 i^2_{n}=\displaystyle\iint_{\omega_t } \big((\varrho_1-\varrho_2)+(p_1-p_2)\big ) ( b - b_n ) \left(\delta\psi_n - c_n \psi_n\right ) , \\[10pt ] i^3_{n}=\displaystyle\iint_{\omega_t } ( \varrho_1-\varrho_2)(\phi(p_1)-\phi_{1,n})\psi_n,\\[10pt ] i^4_{n}=\displaystyle\iint_{\omega_t } \big((\varrho_1-\varrho_2)+(p_1-p_2)\big ) b(c - c_n)\psi_n . \end{array}\ ] ]    limit @xmath285 . uniform bounds and conclusion . our aim is to prove that @xmath286 , @xmath287 , which implies that @xmath269 . for this task we will use some bounds that we gather in the following lemma . [ lem : bounds.uniqueness.scalar.equation ] there are constants @xmath288 , @xmath289 , depending on @xmath243 and @xmath271 , but not on @xmath290 , such that @xmath291    _ proof . _ the first bound is just the maximum principle because @xmath292 is non - negative and @xmath293 uniformly bounded . the second bound is obtained multiplying the equation in   by @xmath294 . after integration in @xmath295 , we get @xmath296 \displaystyle \quad + \int_t^t \int_\omega \left [ -\partial_t c_n \frac { \psi_n^2}{2 } -   \phi_{1,n } |\nabla \psi_n|^2 - \psi_n \nabla \phi_{1,n }   \cdot\nabla \psi_n+ c_n \phi_{1,n } \psi_n^2 + \psi_n \delta g - g c_n \psi_n   \right]\\[4 mm ] \displaystyle\    \leq k\left(1-t+   \int_t^t \|\nabla \psi_n ( s ) \|_{l^2(\omega)}^2\ , ds\right ) , \end{array}\ ] ] where we denote by @xmath297 various constants independent of @xmath290 . we now use the grnwall lemma to conclude our second uniform bound . then we can use again the inequality to conclude the third one . @xmath298@xmath188    we now get @xmath299 & \le&\displaystyle kn^{1/2}\|a - a_n\|_{l^2(\omega_t)}\le k\alpha / n^{1/2};\\[10pt ] \displaystyle |i^2_{n}|&\le & \displaystyle k\iint_{\omega_t}|b - b_n|\ ,    k\|(a_nn^{1/2}/b_n)^{1/2}(b - b_n)\|_{l^2(\omega_t)}\\[10pt ] & \le&\displaystyle kn^{1/2}\|b - b_n\|_2\le k\beta / n^{1/2};\\[10pt ] \displaystyle |i^3_{n}|&\le & \displaystyle\iint_{\omega_t }    k\|\phi(p_1)-\phi_{1,n}\|_{l^2(\omega_t ) } \le k\delta_2/n ; \\[10pt ] \displaystyle |i^4_{n}|&\le & \displaystyle k\iint_{\omega_t } b    \end{array}\ ] ]    once we have proved that @xmath269 , equation   says that @xmath300 = 0,\ ] ] from where the result will follow by taking as test function @xmath301 , using the monotonicity of @xmath13 . when nutrients are taken into account , we are led to problem . we again assume that the pressure field is given by  , with @xmath9 set equal to 1 . our assumptions on the initial data are stronger than for the purely mechanical problem . namely , in addition to , we assume that for some @xmath302 such that @xmath303 , @xmath304 \|{\rm div}(\varrho_m^0 \nabla p_m^0 ) + \varrho_m^0 \ ; \phi(p_m^0,c_m^0)\|_{l^1(\mathbb{r}^n)}\le c , \quad   \|\delta c_m^0 - \varrho_m^0   \ ; \psi(p_m^0,c_m^0)\|_{l^1(\mathbb{r}^n)}\le c. \end{array } \right.\ ] ]    an interesting difference with the purely mechanical model is that when nutrients are not enough the cells might die ( or become quiescent ) , which is represented by @xmath305 for @xmath11 small enough with @xmath3 given . these cells , which are in principle in the center of the tumor , are replaced by cells from the boundary moving inwards by pressure forces . in particular the growth inequalities can not hold true here . a typical choice is @xmath306 , @xmath307 with @xmath308 two constants and @xmath309 a function having the same properties as in the purely fluid mechanical model . as in the case without nutrients , in the limit we obtain a free boundary problem of hele - shaw type . let @xmath13 , @xmath14 satisfy , and @xmath310 satisfy the hypotheses   and  . then , after extraction of subsequences , the density @xmath34 , the nutrient @xmath311 and the pressure @xmath35 converge for all @xmath23 strongly in @xmath36 as @xmath17 to limits @xmath312 that satisfy @xmath313 , @xmath314 and @xmath40 , and @xmath315 \displaystyle \partial_t c_\infty = \delta c_\infty - \varrho_\infty \psi(p_\infty , c_\infty)\quad&c_\infty(0)=c^0 , \end{array } \right.\ ] ] in a distributional sense , plus the relation @xmath316 , where @xmath43 is the hele - shaw graph  . [ th : fbn ]    when we say that @xmath317 is a solution to   in a distributional sense , we mean that @xmath318 \displaystyle \iint_{q } \big\{c_\infty \zeta_t+c_\infty\delta \zeta-\varrho_\infty\psi(p_\infty , c_\infty)\zeta\big\}=-\int_{\mathbb{r}^n}c^0\zeta(\cdot,0 ) , \end{array}\ ] ] for all test functions @xmath319 . as mentioned above , for the system we lose the monotonicity properties , and we are not able to prove the expected continuity of @xmath68 and @xmath320 in @xmath183 . however , we are still able to prove that @xmath321 in @xmath63 . therefore , the equation on the cell density can also be written as a transport equation , @xmath322    if the initial data @xmath29 are compactly supported uniformly in @xmath52 , we can control the supports of @xmath234 uniformly in @xmath52 for each @xmath51 . hence , in the limit problem tumors propagate with a finite speed ( this is not true for the nutrients ) , and a free boundary shows up . this can be proved through a comparison argument with the same barrier functions as in the scalar case . moreover , the constructed solution will fall within a class for which we prove uniqueness . [ thm : uniqueness.system ] there is a unique triple @xmath323 , @xmath324 , @xmath325 , satisfying   in the distributional sense and such that for all @xmath23 :    * @xmath99 is uniformly compactly supported for @xmath100 $ ] ; * @xmath326 ; * @xmath327 , @xmath328 .    as in the purely mechanical model , we can write an equation for the pressure @xmath35 , @xmath329 this suggests that the complementarity formula @xmath330 holds . however , we have not been able to establish it rigorously . the reason is that , in contrast with the case without nutrients , we have failed to control @xmath331 from below by means of a comparison argument , or to prove the strong convergence of @xmath55 in @xmath63 for all @xmath23 . we leave open the question to give conditions on @xmath332 allowing to prove formula  . the rest of this section is devoted to prove the first of these theorems . the uniqueness result is postponed to a later section . @xmath107 bounds for @xmath34 , @xmath35 , @xmath311 . the assumptions on the growth functions   imply , using standard comparison arguments , @xmath333    @xmath110 bounds for @xmath34 , @xmath35 , @xmath311 . let @xmath334 and @xmath335 be two solutions of the system for a fixed @xmath52 , with corresponding pressures @xmath3 and @xmath336 . we subtract the equation for @xmath337 from the equation for @xmath74 , and multiply the resulting equation by @xmath338 , and do an analogous manipulation for the @xmath11 variable . we integrate by parts , and obtain @xmath339(t ) , \\[10pt ] \displaystyle\frac{d } { dt}\|(c-\hat c)(t)\|_{l^1(\mathbb{r}^n)}\le- \int_{\mathbb{r}^n}\big[\big(\varrho \psi(p , c)-\hat \varrho\psi(\hat p,\hat c)\big)\mathop{\rm sign}(c-\hat c)\big](t ) . \end{array } \right.\ ] ] let @xmath340 be a positive constant to be chosen later . we have @xmath341 \quad\displaystyle \underbrace{\int_{\mathbb{r}^n}\big[\big(\varrho-\hat\varrho\big)\big(\phi(p , c)\mathop{\rm sign}(\varrho-\hat\varrho)-\mu\psi(p , c)\mathop{\rm sign}(c-\hat c)\big)\big](t)}_{\mathcal{i}}\\[4 mm ] \quad+\underbrace{\int_{\mathbb{r}^n}\big[\hat\varrho\big((\phi(p , c)-\phi(p,\hat c))\mathop{\rm sign}(\varrho-\hat\varrho ) -\mu(\psi(p , c)-\psi(p,\hat c))\mathop{\rm sign}(c-\hat c)\big)\big](t)}_{\mathcal{j}}\\[4 mm ] \quad+\underbrace { \int_{\mathbb{r}^n } \big [ \hat\varrho \big ( ( \phi(p,\hat c)-\phi(\hat p,\hat c ) ) \mathop{\rm sign}(\varrho-\hat\varrho ) -\mu(\psi(p,\hat c)-\psi(\hat p,\hat c ) ) \mathop{\rm sign}(c-\hat c ) \big ) \big](t ) } _ { \mathcal{k}}. \end{array}\ ] ] let @xmath342 , @xmath343 , @xmath344 . since @xmath345 , if @xmath346 , there are constants @xmath206 independent of @xmath52 such that @xmath347(t)\leq c\|(\varrho-\hat\varrho)(t)\|_{l^1(\mathbb{r}^n ) } , \\[4 mm ] \displaystyle\mathcal{j } \le   \| \hat \rho\|_{l^\infty(q ) } \left(\| \partial_c\phi\|_{l^\infty(a ) }   + \mu \| \partial_c\psi\|_{l^\infty(a ) } \right ) \|(c - \hat c)(t ) \|_{l^1(\mathbb{r}^n ) } \leq c   \|(c - \hat c)(t)\|_{l^1(\mathbb{r}^n)},\\[4 mm ] \displaystyle\mathcal{k}\le\int_{\mathbb{r}^n } \big [ \hat \varrho \big ( - \big|   \phi(p,\hat c)- \phi(\hat p,\hat c ) \big| + \mu \big| \psi(p,\hat c)- \psi(\hat p,\hat c ) \big| \big)\big](t)\le0 . \end{array}\ ] ] we conclude that @xmath348 therefore , @xmath349 this gives uniqueness , and choosing @xmath350 , and @xmath351 , we find the uniform estimates @xmath352    as in the purely fluid mechanical model , from the @xmath353 bounds and , we conclude that , for @xmath163 , @xmath354    @xmath110 bounds on the derivatives of @xmath34 and @xmath311 . we differentiate the two equations of the system with respect to time , multiply the first one by @xmath355 and the second by @xmath356 and use kato s inequality to obtain @xmath357 \displaystyle \partial_t |\partial_t c_m|-\delta(|\partial_t c_m|)\le -\partial_t\varrho_m \psi\mathop{\rm sign}(\partial_t c_m)-\varrho_m\partial_p\psi \partial_t p_m\mathop{\rm sign}(\partial_t c_m)-\varrho_m\partial_c\psi|\partial_t c_m| . \end{array } \right.\ ] ] we integrate in space and add the two equations to obtain , using the monotonicity properties of the growth functions , @xmath358   \int_{\mathbb{r}^n } \left ( |\partial_t \varrho_m(t)| \big(\phi + \mu \psi\big)(0,c_b ) + \| \varrho_m \|_{l^\infty(q ) } \|\partial_c\phi\|_{l^\infty(a)}|\partial_t c_m(t)| \right ) , \end{gathered}\ ] ] and thus , thanks to assumption , @xmath359    we can estimate the space derivatives in the same way , and arrive to @xmath360    estimates on the derivatives of @xmath35 . from the first equation in , using the strict monotonicity of @xmath13 with respect to @xmath3 and the @xmath361 estimates that we have already proved , we get @xmath362 where , as above , @xmath343 . therefore , @xmath363    the control on the space derivatives follows by a similar argument , and we obtain @xmath364    convergence and identification of the limit . the above estimates give strong convergence in @xmath151 for @xmath34 , @xmath365 and @xmath35 . our uniform control of the @xmath110 norms of these quantities in @xmath52 and @xmath100 $ ] shows that the mass in thin initial strips is uniformly in @xmath52 small . therefore , to get global convergence in @xmath36 , we just need to control the tails . proceeding as in the case without nutrients , we get @xmath366 from where the tail control for @xmath34 , and then for @xmath35 , follows easily .    as for the nutrients , we have @xmath367 but we already have a tail control for @xmath34 . hence , for all @xmath154 and @xmath52 large enough , @xmath368 which immediately implies a tail control for @xmath365 . the strong convergence in @xmath36 is enough to pass to the limit and recover the system in a distributional sense with the limiting pressure graph relation . @xmath54 bounds for @xmath55 , @xmath369 . the bound for the gradients of the pressures follows from equation written in the form @xmath370 indeed , integrating in @xmath186 we obtain @xmath371 this is enough to prove that the limit satisfies the transport equation  . the estimate for the gradients of the nutrients , which is needed to prove that the limit falls within the uniqueness class , is even easier . we just have to multiply the equation for the nutrients by @xmath311 and integrate by parts in @xmath186 . since @xmath372 , we obtain @xmath373 we meet two difficulties . on the one hand , as in the case without nutrients , @xmath3 is not a lipschitz , single - valued function of @xmath74 . on the other hand , the density of nutrients is not compactly supported . hence we can not restrict to the case of a bounded domain . to take care of the lack of compact support of @xmath11 , we will use an idea from  @xcite .    dual system . let @xmath374 and @xmath375 be two solutions with the same initial data and fix a final time @xmath243 . since they satisfy in the sense of distributions , we have @xmath376 \displaystyle\iint_{q_t}\left\ { ( c_1-c_2 ) \partial_t \zeta     +   ( c_1-c_2 ) \delta \zeta - ( \varrho_1 \psi(p_1,c_1 ) - \varrho_2 \psi(p_2,c_2 )   ) \zeta \right\ } = 0 , \end{cases}\ ] ] for any pair of test functions @xmath377 such that @xmath378 . denoting @xmath379 and @xmath380 , we can rewrite the above equations as @xmath381 \displaystyle \iint_{q_t }   \left\{(c_1-c_2 )   \left ( \partial_t \zeta   +   \delta \zeta - e   \zeta \right )   - ( \varrho_1- \varrho_2+p_1-p_2 ) \left(a\psi_1    - fb \right ) \zeta \right\}= 0 , \end{cases}\ ] ] where @xmath382 0 \leq \displaystyle c= \ ; -\varrho_2\frac{\phi(p_1,c_1)-\phi(p_2,c_1)}{p_1- p_2 } \le \nu_1 ,   \ & \displaystyle 0\leq d= \varrho_2 \frac{\phi(p_2,c_1)-\phi(p_2,c_2)}{c_1- c_2}\le \nu_2 , \\[12pt ] \displaystyle0\leq e= \varrho_2 \frac{\psi(p_2,c_1)-\psi(p_2,c_2)}{c_1- c_2 } \le \nu_3,\ & 0 \leq \displaystyle f= \ ; -\varrho_2\frac{\psi(p_1,c_1)-\psi(p_2,c_1)}{p_1- p_2}\le \nu_4 . \end{array}\ ] ] adding these two equations we get @xmath383 where @xmath384 \displaystyle\mathcal{b}(\psi,\zeta)=\partial_t \zeta    +   \delta \zeta - e\zeta   + d \psi . \end{array } \right.\ ] ]    let @xmath271 , @xmath385 be any non - negative functions in @xmath386 . if the _ dual system _ @xmath387 admits a smooth solution @xmath388 with a suitable decay , we may use @xmath259 and @xmath177 as test functions to obtain @xmath389 from where uniqueness would follow . unfortunately , the coefficients in the equations defining the dual system are not smooth . even worse , @xmath263 an @xmath264 may vanish : the system is not uniformly parabolic , and a delicate regularization procedure is required to fulfill our plan . regularized dual system . given @xmath271 , @xmath385 as above , let @xmath390 be such that the supports of @xmath391 , @xmath392 , @xmath393 , @xmath394 , @xmath395 are contained in @xmath396 . given any @xmath397 , we introduce the dual system with regularized coefficients , posed in @xmath398 , @xmath399 \displaystyle \partial_t \zeta_{n , r }     +   \delta \zeta_{n , r } - e_n   \zeta_{n , r } + d_n   \psi_{n , r } = h , \end{array } \right.\ ] ] with final and boundary data given by @xmath400 where , thanks to the hypotheses on the solutions , the @xmath401 error in the approximation of each coefficient is @xmath402 , and @xmath403 \| \partial_t c_n \|_{l^1(q_{t})},\ \| \partial_t f_n \|_{l^1(q_{t})},\ \| \nabla \phi_{1,n }   \|_{l^2(q_{t})},\ \| \nabla \psi_{1,n }   \|_{l^2(q_{t } ) } \leq k. \end{array } \right.\ ] ] our aim is to use @xmath404 and @xmath405 , suitably extended by zero , as test functions . since these functions have non - zero derivatives at the boundary , we find yet another difficulty : the lack of smoothness at the boundary of the ball @xmath406 , which requires an extra regularization . avoiding the lack of smoothness at the boundary of the ball . for @xmath407 , we consider a family of cut - off functions @xmath408 such that @xmath409 , @xmath410 \|\nabla\eta_\varepsilon\|_\infty \le \mathcal{c}/\varepsilon,\qquad \|\delta\eta_\varepsilon\|_\infty \leq \mathcal{c}/\varepsilon^2 . \end{array}\ ] ]    now put @xmath411 , @xmath412 in the weak formulation   to get @xmath413 where @xmath414 \mathcal{k}_{n , r,\varepsilon}=\displaystyle\iint_{q_{r , t}}\left\{(c_1-c_2 ) \big(2\nabla\eta_\varepsilon\cdot\nabla\zeta_{n , r } + \zeta_{n , r}\delta\eta_\varepsilon\big)\right\},\\[10pt ] i^1_{n , r,\varepsilon}=\displaystyle\iint_{q_{r , t}}\left\{(\varrho_1-\varrho_2 + p_1-p_2)\eta_\varepsilon b_n\left(1-\frac{a}{a_n}\right ) \left(\delta\psi_{n , r}-c_n\psi_{n , r}+f_n\zeta_{n , r}\right)\right\},\\[10pt ] i^2_{n , r,\varepsilon}=\displaystyle\iint_{q_{r , t}}\left\{(\varrho_1-\varrho_2+p_1-p_2 ) \eta_\varepsilon ( b - b_n)\left(\delta\psi_{n , r}-c_n\psi_{n , r}+f_n\zeta_{n , r}\right)\right\},\\[10pt ] i^3_{n , r,\varepsilon}=-\displaystyle\iint_{q_{r , t}}\big\{(\varrho_1-\varrho_2+p_1-p_2 ) \eta_\varepsilon b [ \psi_{n , r}(c - c_n)+\zeta_{n , r}(f - f_n ) ] \big\},\\[10pt ] i^4_{n , r,\varepsilon}=\displaystyle\iint_{q_{r , t}}\big\{(c_1-c_2 ) \eta_\varepsilon\zeta_{n , r } [ ( e - e_n)+(d - d_n ) ] \big\},\\[10pt ] i^5_{n , r,\varepsilon}=\displaystyle\iint_{q_{r , t}}\left\{(\varrho_1-\varrho_2+p_1-p_2)\eta_\varepsilon a\left((\phi_1-\phi_{1,n})\psi_{n , r}- ( \psi_1-\psi_{1,n})\zeta_{n , r}\right)\right\}. \end{array}\ ] ] the term @xmath415 vanishes , because the supports of @xmath416 , @xmath417 , @xmath395 are contained in @xmath396 . hence we can drop it . we will now take the limit @xmath418 . the only difficult term at this step is @xmath419 . we write @xmath420 since @xmath421 on @xmath422 , we have @xmath423 and , if @xmath424 is the unit outwards normal to @xmath425 , then @xmath426 therefore , since @xmath427 and @xmath428 are bounded , @xmath429    limit @xmath285 . uniform bounds . proving that @xmath430 , @xmath431 , is easy , thanks to the following estimates . there are constants @xmath432 , @xmath433 , depending on @xmath243 , @xmath271 and @xmath385 , but not on @xmath290 and @xmath154 , such that @xmath434 \left\| \left(b_n / a_n\right)^{1/2 } \left ( \delta \psi_{n , r } -c_n \psi_{n , r}+f_n\zeta_{n , r}\right)\right\|_{l^2(q_{r , t } ) } \leq \mathcal{c}_2 . \end{array}\ ] ]    _ proof . _ _ @xmath107 bounds . _ an easy computation shows that the functions @xmath435 satisfy the differential inequalities @xmath436 - n'(t )   \leq   \mathcal{c}(1 + m(t ) ) ,   \\[10pt ]   m(t)=n(t)=0 , \end{array } \right.\ ] ] with @xmath437 , @xmath438 , where @xmath439 is the point where the maximum defining @xmath440 is achieved . thanks to the assumptions on @xmath441 and @xmath442 , there is a constant @xmath443 such that @xmath444 . let now @xmath445 as a combination of the equations on @xmath446 and @xmath447 , we find that @xmath448 indeed , assume that @xmath449 . then , @xmath450 and the bad @xmath451 terms cancel . if , on the contrary , @xmath452 , then the result follows by the harmless equation on @xmath453 .    from we have @xmath454 , hence the result .    _ estimates on the laplacian . _ we multiply the second equation in   by @xmath455 and integrate in space and time to obtain @xmath456 & \le&\displaystyle c\|\partial_t\zeta_{n , r}\|_{l^2(q_{r , t})}. \end{array}\ ] ] hence @xmath457 is uniformly bounded . we now multiply the first equation in by @xmath458 and integrate in space and time . we get @xmath459 \ \displaystyle=-\int_{b_r(0)}\left(\frac{c_n\psi_{n , r}^2}2+f_n\psi_{n , r}\zeta_{n , r}\right)(t)\\[4 mm ] \quad\displaystyle+\int_t^t\int_{b_r(0 ) } \left(-\frac{\psi_{n , r}^2\partial_t c_n}2 + \psi_{n , r}\zeta_{n , r}\partial_t f_n+f_n\psi_{n , r}\partial_t\zeta_{n , r}\right)\\[4 mm ] \quad \displaystyle + \int_t^t\int_{b_r(0)}\big(-\phi_{1,n}|\nabla\psi_{n , r}|^2+\psi_{n , r}\nabla\phi_{1,n}\cdot \nabla\psi_{n , r}+ c_n\phi_{1,n}\psi_{n , r}^2- \phi_{1,n}f_n\psi_{n , r}\zeta_{n , r}\big)\\[4 mm ] \quad \displaystyle + \int_t^t\int_{b_r(0)}\big ( -\psi_{1,n}\nabla\zeta_{n , r}\cdot\nabla\psi_{n , r}-\zeta_{n , r}\nabla\psi_{1,n}\cdot \nabla\psi_{n , r}-c_n\psi_{1,n}\psi_{n , r}\zeta_{n , r}+f_n\psi_{n , r}\zeta_{n , r}^2 \big)\\[4 mm ] \quad \displaystyle + \int_t^t\int_{b_r(0)}\big(\psi_{n , r}\delta g - c_ng\psi_{n , r}+f_ng\zeta_{n , r } \big)\\[4 mm ] \ \displaystyle\le k\left(1-t+\int_t^t\|\nabla\psi_{n , r}(s)\|_{l^2(b_r(0))}\,ds\right ) . \end{array}\ ] ] using the grnwall lemma we get @xmath460 , and then the desired estimate @xmath188    in order to pass to the limit in the @xmath461 term we need a good enough estimate , uniform in @xmath290 , for the decay of the normal derivative .    as @xmath462 , @xmath463 uniformly in @xmath290 . _ as a first step we obtain a more precise estimate for the size of @xmath405 . we first notice that , since @xmath464 , @xmath465 and @xmath385 are bounded and compactly supported in @xmath396 , and @xmath404 is bounded , then @xmath466 for some @xmath467 . on the other hand , given a fixed @xmath468 , if @xmath469 is large enough , the function @xmath470 satisfies @xmath471 & \le&\displaystyle-\mathcal{c } \mathds{1}_{b_{r_0}(0)}. \end{array}\ ] ] moreover , @xmath472 , and @xmath473 for @xmath474 and @xmath475 . therefore , comparison yields @xmath476    if @xmath477 , we consider a function @xmath478 defined by @xmath479 where @xmath480 , @xmath481 satisfy @xmath482 note that @xmath483 on @xmath484 . moreover , @xmath485 g(x)=\zeta_{n , r}(x , t)=0,\qquad&|x|=r,\ 0<t < t,\\[4 mm ] g(x)\ge\zeta_{n , r}(x , t)=0,\qquad&r-1<|x|<r . \end{array}\ ] ] therefore , @xmath486 for all @xmath487 , @xmath475 . since @xmath488 for @xmath474 , @xmath475 , we conclude that @xmath489 and hence @xmath490 the result then follows from the following estimate , @xmath491 & = & \displaystyle o(1)\quad\text{as } r\to\infty . \end{array}\ ] ]    an analogous computation with @xmath492 if @xmath493 , and @xmath494 if @xmath495 , lead to the same estimate  . @xmath188    limit @xmath462 and conclusion . passing to the limit in @xmath154 , we finally obtain that @xmath496 hence @xmath269 and @xmath497 . uniqueness for @xmath3 then follows just taking @xmath301 as test function in  . a typical application of the hele - shaw equations is to describe tumor spheroids @xcite . when nutrients are ignored , the tumor is assumed to fill a ball centered at @xmath499 , @xmath500 the radius @xmath501 of this ball is computed according to the geometric motion rules   and  ; that is , we consider the unique ( and thus radially symmetric ) solution to @xmath502 and evolve the radius according to @xmath503 then , we consider @xmath504 defined as @xmath505 this is indeed a correct solution to our model . let @xmath506 be given . problem  defines a unique dynamic @xmath501 , @xmath507 , @xmath508 , which turns out to be the unique solution to the hele - shaw limit problem   with initial data @xmath509 . for long times it approaches a traveling wave solution with a limiting speed independent of the dimension , @xmath510 [ th : ts ] the limit profile can also be calculated and is one - dimensional .      _ proof . _ _ existence . _ a unique solution to can be defined for every fixed @xmath501 , because the elliptic problem comes with a decreasing nonlinearity . indeed , shooting or iterative methods apply and give a solution with smooth dependency on the parameter @xmath501 . therefore the differential equation can be solved thanks to the cauchy - lipschitz theorem .    _ equivalence with the hele - shaw problem . _ to show that this geometric method gives the solution to , we compute , in the distributional sense , that @xmath511 also , because @xmath61 satisfies @xmath512 in @xmath69 and since @xmath61 vanishes out of @xmath69 , we have , still in the distributional sense in @xmath513 , with @xmath424 the unit outward normal to @xmath514 , @xmath515 . now @xmath516 is solved on the unit ball as @xmath517 by the comparison principle , @xmath516 is increasing in @xmath154 and @xmath518 . passing to the limit in the equation we find that @xmath519 as @xmath520 . it is easy to see that @xmath521 , because we can build sub- and supersolutions of the form @xmath522 with @xmath523 for appropriate values of @xmath206 . by elliptic regularity we conclude that @xmath524 is of order @xmath525 and , since @xmath526 , it is bounded in @xmath527 . as a consequence , @xmath528 is of order @xmath529 . next we multiply equation by @xmath530 and integrate in @xmath531 to obtain @xmath532 from the above @xmath54 estimate on @xmath533 , we conclude that @xmath534 . going back to the dimensionalized variable , follows . _ one - dimensional profile . _ we consider moving coordinates , @xmath535 , @xmath536 . then @xmath537 is positive if and only if @xmath538 . hence , @xmath539 since @xmath540 as @xmath541 , we can pass to the limit uniformly on compact sets @xmath542 to obtain @xmath543      _ remark . _ the asymptotic growth rate   is generic for solutions to  , since one can always sandwich , after a certain delay , any standard initial data between two particular profiles of the form   which asymptotically travel with the same speed  . the first and simplest example is constructed in radial symmetry . it consists of an initial datum which is constant @xmath544 in a ball centered at @xmath499 and of radius 1 and zero otherwise . then the solution will have 0 pressure for a certain time , @xmath545 , in which the density grows exponentially according to equation  , so that @xmath546 at @xmath547 , @xmath68 reaches the threshold level @xmath90 in @xmath548 . it can not happen that @xmath549 for @xmath550 , since otherwise @xmath68 would violate the bound @xmath551 . therefore , @xmath50 satisfies @xmath552 in some ball @xmath553 with @xmath554 . an easy argument using the maximum principle shows that @xmath555 . it means that @xmath50 is equal or larger than the solution @xmath556 of the problem @xmath557 , @xmath558 for @xmath559 , @xmath560 for @xmath561 . taking the limit @xmath562 , we obtain a jump discontinuity in time at @xmath563 for every @xmath561 . note that @xmath50 is a discontinuous function of time with values in any @xmath564 , @xmath565 . the second example is best presented in one space dimension and consists of the evolution of an initial data consisting of two copies of the standard example presented in section [ sec : spheroids ] ( tumor spheroid ) , which have now an interval as support . we locate the supports at a distance from each other of say 1 . for a time interval , @xmath545 they evolve independently . at @xmath563 , the supports meet and an easy application of the maximum principle , shows that the solution is strictly positive in the union of the two intervals , therefore , larger than the solution corresponding to the union of the two intervals at @xmath563 . an easy inspection of the solutions at times @xmath566 and @xmath567 shows that there is a jump discontinuity in the space pressure profiles . this example can be adapted to several space dimensions by replacing disjoint intervals by disjoint concentric annuli . we consider the cell density ( defined for short enough times ) with two discontinuities @xmath568 we claim that the correct dynamics is defined by the speed and values of @xmath569 given through @xmath570 with @xmath61 vanishing outside the ball of radius @xmath571 and satisfying @xmath572 in @xmath573 , with zero dirichlet boundary condition . these rules are simply derived from equation . for @xmath574 we have @xmath67 and thus the equation is reduced to @xmath575 which gives us the evolution of @xmath569 . for @xmath576 , we have @xmath577 , while @xmath578 . using the equation , we get the dynamics for @xmath571 .    * acknowledgments . * authors fq and jlv partially supported by spanish project mtm2011 - 24696 . the paper was started while they where visiting upmc . they are indebted to this institution for the warm hospitality . bp is supported by institut universitaire de france . blank , i.  a. ; korten , m.  k. ; moore , c.  n. _ the hele - shaw problem as a mesa  limit of stefan problems : existence , uniqueness , and regularity of the free boundary_. trans . ( 2009 ) , no .  3 , 12411268 . lowengrub , j.  s. ; frieboes h.  b. ; jin , f. ; chuang , y .- l . ; li , x. ; macklin , p. ; wise , s.  m. ; cristini , v. _ nonlinear modelling of cancer : bridging the gap between cells and tumours_. nonlinearity 23 ( 2010 ) , no .  1 , r1r91 . tang , m. ; vauchelet , n. ; cheddadi , i. ; vignon - clementel , i. ; drasdo , d. ; perthame , b. _ composite waves for a cell population system modelling tumor growth and invasion_. chin . b 34 ( 2013 ) , no .  2 , 295318 .
models of tumor growth , now commonly used , present several levels of complexity , both in terms of the biomedical ingredients and the mathematical description . 
 the simplest ones contain competition for space using purely fluid mechanical concepts . 
 another possible ingredient is the supply of nutrients through vasculature . 
 the models can describe the tissue either at the level of cell densities , or at the scale of the solid tumor , in this latter case by means of a free boundary problem . 
 our first goal here is to formulate a free boundary model of hele - shaw type , a variant including growth terms , starting from the description at the cell level and passing to a certain limit . 
 a detailed mathematical analysis of this purely mechanical model is performed . 
 indeed , we are able to prove strong convergence in passing to the limit , with various uniform gradient estimates ; we also prove uniqueness for the asymptotic hele - shaw type problem . 
 the main tools are nonlinear regularizing effects for certain porous medium type equations , regularization techniques  la steklov , and a hilbert duality method for uniqueness . at variance with the classical hele - shaw problem , here 
 the geometric motion governed by the pressure is not sufficient to completely describe the dynamics . 
 a complete description requires the equation on the cell number density .    using this theory as a basis , we go on to consider the more complex model including nutrients . 
 we obtain the equation for the limit of the coupled system ; the method relies on some bv bounds and space / time a priori estimates 
 . here , new technical difficulties appear , and they reduce the generality of the results in terms of the initial data . finally , we prove uniqueness for the system , a main mathematical difficulty .    * 
 key words * tumor growth ; hele - shaw equation ; free boundary problems ; porous media ; hilbert uniqueness method . 
 * mathematics subject classification * 35k55 ; 35b25 ; 76d27 ; 92c50 .